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# Particle resuspension: challenges and perspectives for future models

Christophe Henry, Jean-Pierre Minier, Sara Brambilla

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Christophe Henry, Jean-Pierre Minier, Sara Brambilla. Particle resuspension: challenges and perspectives for future models. Physics Reports, 2023, 1007, pp.1-98. 10.1016/j.physrep.2022.12.005. hal-03909063

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# Particle resuspension: challenges and perspectives for future models

Christophe Henry<sup>a,*</sup>, Jean-Pierre Minier<sup>b</sup>, Sara Brambilla<sup>c</sup>

<sup>a</sup>Université Côte d'Azur, INRIA, CaliSto laboratory, Sophia-Antipolis, France

<sup>b</sup>EDF R&D, Mécanique des Fluides, Energie et Environnement, 6 quai Watier, 78400 Chatou, France

<sup>c</sup>Los Alamos National Laboratory, Bikini Atoll Rd, Los Alamos NM, 87545, United States

## Abstract

Using what has become a celebrated catchphrase, Philip W. Anderson once wrote that “more is different” (Science, Vol. 177, Issue 4047, pp. 393-396, 1972). First formulated in the context of condensed matter, this statement carries far beyond the sole limits of solid-state physics. It emphasizes that collective behavior can be more than the mere sum of what happens for elementary constituents or the mere collation of the evolution of each degree of freedom. Said otherwise, complex phenomena can arise out of the interplay between multiple sub-phenomena each of which can be relatively simple. The process of particle resuspension, in which discrete particles adhering on a surface are pulled off and carried away by a fluid flow, is another example involving a web of phenomena pertaining to fluid mechanics, particle dynamics and interface chemistry whose cross-effects create an intricate topic.

The purpose of this review is to analyze the physics at play in particle resuspension in order to bring insights into the rich complexity of this common but challenging concern. Following the more-is-different vision, this is performed by starting from a range of practical observations and experimental data. We then work our way through the investigation of the key mechanisms which play a role in the overall process. In turn, these mechanisms reveal an array of fundamental interactions, such as particle-fluid, particle-particle and particle-surface, whose combined effects create the tapestry of current applications. At the core of this analysis are descriptions of these physical phenomena and the different ways through which they are intertwined to build up various models used to provide quantitative assessment of particle resuspension. The physics of particle resuspension implies to hold together processes occurring at extremely different space and time scales and models are key in providing a single vehicle to lead us through such multiscale journeys. This raises questions on what makes up a model and one objective of the present work is to clarify the essence of a modeling approach. In spite of its ubiquitous nature, particle resuspension is still at the early stages of developments. Many extensions need to be worked out and revisiting the art of modeling is not a moot point. The need to consider more complex objects than small and spherical particles and, moreover, to come up with unified descriptions of mono- and multilayer resuspension put the emphasis on solid model foundations if we are to go beyond current limits. This is very much modeling in the making and new ideas are proposed to stimulate interest into this everyday but challenging issue in physics.

**Keywords:** Resuspension, Particle, Surfaces, Roughness

<footnote>*Corresponding author
Email address: christophe.henry@inria.fr (Christophe Henry)</footnote>
# Contents

1 Introduction 3

2 The physics of particle resuspension 5

2.1 Context and applications 5

2.1.1 Illustrations of typical resuspension phenomena 6

2.1.2 The reverse process of particle deposition 13

2.1.3 Terminology suggested and used here 14

2.2 The phenomenology of particle resuspension 16

2.2.1 A competition between motion-inducing versus motion-preventing factors 16

2.2.2 Fundamental physical interactions 17

2.2.3 Key resuspension mechanisms 17

3 Estimating resuspension: experimental measurements and quantities 20

3.1 Indicators for particle resuspension 21

3.1.1 Physical quantities of interest 21

3.1.2 Usage and interpretation of these quantities 24

3.1.3 Summary on physical quantities 28

3.2 Measuring particle resuspension 29

3.2.1 Measurement techniques and protocols 29

3.2.2 Limitations of experimental methodologies 35

4 Modeling particle resuspension 37

4.1 Key forces at play in resuspension 37

4.1.1 Interactions with external fields 37

4.1.2 Fluid forces on particles 38

4.1.3 Non-contact forces 42

4.1.4 Contact forces 46

4.1.5 Summary of forces 50

4.2 Accounting for turbulence and surface roughness 50

4.2.1 Capturing the impact of turbulence on resuspension 50

4.2.2 Surface roughness 54

4.3 A wide range of approaches 58

4.3.1 Dynamic models based on spatio-temporal integration 58

4.3.2 Dynamic PDF approaches 61

4.3.3 Integrated kinetic approaches 65

4.3.4 Force-balance/torque-balance approaches 70

4.3.5 System-scale models based on force/torque balance 71

4.3.6 Empirical formulas 73

4.3.7 A hierarchical view on modeling approaches 74

5 The art of modeling: new insights into existing models 75

5.1 Defining a model 76

5.2 Conceiving a model 76

5.3 Evaluating a model 79

5.4 Summary 79
<table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>6</td><td>Analyzing resuspension models</td><td>80</td></tr><tr><td>6.1</td><td>Accounting for resuspension mechanisms</td><td>81</td></tr><tr><td>6.2</td><td>Fundamental interactions at play</td><td>82</td></tr><tr><td>6.3</td><td>Nature of resolved/required information</td><td>85</td></tr><tr><td>6.4</td><td>Consistency issues</td><td>86</td></tr><tr><td>6.4.1</td><td>Avoiding typical statistical pitfalls</td><td>88</td></tr><tr><td>6.5</td><td>Uncertainty quantification</td><td>90</td></tr><tr><td>6.6</td><td>Summary</td><td>91</td></tr><tr><td>7</td><td>Paving the way for new models and experiments</td><td>91</td></tr><tr><td>7.1</td><td>Beyond spherical particles: towards more realistic descriptions</td><td>92</td></tr><tr><td>7.1.1</td><td>Resuspension of particles of complex shapes</td><td>92</td></tr><tr><td>7.1.2</td><td>Resuspension from complex surfaces</td><td>99</td></tr><tr><td>7.2</td><td>Beyond individual resuspension events and collection motion: towards unified descriptions</td><td>103</td></tr><tr><td>7.2.1</td><td>Bridging the gap between monolayer and multilayer deposits</td><td>103</td></tr><tr><td>7.2.2</td><td>The role of collective motion</td><td>107</td></tr><tr><td>7.3</td><td>The road ahead: a personal viewpoint on future models</td><td>110</td></tr><tr><td>8</td><td>Conclusions and perspectives</td><td>111</td></tr></tbody></table>

# 1. Introduction

Let us imagine the following situation: a set of particles are lying on a surface. Then, a fluid flow blows them off the surface. In a nutshell, this simple image captures the notion of particle resuspension.

In a more technical formulation, resuspension refers to the physical process by which discrete elements resting on a surface are, first, detached and, then, entrained away through the action of a fluid flow. It follows that this process occurs in wall-bounded two-phase flows, i.e., flows composed of a fluid phase and a discrete phase both contained in a domain bounded by at least one wall. The fluid phase involved in resuspension can be either a liquid (e.g., water, oil, milk) or a gas (like air or any other gas mixture). The discrete elements can be air bubbles, water droplets, organic elements or inorganic solids, and can exhibit a wide range of forms and properties. In this review, we have chosen to focus essentially on inorganic or organic solid particles. This means that some typical features are not addressed, such as the formation of liquid films when droplets merge to form a thin continuous coating along a wall. Nevertheless, the focus on solid particles does not imply an overly simplified problem since a wealth of intricate phenomena does take place. Indeed, as these particles interact with a solid boundary, they can form deposits of various shapes and sizes, and these deposits, or even individual particles, can be complex mechanical objects (e.g., deformable). Deposits with a significant size can alter the fluid flow, and fluid motion over such large-scale deposits can lead to the detachment and entrainment of single particles, or even groups of particles, implying a rich and complex range of physical issues. This corresponds to the all-important topic of particle resuspension.

Particle resuspension is a process typically encountered in our daily activities. One spectacular example involves sand, dust or leaves, deposited on the ground: these particles can be blown away by a local airflow (e.g., due to the wind, a person walking nearby or even a door being opened/closed) [1]. Another notable example is related to the quality of drinking water [2]: it can be compromised by the release of tiny elements that accumulate on the interior of pipes in water distribution systems (e.g., bacteria, nitrates and/or trace metals such as iron or copper). Particle resuspension is a remarkable phenomenon in environmental situations, for instance in hydrology (with the initiation of sediment transport by rivers) [3] or in oceanography (e.g., re-entrainment of sediments generated by waves) [4]. Many industrial processes are also concerned by particle resuspension. To name a few examples, resuspension is an undesirable phenomenon when dealing with potentially harmful particles in medical applications (e.g., resuspension of contaminated particles from hospital floors due to human activities or from ventilation systems [5]), in the energy industry (e.g., radioactive particles blown away by strong wind following nuclear incidents [6]) or
even in the automotive industry (e.g., road resuspension of soot particles from exhaust emissions as well as abrasion-derived particles stemming from tires or brakes [7]).

These selected examples bring out that resuspension is a highly multiscale process with challenging and complex issues. More specifically, it involves a variety of flow conditions (e.g., in the atmosphere, rivers, oceans or confined systems) as well as particle properties. For instance, particles can have very different geometrical (e.g., shape, size), physical (e.g., composition, density) as well as mechanical properties (e.g., rigid, extensible, deformable, or flexible). Moreover, resuspension is a process taking place over a wide range of temporal and spatial scales, typically from biofilm resuspension over a few seconds in millimeter-size pipes to sand resuspension over a few minutes/hours in desert dunes. Finally, it is worth pointing out that resuspension has a two-side nature since it can either be required (e.g., to limit the growth of limescale deposits within pipes) or has to be avoided (e.g., when particles are contaminants).

This brief overview of applications indicates that resuspension is a relevant process in several scientific domains. A first consequence of this ubiquity is that it has led to a rich terminology developed separately and used to refer to what is basically the same physical issue: for example, depending on the application domain, it is called re-entrainment, removal, detachment, reaerosolization, remobilization or even winnowing. A second consequence is that our familiarity with the subject can create the misleading belief that particle resuspension is either a straightforward physical question or that it has been thoroughly investigated, leaving no areas on which to shed new light. However, particle resuspension encompasses different physical mechanisms such as: adhesion between bodies, rupture of static equilibrium, particle motion along and near surfaces, response to near-wall turbulent motions, inter-particle collisions and collective motion. In turn, these mechanisms are a combination of three fundamental interactions, namely particle-fluid, particle-particle, and particle-surface interactions. This means that, despite the complexity of this process and the richness of the applications involved, a more general picture can emerge to describe particle resuspension. Yet, this general picture can only arise from a multidisciplinary and multiphysics approach, since the process is at the crossroads between a range of physics-based fields, from fluid dynamics to physico-chemical interface forces and tribology. New insight is indeed needed to clarify the interplay between these fundamental interactions. Furthermore, it is important to realize that such a description is expressed essentially through the formulation of models. Indeed, we do not have a self-contained and closed set of equations, operating at the same level of physical description, to address particle resuspension. There are subtle quantum effects, as in the inter-molecular origin of van der Waals forces between bodies, while fluid flows are described by continuous mechanics, such as the Navier-Stokes equations and even coarser versions of them for high Reynolds-number turbulent flows. In other words, modeling is an intrinsic and central part of any physics-based description of resuspension.

It follows that modeling particle resuspension represents a multiscale task with, on the one hand, macroscopic effects (such as in fluid mechanics) and, on the other hand, microscopic ones (e.g., physico-chemical forces at the nanoscale between adhering bodies). In fact, modeling particle resuspension remains an open challenge in Statistical Mechanics in relation with dispersed two-phase flow studies [8, 9]. More than in fields where a well-established physics-based modeling framework is already available providing consistent safeguards for the assessment of specific models, it is here important to revisit carefully what a model represents, what is captured, and on which underlying assumptions a given formulation relies on.

Over the years, a range of models have been proposed [10, 11], often with the sole purpose of directly capturing the overall particle resuspension rate. This resuspension rate is a measure of the average number of particles removed from a surface per unit time. As such, it is a very common output of models but which results from the combination of physical phenomena that need to be distinguished and addressed separately before being combined together to provide a comprehensive description of a specific situation. Trying to integrate or bridge over these intricate physical mechanisms without careful consideration can generate confusion or uncertainty with respect to what is truly modeled and, more specifically, to what is actually predicted or assumed in a model. In that sense, a first challenge is to come up with an unambiguous description of what a model of particle resuspension stands for and what we are to expect from well-posed formulations. In addition, uncertainty quantification is currently missing but paramount as the consequences of a particle being resuspended may be non-linear. For instance, infectious agents spread has usually an exponential growth meaning that a certain error in the resuspension prediction can have a much larger error
in the forecast of sick population.

In this review, we take up recent experience-based physical analyses of particle resuspension [10, 3] and concentrate on the modeling step, revisiting past and present efforts from a mechanistic standpoint to provide insights into the state of the art. This is helpful to pave the way for extended formulations able to address new problems involving complex-shaped particles while remaining tractable for simulations in realistic situations where resuspension needs to be quantified along with its counterpart phenomenon of particle deposition. More precisely, the objectives of this work are to:

(1) Clarify the physical mechanisms and fundamental interactions involved;

(2) Provide a general framework to define different classes of models;

(3) Distinguish what is assumed and what is predicted by each class of models;

(4) Reveal and analyze the origin as well as the physical nature of model inputs;

(5) Point out directions of progress where developments are needed along with new well-targeted experiments.

The first three objectives will be used as guidelines to come up with a new classification of existing resuspension models, which will guide interested readers into choosing adequate models depending on the system of interest and the expected results.

To reach these objectives, the paper is organized as follows. The physics of particle resuspension is first recalled in Section 2: an overview of the spectrum of situations and physical contexts where particle resuspension plays an important role is given before introducing a more precise terminology and discussing the physical mechanisms as well as fundamental interactions at play. In Section 3, existing experimental methods are briefly reviewed together with the corresponding quantities that can be extracted from such measurements. In Section 4, the existing models used to measure or predict these quantities of interest are introduced. Then, following basic guidelines for model development recalled in Section 5, we propose and analyze existing models with respect to a new set of criteria in Section 6, which provide insights into the contents of particle resuspension models (e.g., inputs, outputs, assumptions). Apart from assessing existing propositions, one of the main interests of the approach developed in the first six sections of this review is to pave the way for future extensions. This is done for models in section 7 when more complex particle shapes and particle collective effects are addressed. In this section, the needs to design new experiments aiming at validating key elementary aspects of models and of the physical mechanisms involved are also discussed together with the development of experimental techniques that have the potential to provide new insights into the physics of particle resuspension. Finally, conclusions and perspectives are proposed in section 8.

## 2. The physics of particle resuspension

The purpose of this section is to set the stage for the discussions to come by introducing the key notions and terminologies related to the physical aspects of particle resuspension. Since several reviews on particle resuspension are already available (see for instance [3, 10, 11, 12, 13, 14, 15]), only the most relevant aspects are detailed here.

This section is organized as follows: a range of practical applications is first described in Section 2.1, including a presentation of the terminology used throughout the rest of the paper; second, the phenomenology of particle resuspension is introduced in Section 2.2, providing an overview of the key interactions at play as well as of the fundamental mechanisms.

### 2.1. Context and applications

Etymologically, re-suspension is formed by the combination of the prefix “re-” (meaning again/anew) and “suspension” (meaning to disperse throughout the bulk of a fluid). Hence, resuspension corresponds to the process that induces a renewed suspension of particles. To put it differently, it means that solid discrete elements, previously at rest on a surface, are pulled off the surface and brought back into the bulk of the fluid in a suspended state. In that sense, this process differs from abrasion or erosion, which may appear
very similar but imply the release of tiny elements that were previously parts of larger objects (like freshly formed dust resulting from the erosion of rocks).

According to this definition, it is evident that particle resuspension is one of the processes at play in bounded dispersed two-phase flows. The reverse process is particle deposition, i.e., the accumulation of particles initially suspended on a surface. The deposition process will only be briefly discussed in Section 2.1.2 and is not considered in great details. Concentrating on resuspension, the above definition raises the following issues:

(Q1) Where does resuspension take place?

(Q2) What are the conditions required for resuspension to occur? More specifically, what are the nature of the fluid, of the particles, and of the surfaces involved?

(Q3) What are the driving mechanisms allowing particles to resuspend from surfaces?

(Q4) What happens to the resuspended particles after being pulled off the surface?

In the following paragraphs, the key features of particle resuspension and the challenges associated with them are briefly introduced starting from a selection of practical situations taken from natural and industrial applications (see Section 2.1.1). These selected examples illustrate the issues (Q1), (Q2) and (Q4) (see Section 2.2 for details about the underlying mechanisms related to the issue (Q3)). The reverse process of particle deposition is briefly covered in Section 2.1.2 while the terminology used in the rest of the manuscript is detailed in Section 2.1.3.

### 2.1.1. Illustrations of typical resuspension phenomena

**Resuspension in the air:** When thinking about particle resuspension, one of the first examples that comes to mind is the resuspension of sand/dust through the action of the wind [1, 16]. This process is typically at play in the winnowing of a sand dune [17], in which sand particles located in the upper layer of a dune become suspended when exposed to a strong-enough wind (see Fig. 1a). In the case of sand dunes, this process is generally an intermittent one, meaning that sand particles are brought back in suspension only by intense events due to strong but short-lived fluctuations in the wind velocity and direction. Yet, in the case of extreme wind speeds (as in storms), severe resuspension episodes persist for a significant amount of time, leading to high concentrations of particles in the atmosphere. Such persistent processes are responsible for either very localized phenomena (e.g., dust devils due to local vortices, see Fig. 1b) or broader spatial phenomena (e.g., dust storms as shown in Fig. 1c).

<figure><img src="image_2.png" /><figcaption>(a) Wind-induced resuspension of sand on a dune. Source: Pixabay.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(b) Dust devil in a field. Source: NASA</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(c) Sand-storm over a desert. Source: Pixabay.</figcaption></figure>

<caption>Figure 1: Illustrations of wind-driven resuspension of sand by various types of flows: normal wind conditions (left), whirlwinds (middle) and storms (right).</caption>

Another interesting feature of sand/dust resuspension is that the fate of the resuspended particles depends on both particle properties (size, density, etc.) and fluid flow characteristics (velocity, intermittency, etc.). For instance, as sketched in Figure 2, large and heavy sand particles can perform short hops above the surface (called reptation) or, alternatively, roll on the surface without actually getting detached from it (creeping). These intermittent, yet persistent, processes are responsible for the constant evolution of sand dunes or the
formation/erasure of ripples under the wind. Meanwhile, smaller and lighter particles can remain suspended for much longer times before impacting again the surface: this specific behavior is referred to as saltation. Upon colliding with the surface, saltating particles can eject other particles due to a transfer of kinetic energy. When small dust/sand particles reach high-enough altitudes in the atmosphere, these dust/sand grains can even move across continents (e.g., Sahara sand reaching south-European countries or even South-America). These different modes of transport for wind-blown sand/dust particles are the subject of dedicated studies in the field of geomorphology (interested readers are referred to existing books [18] or reviews [1, 3]). Similar features are also reported on Mars, where dust resuspension occurs under locally strong whirlwind or turbulence [19]. Another serious concern related to the exploration of extra-terrestrial planets is the potential contamination of the planet environment by biological particles from Earth, which can resuspend from the rover surface depending on the local atmospheric conditions [20, 21].

<figure><img src="image_5.png" /><figcaption>Figure 2: Representation of the different modes of wind-blown sand transport, namely: suspension (i.e., particles remaining suspended in the flow), saltation (long hops above the surface), repletion (short hops above the surface) and creep (migration on the surface). Reprinted with permission from [22]. Copyright 2021, Elsevier.</figcaption></figure>

Studies on particle resuspension are not limited to particles of natural origin. For instance, many works concern the resuspension of radioactive particles that can accumulate on the ground following nuclear accidents (see for instance [23, 24, 25]). In this case, particle resuspension is shown to expand the contaminated zone under strong wind conditions. Besides, since even tiny amounts of radioactive particles can jeopardize human health, resuspension can prolong the threat of exposure to radioactive particles even beyond the passage of the initial particle-laden plume. In fact, even though the plume containing radioactive particles is emitted over a limited amount of time following such accidents, these primary particles can deposit on soils and be resuspended later over much longer times. However, quantifying such phenomena is not easy since it requires measuring rare (extreme) events. This is due to the fact that the amount of particles present on a given surface decreases as time goes by, since the initial particle deposit is not replenished with a fresh source of particles. It is indeed observed that loosely-adhering particles are quickly blown away by the wind at short exposure time. Hence, at longer exposure times, only particles adhering strongly to the surface are left on this surface. Removing such particles requires higher wind conditions, which occur rarely. This explains the typical trend observed in experiments, where the amount of resuspended particles is shown to decrease with the inverse of time at long exposure times (e.g., [24]).

When both natural and artificial particles are present, more complex phenomena can take place when these particles interact with each other to form new compounds or aggregates (called secondary particles). In such cases, resuspension can be significantly altered compared to the case with only primary particles since it is not driven by the primary particle properties but rather by the aggregate characteristics (e.g., size, shape, composition). Such phenomena are observed when dealing with artificial hazardous particles that accumulate in soils. For instance, water-soluble $^{137}$Cs ions like those released in the Fukushima accident in 2011 reacted with airborne aerosols and were found attached to mineral particles. After deposition,
they were reintroduced into the atmosphere as radioactive dust [26]. Identifying the host particles is not always straightforward as many chemical reaction pathways can coexist depending on the soil composition and conditions (e.g., plutonium compounds [27]). Particles can also be absorbed by vegetation and be resuspended within pollen [26] or during wildfires from the combustion of vegetation [28]. Similar issues arise in studies of lead poisoning for children in urban environments [29], where lead has accumulated in urban soils due to the extensive use of Pb additives in both gasoline and house-paints in the past.

Still referring to atmospheric flows, another example of interest is the resuspension of particulate matter (PM) accumulated on various surfaces. Air pollution from PM has been widely studied in the aerosol science community [30] due to their potential impact on human health when inhaled [31, 32]. This includes a large variety of particles, such as spores [33], road dust [15], exhaust emissions of combustion engines [34] or abrasion-derived particles from cars (e.g., tires, brake, road-wear) [7, 35]. Conversely to primary radioactive particles, these pollutants are continuously released in the atmosphere and deposited, basically providing an “infinite” source of particles to be resuspended. In-situ observations show that resuspension is affected by environmental factors (e.g., local/seasonal variations in wind speed, precipitation, humidity) but also by human activities. For instance, as displayed in Fig. 3a, the re-emission of road dust is significantly higher in the wake of a vehicle (see for instance [7, 36, 37]). This so-called traffic-induced resuspension of road dust is estimated to contribute to 9-59% of the PM10 mass concentration from non-exhaust emissions of vehicles (see the recent review [15] and references therein). This makes it one of the predominant sources of particulate matter emission, possibly reaching similar levels as the traffic exhaust emissions [34].

<figure><img src="image_6.png" /><figcaption>Figure 3: Illustrations of dust resuspension in the air induced by human activities (traffic on the left, walking on the right).</figcaption></figure>

<figure><img src="image_7.png" /></figure>

This last example on traffic-induced resuspension brings out the question of the pathways followed by particle resuspension in the air. In fact, resuspension events are usually classified in two main categories in the literature [5, 39]: resuspension induced by an airflow and resuspension induced by human activities. These various pathways have been extensively studied in indoor environments, such as home or office rooms [5], due to the growing concerns over air quality over the last few decades [40, 41]. This includes both studies on particulate matter (e.g., PM2.5, PM1.0, PM0.1 or lower) [5] as well as hazardous biological/chemical particles (e.g., anthrax [33] or even flour dust [42, 43]). These studies have allowed to identify a number of sub-categories briefly listed below (see also Fig. 4 and previous reviews [5, 12]):

* Resuspension induced by an airflow:

- Wind-induced resuspension in indoor environments is very similar to the outdoor one. One of the differences is that indoor surfaces can be very diverse (to name a few: wooden floor, concrete floor, metallic surfaces, vinyl floor or carpets) [5]. As a result, the amount of particles available in each surface can vary significantly while their resuspension depends highly on the local configuration (e.g., lower exposure to the airflow in fibrous environments like carpets, shielding mechanisms). In addition, recent studies have confirmed that the amount of resuspension depends on particles
<figure><img src="image_8.png" /><figcaption>Figure 4: Representation of the different pathways for indoor airborne resuspension, showing resuspension induced by ventilation, crawling, walking, wind/airflows as well as human interactions with surfaces (e.g., scratching).</figcaption></figure>

properties, including their size, their shape and their composition and origin (see for instance
recent comparisons between dust, cat or dog fur, or bacterial spores [44]).

- Ventilation-induced resuspension involves two different aspects related to the impact of ventilation systems. First, some particles can deposit within the ventilation ducts and, as air is flowing through the ducts, these particles can be resuspended and emitted into a room (provided they are not captured by filters) [45, 46]. Second, the airflow induced by a ventilation system can affect the amount of particles resuspended from nearby surfaces. This second effect depends naturally on the orientation and flow rate of the ventilation system. For instance, a recent study has shown that the resuspension in impinging jet ventilation (i.e., high-speed airflow perpendicular to the ground with a vent placed at a given height) can be two to three times larger than displacement ventilation (i.e., low-speed airflow parallel to the floor introduced near the ground through a vent), and more than 10 times larger than mixed ventilation (i.e., low-speed airflow through a vent close to the ceiling) [47].

* Resuspension induced by human activities differs from resuspension induced by airflows since it usually involves sudden changes in the flow conditions (e.g., velocity, turbulence) as well as extra vibrations, mechanical contact and possibly shocks that can enhance particle resuspension. In the following, various indoor human activities are listed following the usual classifications used in the literature (see for instance [39]):

- Walking-induced resuspension is triggered by a combination of several factors (see the recent review [48]). As displayed in Fig. 3b, the upward and downward foot movement during walking can generate complex air movements in the immediate surroundings and cause particles to resuspend [37, 38]. Walking motion also induces surface vibrations and electrostatic forces acting between the particles and the shoe bottom surface, which can further affect particle resuspension. In addition, both the pace and style of walking were observed to impact resuspension [49]. More recently, a study of infant exposure to respiratory pathogens has provided evidence that the type of socks worn by children while walking has profound consequences on resuspension [50].

- Crawling-induced resuspension occurs due to the crawling motion of children on the floor, especially on carpets or other fabrics. It has been increasingly studied in the past few years, since children are exposed to higher concentrations of PMs due to their proximity to the surface. In
fact, chamber experiments show that infants are exposed to contamination levels that can be one order of magnitude higher than adults [51, 52]. These levels are also shown to be lower during walking rather than crawling.

- More generally, physical activities are reported to induce higher particle resuspension in indoor environments. It is observed that resuspension intensifies with an increase in the intensity of physical activities (see for instance sports in school gyms [53], walking intensities [54], dancing [55] or even motion while sleeping [56]). This can be related to a combination of effects, including aerodynamic resuspension, surface vibrations, mechanical abrasion, and contact transfer. Besides, particles can resuspend from a variety of surfaces including clothes worn by humans during their activities [57] and surfaces with which they interact (e.g., mattresses while sleeping [56]). Recent measurements on clothing have also highlighted that resuspension depends highly on the clothing texture (i.e., roughness, weave pattern) and on the type of motion imposed on the clothes (e.g., stretching with various intensity) while it seems to be slightly affected by the fabric material [55].

- Cleaning activities can also lead to a sudden and drastic change in the amount of particles resuspended. A counter-intuitive example is the case of vacuum cleaners. Vacuum cleaners are designed to collect dust from the floor using a strong airflow that favors their resuspension followed by their aspiration in the device. However, when switched on, vacuum cleaners can also induce PM10 concentrations that can be up to 3 or 4 times higher than under normal conditions [58]. This is explained by the substantial increase in the number of resuspended particles from nearby carpet fibers due to the high-velocity airflow induced around the vacuum cleaner and also to mechanical motion induced by the contact between the wheels and the carpet. In addition, some of the particles captured by the collectors within the vacuum cleaner can be re-entrained during or after the use of the device [59]. Such re-entrainment actually depends on the type of collectors used (such as filter bags, wet dust collectors or cyclonic collectors). An alternative cleaning technique is based on mechanical resuspension, as for sweeper trucks with rotating brushes that are used to remove leaves from gutters before aspirating them.

- Another way to remove particles from a surface is contact transfer. This happens when particles on the host surface are picked up by a different surface during contact between two surfaces. Contact transfer can occur between objects (e.g., a stack of paper placed on a table) but also between the host surface and a person's skin and clothing. This mechanism is only seldom studied in the context of resuspension because particles are not immediately re-emitted into the air. Nevertheless, this process is specifically identified here since it allows particles to be picked up from a contaminated surface in a certain environment and, thanks to their transfer to another surface, to be resuspended at a later time in a different environment (e.g., from one room to the next). This whole process is actually known as translocation. For instance, higher concentration compared to ambient air pollution were observed when people came back from smoke breaks [57]. Alternatively, contact transfer can also be used to detect the presence of trace explosive particles from fibrous surfaces (e.g., cotton, polyester) [60, 61].

**Resuspension in marine systems:** Particle resuspension has also been extensively studied when the carrier phase is a liquid. From a historical perspective, research on particle resuspension has emerged more or less at the same time to address sand transport in the air [16] and sediment transport in rivers [62]. Sediments at the bottom of a river are formed by the accumulation of various particles, including natural ones (e.g., clay, sand, gravel) and contaminants (e.g., heavy metals, plastics). As the river discharge fluctuates, various amounts of sediments are resuspended from the riverbed and transported further downstream following similar modes as for airborne particles (i.e., reptation or creep as shown in Fig. 2). In fact, the combined deposition and resuspension of sediments in rivers plays a key role in the morphodynamics of rivers (i.e., changes in the river path over geological times) and in river deltas. As discussed in recent papers [63, 64, 65]), decades of research on bedload river transport have revealed that the threshold velocity above which particles start to move depends on a number of parameters, including the particle size and the fluid-to-particle density ratio, thus confirming the early findings of Shields in 1936 [62].
<figure><img src="image_9.png" /><figcaption>(a) Picture showing the complex deposit formed by gravels at the bottom of rivers or lakes (which can be later on resuspended by strong flows). Reprinted with permission from [66]. Copyright 2018, John Wiley and Sons.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) Visualization of internal wave propagation in stratified flows: left images showing the density fields and right images displaying the seed particles being detached from the surface due to internal waves. Reprinted with permission from [67]. Copyright 2009, John Wiley and Sons.</figcaption></figure>

Figure 5: Illustrations showing the sediment beds formed in rivers (a) their resuspension due to waves within lakes/water (b).
Resuspension of sediments in marine systems can originate from either natural or artificial processes. For instance, natural resuspension processes in marine systems encompass river currents [18], tidal currents or storm surge in oceans [68, 69], the action of surface waves [70, 71] or even internal solitary waves which occur in stratified fluids (see Fig. 5b, and the recent review [4]). Among the various anthropic activities that impact particle resuspension, dredging, shipping and trawling [72] have been confirmed to modify significantly the amount of resuspended sediments. Similarly, other animals like fishes can modify sediment resuspension due to the waves induced by their motion [73]. These selected examples imply that resuspension can be intermittent in nature, since it can take place over different temporal scales (from less than one hour for dredging operations to weeks for storms ([68]). Besides, as for resuspension of airborne particles, various types of particles can be involved (including natural sediments [3] or plastic contaminants in riverbeds [74]).

Studies on sediment transport in rivers have also highlighted the role of the deposit morphology in the resuspension process. In fact, sediments naturally form a multilayer deposit, where particles pile up to form large clusters on the underlying surface (see Fig. 5a). In that case, the resuspension process differs from measurements made with sparse particles on the surface. In sparse monolayer deposits, isolated particles tend to resuspend independently of each other, while in multilayer deposits particles collide with each other, possibly leading to cascade phenomena as in avalanches. Hence, resuspension by collision is one of the main modes of resuspension in multilayer systems. This has recently led to a distinction between fluid-driven motion and entrainment-driven motion which are both occurring but with various relative importance depending on the situation considered [3, 75]. Collision-induced resuspension has been observed for river sediments [3] and also for sand particles (where saltation is predominant) [16]. Since the number and frequency of collisions between particles are driven by the deposit structure, many studies have assessed the role of sediment size, size distribution [76], inter-particle cohesive forces (see for instance [3, 10] and references therein). In addition, since very large millimeter-size particles are harder to resuspend than micrometer particles, armoring effects can occur: this corresponds to the presence of less mobile particles preventing resuspension of smaller particles located underneath or in their wake, thereby acting as a sort of shield for nearby mobile particles. Such phenomena have been observed in multilayer deposits formed by particles with a large size distribution [66]. In such cases, the amount of resuspension can display sudden changes with time: it can be initially important until the small particles on the top of the deposit are removed and only large ones are left exposed to the fluid, at which point these large particles act as shelter for the layers of deposit below, thereby severely reducing the resuspension rate. Alternatively, consolidation effects can also be responsible for a decrease in the resuspension at longer times. These processes are actually related to a restructuring of the deposit and sometimes to modifications of the contact area between individual particles, which leads to the formation of more compact and cohesive deposits that are harder to resuspend [77].

**Resuspension in confined systems:** Last but not least, resuspension occurs in many constrained environments, such as pipes or porous media. As discussed in a recent review [2], this is a matter of concern in drinking water systems, where a biofilm grows on the interior surface of the pipes (see also Fig. 6). When the flow through the pipe is strong enough, deposited particles can detach from the surface and be transported in the pipe network. This can lead to the presence of unwanted contaminants in the water (e.g., chemical or biological particles), thus giving rise to poor water quality and increased health hazard for customers [78]. In such cases, the objective is often to limit the accumulation of a deposit on the surfaces (e.g., by imposing high velocities in pipe networks [79]) and, when it cannot be prevented, to limit its resuspension. The difficulty is that the particles involved often span a wide range of sizes [80]. Similar issues arise when studying groundwater quality, which can be affected by the mobilization of contaminants present in the ground (see for instance [81, 82]). Resuspension in pipes has been also explored in industrial applications. For instance, corrosion products can accumulate on the interior surface of pipes within heat exchangers (leading to fouling), thereby affecting the normal operating conditions [83]. Resuspension then appears as a desirable process: it has led to the development of cleaning procedures using ultrasonic methods, where cavity bubbles are used to break the deposits formed and then a strong flow through the pipe brings the detached particles out of the system [84] (similarly to resuspension in other three-phase flows [85]).
Resuspension is also a matter of concern in nuclear safety [86]. For example, in nuclear fusion reactors, radioactive particles (like dust or graphite) can be resuspended and released in the environment during a LOss of Coolant Accident (LOCA, i.e., when the pipe containing the coolant is damaged and fluid is leaking [87, 88, 89, 90]) or during a Loss Of Vacuum Accident (LOVA, i.e., when the vacuum vessel is damaged [91, 92, 93]).

<figure><img src="image_11.png" /><figcaption>Figure 6: (a) Images showing biofilm growth on different pipe materials. Reprinted with permission from [94]. Copyright 2015, Springer Nature. (b) Representation of a biofilm life cycle, including their possible resuspension. Reprinted with permission from [2]. Copyright 2016, American Chemical Society.</figcaption></figure>

**Summary:** Based on these selected examples, several conclusions can be drawn. First, resuspension is observed in a large array of environmental and industrial situations. Second, resuspension involves a variety of fluids (such as liquids, gas or even a mixture of both), surfaces (from smooth glass/metal substrates to fibrous clothes or carpet fibers), and particles (aerosols like spores or radionuclides and sediments). The involved particles have a range of origins (e.g., natural or artificial), compositions (e.g., chemical, biological), shapes (e.g., spheres, spheroids, fibers), and sizes (from nanometric aerosols to millimeter-size sediments). In addition, deposited particles can be arranged either in a monolayer fashion, with isolated individual particles, or in a multilayer configuration with complex particle arrangements. This leads to a distinction between independent resuspension events and correlated resuspension events (possibly with collective motion). Third, resuspension is a highly multiscale process, spanning a wide range of temporal and spatial scales, from the resuspension of biofilms over a few milliseconds in millimeter-size pipes to radioactive resuspension across kilometer-size fields over years. Fourth, resuspension is a multifaceted problem, since the process can occur through a variety of pathways (including flow-induced or collision-induced modes as well as be induced by human activities). This also implies that resuspension is a highly multidisciplinary topic. In fact, the mechanisms and forces at play in resuspension involve physical, chemical and/or biological effects. Hence, studying particle resuspension requires a knowledge at the crossroad between several fields, such as fluid mechanics, interface chemistry, surface sciences, and possibly biology.

A consequence of this multifaceted, multidisciplinary, and multiscale process is that the important mechanisms and forces at play in a given situation are not necessarily the dominant ones for a different application. This explains why many studies have addressed resuspension within a specific sub-domain and avoided tackling the overall phenomenon. As will be seen later, this represents a challenge when trying to come up with generic models (see Section 4.3). Nevertheless, despite these difficulties, a common and universal description of the resuspension process is slowly but surely emerging (see also the recent review [3]).

### 2.1.2. *The reverse process of particle deposition*

Previous examples have revealed that resuspension occurs often concomitantly with the reverse process of particle deposition. As illustrated in Fig. 7, the deposition process corresponds to the migration of suspended
particles toward a surface on which they can adhere and eventually come to rest. From a physical point of view, it is best to describe particle deposition as a two-step process (see for instance [95]): first, a transport step where suspended particles are carried by the flow towards the immediate vicinity of a surface and, second, an attachment step in which these particles can stick to the surface or rebound from it. As such, this process is the opposite of the resuspension process, where particles adhering to a substrate are brought back in suspension and transported by the action of a fluid flow.

<figure><img src="image_12.png" /><figcaption>Figure 7: Illustration of the resuspension process and of the reverse deposition process</figcaption></figure>

In fact, deposition and resuspension are deeply intertwined. For example, once particles are detached from a surface and resuspended, their subsequent transport by the flow can either lead them to remain suspended in the fluid or to redeposit later on the surface (as in reptation and saltation). This continuous switch from one process to another is responsible for the slow and intermittent migration of sand dunes [1, 18]. Another example of the intricate coupling between these two processes is the formation of large complex-shaped deposits on a surface. This happens due to the continuous accumulation of deposited particles on a surface, which leads to the formation of multilayer deposits (e.g., sand dunes in desert, sediments in riverbeds or fouling of heat exchangers by corrosion products). In turn, these deposits can modify the velocity of the flow near the surface, especially when their size is large enough (typically with a size comparable to the smallest scale in the fluid, i.e., the Kolmogorov length scale $\eta_K$ in a turbulent flow). This modified fluid velocity can then lead to spatial variations in the amount of resuspended particles, giving rise sometimes to regular shapes (e.g., ripples in sand dunes).

Since particle deposition usually occurs before resuspension start to kick in, it has been extensively studied in the multiphase flow community and interested readers are referred to existing reviews on particle deposition for further information (see for instance [96, 97]). Given their correspondence, it is not surprising that the key factors affecting deposition are similar to the ones identified for resuspension, namely:

* Particle characteristics, including: particle size distribution, density, shape, surface area, hygroscopicity and hydrophobicity. For instance, atmospheric aerosols can be highly hygroscopic and absorb water vapor at high relative humidity thus changing dimension, density, and optical properties ([98]).

* Surface characteristics, including: surface roughness, adhesiveness with particles, electric charge or magnetic properties. For instance, roughness features much smaller than the particle size can significantly modify the adhesive forces, while roughness features with a size comparable or bigger than the particle size can lead to sheltering effects [97]. When the size of roughness features becomes comparable or larger than the smallest fluid scale, it also modifies near-wall fluid velocities (and, consequently, particle transport in the vicinity of the surface).

* Flow characteristics, including: fluid properties (like density, viscosity), turbulence intensity (e.g., intermittency, coherent structures near boundaries). For deposition in the atmosphere, atmospheric stability is also important and in general, the ratio of the deposition velocity and the friction velocity appears to be smaller in stable atmospheric conditions ([98], [99]).

### 2.1.3. Terminology suggested and used here

To depict accurately any field of physics and develop sound physical analyses as well as proper theories, it is important to start by naming precisely objects and, especially, processes. In the field of particle resuspension, this is not a step to overlook. Indeed, from the various examples introduced in Section 2.1.1, it
appears that the terminology currently in use varies depending on the specific application which is addressed. To provide a few instances, this includes terms like reaerosolization of airborne particles (such as viruses, bacteria, spores) [100], remobilization of plastics in rivers [74] or during floods [101], winnowing of sand in ripples and mega-ripples [102], etc. Particle resuspension is also frequently named particle removal [60], particle detachment [103, 104] or particle re-entrainment [105]. Alternatively, in the context of sediment in rivers, it is common to find studies focused on the onset of motion or the incipient motion of particles [3]. Obviously, this terminology does not provide a clear-cut indication as to which physical mechanism is being addressed specifically. In the literature, the only distinction currently available has been suggested in both aeolian transport and sediment transport: it consists in trying to identify specifically the subsequent transport of particles once they start moving. More precisely, this has led to the definition of creep motion (i.e., continuous or near-continuous contact with the surface through rolling/sliding motion), reptation (particles undergoing short hops above the surface), saltation (i.e., particles undergoing long hops with splashing during subsequent collision events) and suspension (i.e., particles which do not deposit on the surface, at least during the observation time). Clearly, this is not a satisfactory state of affairs: such vague formulations can quickly lead not only to confusion but blurred physical descriptions.

To avoid these pitfalls, we introduce here specific definitions to distinguish between the different phenomena at play. These notions are then used throughout the present paper regardless of the application domain. First, we start by making a distinction for particles that are in contact with a surface or a deposit:

(i) Sticking/deposited/adhering particles correspond to particles that are in contact with a substrate (i.e., either a surface or other deposited particles) at a given location (no motion).

(ii) Attached particles are particles in contact with the substrate but that can be moving.

(iii) Detached particles refers to previously attached particles whose contact with the surface has been broken.

(iv) Adhesion forces refer to the contact forces between a particle and a surface.

(v) Cohesion forces designate the contact forces between particles.

In addition to these fundamental definitions, we introduce another distinction regarding the resuspension process (see also Fig. 8):

(def-1) **Particle incipient motion** refers to the moment at which a sticking particle starts moving. It is related to the rupture of the balance between forces preventing its motion and forces inducing its motion. The dislodgement of sticking particles triggers their motion either on the surface (attached particles) or in the flow (detached particles).

(def-2) **Particle migration** is related to the motion of attached particles until they are actually **detached** from the surface.

(def-3) **Particle re-entrainment** refers to the motion of particles after detachment, i.e., in the near-wall region.

(def-4) **Particle resuspension** is a generic term that encompasses all phenomena (i.e., incipient motion, detachment, migration and re-entrainment).

Hence, particle resuspension can be defined here as the phenomenon that leads to the detachment and sub-sequent re-entrainment of a sticking particle from a surface.

These definitions allow to distinguish between attached and detached particles while making a clearer separation between the three motion encountered in particle resuspension (i.e., incipient motion, migration and re-entrainment). These definitions remain broad enough to encompass phenomena observed in other fields. For instance, in the context of geological flows, abrasion refers to the erosion of a surface by exposure to scraping or other mechanical constraints on the surface [106, 107]: it can thus be interpreted here as a specific case of particle detachment due to hydro-mechanical forces. Yet, it does not necessarily imply resuspension (unless particles were suspended previously). More generally, an interesting characteristic of this terminology is that it allows to bring out the difference between particle resuspension and particle release (also called seeding for pollens). In fact, the two processes might sometimes appear to be similar (e.g., release of pollen by plants due to the wind). However, according to the present definitions, particle release does
<figure><img src="image_13.png" /><figcaption>Figure 8: Sketch summarizing the terminology used in the present paper for particle resuspension: the incipient motion corresponds to the instant when a particle starts moving, leading to its detachment from the surface (possibly with migration on the surface beforehand) and its subsequent re-entrainment by the flow in the near-wall region. This makes up the whole resuspension process.</figcaption></figure>

not involve particles that were previously suspended, while particle resuspension does. In addition, these
definitions encompass the previous distinctions introduced earlier between creep and saltation/reptation.
More precisely, creep corresponds now to a specific case of particle migration (in which particles remain
in contact with the surface and perform rolling/sliding motion) while reptation and saltation correspond
now to particle resuspension. It is worth noting that these definitions do not allow to distinguish between
reptation and saltation. The main reason for this choice is that they both involve particles that have
resuspended and, after a certain time in suspension, collide with the surface again. While reptation does
not trigger the resuspension of other particles during this collision, saltation leads to multiple subsequent
resuspension events through splashing (see Section 2.2 for more details). This is sometimes referred to as
particle entrainment in the aeolian and fluvial communities [3] (in the sense of motion induced by collisions
as in avalanche processes). However, we do not wish to introduce such a distinction here since similar
phenomena can occur for particles that were simply in suspension and collide with the surface (i.e., not
necessarily resuspended). Hence, we believe that such a distinction should rather be handled by proper
definitions introduced in the description of the deposition phenomenon.

## 2.2. The phenomenology of particle resuspension

The overview of applications in Section 2.1 was meant to illustrate the relevance of particle resuspension in an array of practical situations as well as to point to the intricate physical phenomena which are involved. Drawing on these observations, the questions that now come to mind are:

(a) How are deposited particles resuspended from a surface?

(b) What are the fundamental interactions acting in this process?

(c) What are the underlying mechanisms?

The purpose of this section is to provide answers to these questions. A simple picture meant to clarify
point (a) is first given in Section 2.2.1, after which an outline of the fundamental physical interactions with
respect to point (b) is provided in Section 2.2.2 (before being developed later in Section 4.1). Then, the
underlying mechanisms mentioned in point (c) are analyzed in Section 2.2.3.

### 2.2.1. A competition between motion-inducing versus motion-preventing factors

As it transpires from the terminology introduced in Section 2.1.3, particle resuspension results from
a subtle balance between forces that keep particles attached to a deposit and forces that induce motion
through detachment/migration. Therefore, in spite of its inherent complexity, it is worth starting with a
simple but leading image. In this crudely cut image, it can be said that particle resuspension boils down
to a competition between, on the one hand, elements preventing particle motion and, on the other hand,
factors acting to dislodge particles from the deposit (either the wall surface or layers of deposited particles).

2.2.2. Fundamental physical interactions

In turn, these two main categories of forces, acting either to counteract or facilitate particle motion, result
from three fundamental physical interactions, namely **particle-fluid**, **particle-particle** and **particle-wall**
interactions. As will be seen later in Section 4.1, these interactions are mostly treated using a continuum
assumption, i.e. considering that materials/fluids are continuous media (by contrast with discrete assump-
tions made in molecular descriptions). We are therefore dealing with a process at the crossroads between
different subjects of physics: this includes turbulence, interface chemistry and material properties, whose
combined effects create the tapestry of observed phenomena. Actually, by gathering particle-particle and
particle-wall forces into a general **particle-surface** interaction, it is possible to consider two main types of
interactions. This does not mean, however, that there is a simple one-to-one correspondence with the two
categories of motion-inducing and motion-preventing factors. Indeed, while particle-fluid interactions are
essentially acting to induce particle motion, gravity forces (when present) can either be motion-inducing or
motion-preventing depending on the geometry considered. This means that we need to describe specifically
the key resuspension mechanisms at play in particle resuspension to better grasp this complexity.

2.2.3. Key resuspension mechanisms

To describe the key physical mechanisms at play in the resuspension process, we can rely on a range of
experimental observations which have provided ample evidence over the years. Since comprehensive reviews
have already described these mechanisms (see for instance [3, 10]), we limit ourselves to recalling key points,
with the difference that these mechanisms are summarized here following an original presentation that
paves the way for a unified description of particle resuspension. In this new presentation, we distinguish
between individual resuspension events, in which particle-particle interactions are negligible, and collective
resuspension events, in which particle-particle collisions and forces play a role. Note that this is different
from considering resuspension from mono- and multilayer deposits, which explains why collective events are
further divided into two effects.

**Individual resuspension events** First, studies of individual particles deposited on a surface have
revealed that the incipient particle motion is triggered by a rupture of balance between forces/torques acting
on the particle. This rupture of balance can lead to three different types of incipient motion sketched in
Fig. 9a: **sliding** (a type of surface migration in which the particle has a purely translational motion along the
surface), **rolling** (another type of surface migration where the particle has a purely rotational motion along
the surface) and **lift-off** (when the particle is immediately detached from the surface due to pulling forces).
These three modes induce a variety of complex particle motion near the surface that have been observed
experimentally (see Fig. 9b): a particle can start rolling/sliding on the surface before being actually detached
from it, while another one can be immediately detached from the surface due to direct lift-off [109]. Once
resuspended, particles can collide with the surface again, where they can be recaptured (either deposited or
migrating on the surface) or just bounce on it.

Following extensive experimental observations, there is now an emerging consensus in the multiphase
flow community on how such particle dynamics depends on particle and fluid properties. This is formulated
in a recent review [10] which has shown that particles small enough to be fully embedded within the viscous
sublayer are more prone to display rolling/sliding motion, while larger particles are more inclined to be
directly lifted-off the surface as they interact with the near-wall turbulent structures which exist only at
some distances away and not in the immediate vicinity of the wall (where viscosity effects smooth things
out). Such individual resuspension events are thus induced by the interaction between the fluid flow and
deposited particles (through the drag and lift forces discussed in more details in Section 4.1), and involve
only particle-fluid and particle-wall forces but not particle-particle interactions.
<figure><img src="image_14.png" /><figcaption>(a) Illustration of single particle entrainment from a surface: (a) rolling mode, (b) sliding mode and (c) lifting mode.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) Snapshots showing the different types of induced motion for micrometer-size glass particles on a ceramic substrate: rolling/bouncing (top), rolling/bouncing and potential liftoff (2nd), rolling/bouncing and late liftoff (3rd) and immediate liftoff without rolling/bouncing (bottom). Reprinted with permission from [108]. Copyright 2013, Taylor & Francis.</figcaption></figure>

Figure 9: Sketch of the fundamental mechanisms for the resuspension of individual particles from surfaces and the corresponding images obtained from experimental observations.

**Collective resuspension events** Second, more recent studies on dense monolayer deposits have highlighted the role of inter-particle collisions in resuspension events [110, 111]. As shown in Fig. 10, once a particle starts migrating on the surface or moving near it, it can collide with other deposited particles, thereby setting these other particles into motion. As a result, when the distance between deposited particles is small enough, the incipient motion of a single particle is sufficient to trigger the motion of several other particles due to a series of inter-particle collisions (as collision propagation in avalanches). In such cases, this means that the resuspension of individual particles are not independent of each other but appear rather as correlated events. Such studies have bridged the gap between studies of individual particle resuspension from sparse monolayer deposits and studies of the collective resuspension events in multilayer deposits.

Third, decades of experimental observations on resuspension from complex multilayer systems have been performed within the fluvial and aeolian transport communities. Recently compiled in a review paper [3], these data provide a unified description of the resuspension process in both aeolian and fluvial research. In particular, they emphasize that, when the flow intensity is small enough, only a few particles located on the top of the deposit start moving over relatively short distances. This gives rise to the so-called creep motion (where moving particles remain in contact with the rest of the deposit). According to the current terminology given in Section 2.1.3, this type of motion is thus a sub-category of surface migration. Such resuspension events are actually intermittent (due to near-wall turbulence) and uncorrelated to each other. In that sense, this is similar to the hydrodynamic-induced resuspension over sparse monolayers, the only difference being the type of surface upon which the particles are moving. At higher fluid velocities, particles start displaying long-lasting rolling/sliding/hoping motion over the surface since near-wall fluid velocities are large enough to counteract the potential loss of energy that can happen upon colliding with the surface (for repeating particles), leading to the so-called continuous rebound mechanisms sketched in Fig. 11a. During such sustained motion, the dynamics of particles becomes more and more correlated as the particles are moving closer to each other. Finally, when the flow intensity is even stronger, the kinetic
<figure><img src="image_16.png" /><figcaption>(a) Sketch of collision-induced resuspension.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) Consecutive snapshots of 40 µm glass beads showing how collisions trigger a series of resuspension events in dense monolayer deposits. Reprinted with permission from [110]. Copyright 2021, American Physical Society.</figcaption></figure>

<caption>Figure 10: Illustration of the role of inter-particle collisions in triggering successive resuspension events.</caption>

energy of particles impacting the surface becomes high enough to trigger the resuspension of other deposited particles. In fact, resuspension becomes sustained by the impaction of moving particles on the surface, which can lead to splashing events (where several particles are resuspended following the collision of a single one, as displayed in Fig. 11b). This has been referred to as a threshold for impact entrainment, in the sense that new resuspension events are continuously triggered by particles impacting the surface due to their inertia. In that case, the resuspension events are largely correlated, resulting in a continuous transport of the upper layers of the bed.

The phenomena related to impact entrainment imply that the deposit morphology and structure should be carefully characterized. In fact, as depicted in Fig. 12, the deposit structure has a direct impact on:

a - **Cohesion**: The geometrical arrangement of particles is responsible for the existence of cohesion forces within the deposit whose intensity vary depending on the loose or compact nature of the deposit. Indeed, at the particle level, the number of contact of a single particle with its neighbors (which is referred to as the coordination number) varies depending on the local arrangement of nearby particles. This typically makes loose deposit easier to remove than compact deposits.

b - **Shielding effects**: This happens when a given particle is in the wake of another one placed upstream, thereby reducing its exposure to the fluid flow (see Fig. 12). Such shielding effects can become significant when the upstream particle/cluster has a size larger than, or comparable to, the Kolmogorov scale, meaning that the fluid motion is modified by its presence.

c - **Armoring effects**: This occurs when a layer of particles harder to resuspend is located on top of the deposit, thereby protecting layers located underneath (even if they contain particles easier to resuspend). Such effects have been observed when dealing with particles having a range of sizes (i.e., polydisperse) [66]. In such cases, since large particles are harder to remove than small ones, the deposit structure evolves with time as follows: the small particles are quickly resuspended, leaving mostly large particles in the upper layer of the deposit after a finite time. At that point, the number of resuspension events drops since large particles act as a shelter for all the particles located underneath.

d - **Restructuration**: This can occur when a deposit is exposed to a strong enough flow, leading to the rearrangement of particles within the deposit. Restructuring can help form more compact deposits over time, meaning that particles become harder to remove due to the increased cohesion within the deposit. Such restructuring has been mostly characterized in the context of nanoparticle aggregates due to its impact on the fragmentation process (see for instance [113, 114]).
<figure><img src="image_18.png" /><figcaption>(a) Sketch of the continuous rebound mechanism: the motion of a particle (yellow lines) can either lead to periodic hopping motion (when the kinetic energy gained after each impact $E_↑$ compensates the energy lost during the collision $E_c$) or to the particle stopping on the surface after a few collisions (when the energy loss during each rebound $E_c$ overcomes the energy gained during each hop $E_↑$). Reprinted with permission from [3]. Copyright 2020, John Wiley and Sons.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) Consecutive snapshots of 6 mm PVC beads showing how impacting particles can induce splashing events (i.e., multiple deposited particles are resuspended). Reprinted with permission from [112]. Copyright 2007, American Physical Society.</figcaption></figure>

<caption>Figure 11: Illustrations of the fundamental mechanisms at play in particle resuspension from multilayer deposits.</caption>

**Overall picture of resuspension mechanisms:** Drawing on the similarities between the phenomenology of resuspension from monolayer and multilayer deposits, a new overall picture is now proposed. This is sketched in Fig. 13, which represents the mechanisms at play in particle resuspension as a function of both the deposit structure (monolayer versus multilayer deposits) and of the flow intensity. In this attempt towards a unified description of particle resuspension, the main types of incipient motion for particles (namely rolling, sliding or lifting) appear as predominant in the case of sparse monolayers, where particles are so distant from each other that each event is independent from other ones (i.e., we are dealing with uncorrelated events). As the density of deposited particles is increased, these types of incipient motion are still relevant but the subsequent motion of particles on/near the surface can trigger additional resuspension events due to inter-particle collisions. In the case of multilayer systems, the impaction of particles with a high enough inertia can result in more complex splashing events (where several particles are resuspended). When the resuspension becomes highly correlated (due to a combination of hydrodynamic-induced events and sustained collision-induced effects), the upper layers of the deposit can display complex collective motion.

At this stage, it is worth mentioning that Figure 13 only pictures the evolution of the type of mechanisms as a function of the deposit structure and of the flow intensity. In reality, it also depends on the particle properties (size, shape, density). For the sake of clarity, only a 2D sketch has been displayed here but similar drawings can be made to show the evolution with respect to other parameters (like the particle size, shape, fluid to particle density ratio).

### 3. Estimating resuspension: experimental measurements and quantities

Having introduced the basic physical concepts behind the resuspension process in the previous Section, the purpose of the present section is to describe how particle resuspension can be estimated. To that effect,
<figure><img src="image_20.png" /><figcaption>Figure 12: Illustration of the possible roles of the deposit structure on particle resuspension: the strength of cohesion within clusters is related to the deposit morphology (left); upstream particles can act as shelters for other particles in their wake (2nd image); large and heavy particles can prevent the resuspension of smaller particles located beneath (3rd image); the deposit structure can evolve with time (4th image).</figcaption></figure>

the various quantities used to quantify particle resuspension are first introduced in Section 3.1. Then, the experimental techniques developed to measure such quantities are reviewed in Section 3.2.

### 3.1. Indicators for particle resuspension

When trying to quantify particle resuspension, two main questions appear:

* What are the physical quantities that characterize the resuspension process?

* What is the information contained in these quantities (especially the level of information)? How are they related to each other and how can they be interpreted?

The aim of the next paragraphs is to set forth a list of the physical quantities used to measure resuspension (see Section 3.1.1), then to analyze the meaning / level of information contained within these quantities (see Section 3.1.2).

#### 3.1.1. Physical quantities of interest

A basic, yet efficient, way to characterize resuspension can be inferred from its definition. Recalling that resuspension corresponds to the process by which particles adhering to a surface are detached and then entrained away through the action of a flow, a crude quantification would consist in estimating the amount of particles that are detached from a surface. In practice, this can be achieved by monitoring the evolution of various quantities with time, such as: the number of particles deposited on a surface, the motion of particles near the surface, or even the amount of detached particles that are collected downstream of a deposit. In the literature, this translates into a number of quantities listed in the following (see also Fig. 14):

* Quantities based on surface fluxes/rates:

To quantify resuspension, the first idea that comes to mind is to measure the flux of particles leaving the surface. For that purpose, one can count the number of deposited particles $N_p$ on a given surface $S$ before and after exposure to a given fluid flow 'intensity' $\mathcal{F}$ for a certain time $t$. The notion of the fluid flow intensity actually encompasses various properties (such as the friction velocity, the shear rate, the flow rate or the turbulent kinetic energy). By comparing this number to the initial amount of deposited particles $N_p(t = 0)$, one obtains the number fraction $n_p(t) = N_p(t)/N_p(t = 0)$ from which various quantities can be extracted, including:

- The instantaneous resuspension rate $\mathcal{T}_r(t)$

This is a measure of the rate of change of the number of particles deposited on a surface per unit time (in s⁻¹). It is defined by the derivative of the particle number fraction at a specified time $t$:

$$ \mathcal{T}_r(t) = \frac{dn_p(t)}{dt} \Big|_{S, \mathcal{F}=cte} \qquad (1) $$
<figure><img src="image_21.png" /><figcaption>Figure 13: Sketch showing how the various mechanisms at play in particle resuspension evolve as a function of the deposit structure and flow intensity. For the sake of simplicity, this illustration considers a fixed particle size, shape and density (which could be added to form N-dimensional graphs instead of this 2D representation).</figcaption></figure>

Alternatively, it can be expressed as a mass rate (i.e., resuspended mass per unit time, in kg s⁻¹), or even as a flux (i.e., the resuspended mass per unit time and per unit surface area, in kg m⁻² s⁻¹).

- The time-averaged resuspension rate $\tau_r^{\Delta t}(t)$

This corresponds to the rate of change of the particle number fraction due to resuspension over a certain time interval $\Delta t$ (in s⁻¹):

$$ \tau_r^{\Delta t}(t) = \frac{n_p(t + \Delta t) - n_p(t)}{\Delta t} \Big|_{S, \mathcal{F}=\text{cte}} \quad (2) $$

It is usually called the resuspension rate (without referring to the averaging process over a time interval $\Delta t$), and is related to the instantaneous resuspension rate by simple integration:

$$ \tau_r^{\Delta t}(t) = \frac{1}{\Delta t} \int_t^{t+\Delta t} \mathcal{T}_r(s) ds \quad (3) $$

Alternatively, as for the instantaneous resuspension rate, it can be expressed as a mass rate (in kg s⁻¹) or as a flux (in kg m⁻² s⁻¹).

- The resuspended fraction $f_r$:

This refers to the fraction of particles that are resuspended after a given exposure time $t$ [115]:

$$ f_r(t) = n_p(t) \Big|_{S, \mathcal{F}=\text{cte}} \quad (4) $$

It can also be referred to as the release fraction [93], the resuspension efficiency [116], or sometimes the resuspension factor [117].
<figure><img src="image_22.png" /><figcaption>Figure 14: Sketch summarizing some of the physical quantities of interest in particle resuspension.</figcaption></figure>

- The remaining fraction $f_l$:

This indicates the fraction of particles left on the surface after a given exposure time $t$. It is the opposite of the resuspended fraction:

$$f_l(t) = 1 - f_r(t) \qquad (5)$$

- Median critical velocity $u_{r=50\%}$:

A median critical velocity is sometimes defined as the fluid velocity at which 50% of the particles are removed from the initial deposit [20, 21]. This means that:

$$f_r(u_{r=50\%}) = 50\% \qquad (6)$$

More generally, using the notion of quantiles, it is possible to define a critical velocity $u_{r=k\%}$ at which $k$ percent of the particles have been removed from the initial deposit after a fixed exposure time $t$ (i.e., such that $f_r(u_{r=k\%}) = k$).

In most experiments, this fluid velocity corresponds to the friction velocity (also called the shear velocity) $u_\star$, which is related to the shear stress near the surface $\tau_p$ through $u_\star = \sqrt{\tau_f/\rho_f}$ ($\rho_f$ is the fluid density). Alternatively, critical shear stresses or critical Reynolds numbers can also be defined instead of relying on a criteria based on the fluid velocity.

- Quantities based on particle motion:

Another way to detect resuspension is to record the motion of particles initially deposited on a surface. This can be achieved using high-speed cameras (see the measurement techniques described in Section 3.2.1). By analyzing the images acquired, the following quantities can be extracted:

- Threshold fluid velocity for onset of motion $u_\perp$

This corresponds to the fluid velocity at which a particle starts moving. To put it differently, it is the extremum (minimum) value of the fluid velocity which sets a particle in motion [118]. Similarly to the median critical velocity $u_{r=50\%}$, most experimental studies provide information on the shear velocity $u_\star$. Hence, this threshold criteria $u_\star, \perp$ is largely based on the shear velocity at which a particle starts moving and is expressed as follows:

$$\| \mathbf{U}_p(u_\star, \perp) \| > 0 \qquad (7)$$

where $\mathbf{U}_p$ is the particle velocity. This quantity is also called the threshold velocity for incipient particle motion.

- Median threshold fluid velocity for onset of motion $u_\perp=50\%$

When studying a large number of particles, the velocity at which each particle starts moving can vary from one particle to another. This can be due to variations in the instantaneous hydrodynamic force, to local differences in adhesive forces or in the particle shape and size. As a result,
similarly to the median critical velocity $u_r=50\%$, one can define a median threshold velocity at which 50% of the particles start moving $u_\perp=50\%$.

More generally, using the notion of quantiles, it is possible to define a threshold velocity $u_\perp=k\%$ at which $k$ percent of the particles start to move.

- Particle velocity on/near the surface $U_p$

Recent advances in experimental techniques allow to extract directly information on the particle velocity and acceleration throughout time. By doing so, it becomes possible to distinguish between surface migration (such as rolling/sliding motion that can occur during creeping) and particles undergoing hops/jumps above the surface (such as saltating particles in aeolian research [37]). In the latter case, one can even extract statistical information on saltation events (like the average saltation time, distance or kinetic energy upon re-impact on the surface).

### Quantities based on volume concentrations/fluxes:

Another method for estimating resuspension is to measure the concentration of particles downstream of a deposit (in close proximity to it). By doing so, the following quantities can be extracted:

- Resuspension factor $R_f$:

This is a measure of the volumetric to surface (or soil) concentration of particles (in $m^{-1}$). To put it differently, it is defined as the ratio between the airborne mass concentration of particles sampled in a region near the surface $\rho_V$ and the surface (mass) concentration of particles initially on the surface $\rho_S$ [119, 13]. Therefore, it provides an estimate of the concentration of resuspended particles in the vicinity of a deposit conditioned on the amount of particles initially present on the surface.

- Flux of moving particles $\Phi_p$:

By sampling the particle concentration at various distances from the initial deposit, it is possible to reconstruct the flux of particles around the deposit [120] (as in airborne concentration measurements). Hence, such measurements provide information on the average particle motion along the streamwise, wall-normal, and tangential directions.

These fluxes can be expressed either as mass fluxes (based on the mass concentration in $kg m^{-3}$), as volumetric fluxes (based on the volume concentration) or as numeral fluxes (based on number concentration in $m^{-3}$). When dealing with resuspension induced by human activities (e.g., dancing, walking, driving), these fluxes can also be related to the intensity of the activity [121, 117].

In the rest of the paper, unless explicitly specified, the terminology “resuspension rate” and “resuspended fraction” is used to refer to measurements based on numeral fluxes. It is worth noting that these numeral rates and fractions are the same as those obtained based on mass flux or area flux when dealing with monodispersed particle sizes. However, significant differences would appear when dealing with polydispersed particle sizes. This immediately brings out the following question: which quantity is best suited to study particle resuspension? The next paragraphs provide some guidelines to help selecting a relevant quantity based on the context and objectives of each study.

### 3.1.2. Usage and interpretation of these quantities

When choosing the relevant quantity to characterize particle resuspension in a given study, one has to keep in mind the constraints coming from the objectives of that study. In particular, this choice is driven by a combination of factors in relation with the level of information required as well as existing technical restrictions (e.g., in-situ versus laboratory measurements). In the following, we illustrate some of the choices made by presenting the evolution of the measured quantities from a historical point of view.
**Threshold velocities:** The pioneering studies carried out in the early 1940's on river transport [62] and aeolian transport [16] were conducted by measuring the threshold fluid velocity for incipient motion $u_{\star, \uparrow}$. At that time, this threshold velocity provided the first insights into the necessary condition for particle motion and this helped to bring out the following picture: a particle starts moving when the forces inducing motion exceed the resisting ones [3]. Ultimately, this led to the introduction of the well-known Shields number $\Theta_c$ [62], defined as:

$$ \Theta_c = \frac{\tau_{\uparrow}}{(\rho_p - \rho_f) g d_p} \qquad (8) $$

where $\tau_{\uparrow}$ is the shear stress for the incipient motion of a particle, $\rho_f$ (resp. $\rho_p$) the fluid density (resp. the particle density), $g$ the gravity constant, and $d_p$ the particle diameter. The Shields number measures the ratio between the fluid shear stress needed to move a particle and the particle's immersed weight. By plotting the evolution of the Shields number with respect to the particle-based Reynolds number $Re_p = u_{\star}d_p/\nu_f$, one obtains a master curve, the so-called Shields diagram (see Fig. 15).

<figure><img src="image_23.png" /><figcaption>Figure 15: Shields diagram showing the evolution of the critical shear stress $\Theta_c$ (or Shields number) as a function of the particle-based Reynolds number (here $R_{\star c}$). Reprinted with permission from [122]. Copyright 2018, AIP Publishing.</figcaption></figure>

The main difficulty is that the threshold for incipient motion is not exactly the same for two identical particles exposed to a given fluid flow intensity. This is due a combination of two factors: first, motion-inducing hydrodynamic forces fluctuates in time and space (especially due to the intermittency of turbulent flows [123]); second, motion-preventing forces depend on the local configuration (roughness features on the surface at scales much smaller than the particle size, as well as the deposit structure in multilayer deposits). As a result, the threshold value for the incipient motion of identical particles is not unique but exhibits instead a range of values, even when fluid flow characteristics and particle properties are fixed. Therefore, this questions the significance of a threshold value for a group of identical particles exposed to a similar flow (see also [3]). At this point, since no unique value exists, one can consider statistical quantities, such as the median threshold value $u_{\uparrow=50\%}$ (or more generally any quantile threshold value $u_{\uparrow=k\%}$). However, regardless of the threshold value chosen, one has to remember that such quantile threshold values do not provide information on the whole distribution of values where particles start moving. Hence, by resorting to such threshold values, some of the information about resuspension is definitely lost.

Another issue corresponds to the time-dependence of these threshold values. In fact, the number of resuspended particles changes with time in turbulent flows. This is due to the intermittency in turbulence and especially to near-wall turbulent structures, which can lead to particles being exposed to intense structures for a short period of time. This means that particles with the lowest motion-preventing forces tend to remain deposited on the surface only for a short time. Meanwhile, particles with the highest motion-preventing forces can remain deposited for much longer times. To avoid the dependency of these threshold values on the observation time, one can measure the number of particles set in motion after an observation time long-enough to reach a stationary value. This is all the more relevant since several studies have shown a
quick and intense resuspension at short-term (where all loosely deposited particles are resuspended) followed by a much slower resuspension at longer times (with a resuspension rate decaying with time) [124, 125], as displayed in Fig. 14 (left panel).

As more phenomena related to particle resuspension were unveiled by these pioneering studies, new threshold criteria emerged. For instance, a distinction was introduced between the incipient motion and the cessation of motion (i.e., when particles stop moving as the fluid velocity decreases). In fact, it is observed that, due to their inertia, resuspended particles continue their motion even when the fluid velocity is decreased below the threshold for incipient motion. As a result, this leads to a sort of hysteresis effect. More recently, specific thresholds have been suggested to distinguish between rolling, sliding, and lifting motion [122]. In the case of multilayer deposits, another threshold has been introduced: the threshold for impact entrainment, which corresponds to the velocity at which saltating particles induce extra resuspension upon colliding with the deposit [126]. This threshold allows to quantify when collisions between particles result in enhanced resuspension rates, similar to collision propagation effects observed in avalanche phenomena. Recently, it has even been argued that most of the so-called Shields diagrams do not plot the evolution of the threshold for incipient motion but rather the threshold for impact entrainment [3]. This is attributed to the measurement protocol used but the actual phenomena involved depend also on the definition of the threshold value. To be more specific, let us consider a multilayer bed composed of a few hundreds of sediments resting on a bed and exposed to a fixed flow rate. For the 10% quantile threshold $u_{\Upsilon=10\%}$, a few particles set in motion are enough to reach that quantile. In that case, particle motion is essentially triggered through individual, and independent, events. This means that the threshold measured correspond to the incipient motion of individual particles. However, if one considers the median threshold, many more particles must bet set in motion to reach this threshold: this is probably achieved by a combination of individual resuspension events and collision-induced effects. Therefore, the choice of a given threshold value leads to different information being accessible. In other words, depending on the selection of a given threshold value, we may not measure the same resuspension mechanism.

In spite of these issues, researchers in fluvial and aeolian transport are still relying on threshold values which are easy to use and remain convenient in many practical applications. This explains why, since the pioneering work of Shields, many studies have evaluated how such threshold values depend on fluid properties (such as its density or velocity) and on particle properties (e.g., diameter and density).

**Surface fluxes/rates:** By counting/weighting particles on a given surface (see Section 3.2.1 for measurement techniques), one can determine the number/mass of particles remaining on the surface after being exposed to a given flow for a certain amount of time (see the left panel of Fig. 14). Then, by fixing the observation time $\Delta t$, one can reconstruct the evolution of the remaining fraction $f_r(t)$ as a function of the fluid velocity (see also the right panel of Fig. 14). As displayed in Fig. 16, surface fluxes are widely used in the multiphase flow community (see for instance [10, 11] and references therein).

The main interest of these measures is to provide information on the whole distribution of velocities required to resuspend particles from a substrate. This information is complementary to the one contained in threshold values for incipient motion. In the case where all particles that start moving get resuspended, the threshold value for incipient motion can actually be extracted from the evolution of the fraction remaining with the fluid velocity (as shown in the right panel of Fig. 14). The information can be further simplified to give a critical velocity (like $u_r=50\%$), which is commonly used in studies dealing with potentially harmful particles (like pollen, allergens, radioactive particles) [48].

Yet, surface fluxes/rates should be handled carefully due to a number of issues:

* These quantities depend both on the observation time $\Delta t$ and on the time $t$. More precisely, this means that the time-averaged resuspension rate $\tau_r^{\Delta t}(t)$ changes with time when the observation time $\Delta t$ is fixed, and that it also varies with the observation time $\Delta t$ for a fixed time $t$. In fact, laboratory experiments have shown that, when a fresh deposit is exposed to a constant flow, a large quantity of particles is resuspended from the surface over a very short period of time (typically less than a second, as sketched in the left panel of Fig. 14). This intense short-term resuspension rate is attributed to the resuspension of all particles that are loosely adhering to the surface and, hence, are easily resuspended
<figure><img src="image_24.png" /><figcaption>Figure 16: Comparison of the fraction of particle remaining on the surface *fr* against the fluid friction velocity *u*\* for various experiments. It shows the typical decay of *fr* as the velocity increases (with a sharp decrease near the median velocity *u*\*<sub>r=50%</sub>). Reprinted with permission from [127]. Copyright 2015, Elsevier.</figcaption></figure>

by the flow. At longer exposure times, the number of particles being resuspended is much lower
since most of the particles likely to be resuspended have already left the surface (in reality, long-
term resuspension is related to the interaction with high velocity turbulent structures). Experimental
measurements have indicated that this long-term resuspension rate is usually inversely proportional
to time [124].

This distinction between short-term and long-term resuspension is widely used in practice. In fact, after a few seconds of exposure to a given flow, the long-term resuspension rate is reached, meaning that the number of resuspended particles evolves very slowly. This gives the impression that a nearly steady state is reached once all the particles that can resuspend at this fluid velocity are gone (see right panel of Fig. 14). Drawing on such observations, measurements in the multiphase flow community are usually performed by evaluating this nearly steady-state value [128] (i.e., typically after a few seconds of exposure to a given flow).

* Surface fluxes/rates do not necessarily provide information on the type of motion of each individual particle. Hence, the only information that can be extracted is the total amount of particle resuspended, regardless of the mode involved (e.g., rolling, sliding, or lifting). More recently, new experimental techniques have allowed to classify particles with respect to their mode of motion [129]: this allows to precisely know how much particles are resuspended through rolling, sliding or lifting motion.

* Another issue is that the value obtained can overestimate the real resuspension rate. In fact, unless specific techniques are used, it is impossible to distinguish between particles that have been resuspended from the surface (i.e., detached from the surface) and migrating particles that have left the measurement area (e.g., through rolling or sliding motion). Therefore, when counting techniques are used, all particles leaving the observation area are counted as resuspended particles (including migrating ones).

**Volume fluxes:** The main interest of these quantities is that they deliver information on where the resuspended particles can be transported once they are detached from the surface. For instance, by placing sensors at a target height representative of an adult person, one obtains direct access to the exposure of an individual to potentially harmful particles (e.g., pollution from roads [15], pollen from plants, spores from soil [130], dust from clothing [117]). A typical result is displayed in Fig. 17: it shows the evolution of the concentration of particles in the bulk air (i.e., far above the floor) and in the breathing zone of a crawling infant as a function of time and human activities. It appears that the particle concentration increases by two orders of magnitude in the infant breathing region when infants are crawling on the carpet. Meanwhile, it only increases by one order of magnitude when sampled at 1.5 m from the floor (labeled here bulk air but it can also be seen as the adult breathing zone).
<figure><img src="image_25.png" /><figcaption>Figure 17: Evolution of the mass concentrations of particulate matter ((PM<sub>0.3–10 µm</sub>, i.e., with sizes between 0.3 and 10 µm) monitored in the infant breathing zone (BZ) and in the bulk air (at 1.5 m above ground). It shows the increase in the concentration when an infant crawls on a carpet and that the concentration can be up to one order of magnitude higher in the infant breathing zone than in the bulk air region. Reprinted with permission from [51]. Copyright 2018, Springer Nature.</figcaption></figure>

This simple example actually brings out the difficulties encountered when using such quantities:

* Since particles are often naturally present in environmental flows, the actual information of interest is not exactly to measure a volume concentration but rather variations of this volume concentration as a function of time or environmental conditions (e.g., wind speed, human activities, etc.). These variations are usually obtained by comparing the values at two different times and/or two different conditions.

* Another drawback is that volume concentrations depend heavily on the measurement protocol (see Section 3.2). For instance, it is evident that the concentration of particles resuspended from the floor varies with respect to the distance from the floor. Hence, sensor placement is key when designing such experiments and it is hard to compare two experiments relying on different protocols. In addition, an implicit assumption is that the measurement at a certain height represents the concentration of the plume of airborne particles from the deposit, in a top-hat fashion. The example in Fig. 17 challenges this assumption, which may lead to an over- or under-estimation of the amount resuspended.

* Finally, volume fluxes actually contain information on the environmental conditions that act both on particle resuspension and on their subsequent transport (e.g., terrain topology, target height, intensity of wind and/or human activities) [13]. For that reason, it is very common to express such volume fluxes relative to the intensity of the activity [121, 117]. For example, in road dust resuspension [117], the emission factor is defined as the mass of particles resuspended due to the passage of a single vehicle per unit distance traveled by the vehicle (expressed in $g \text{ km}^{-1} 	ext{ veh}^{-1}$).

**Individual particle tracking** More recently, the development of high-frequency recording techniques has paved the way for precise measurements of particle motion near the surface. While such measurements are still somewhat limited to large millimeter-size particles (see for instance [129]), they provide very detailed information on the statistics of particle properties (such as the translational or rotational velocities, the acceleration). The main interest of such measurements is that they contain a large amount of information: it is possible to reconstruct all the previous threshold quantities and surface fluxes/rates (volume quantities can be accessed depending on the location of the last tracked particle position with respect to the initial deposit). Yet, such quantities require high capacity storage facilities due to the large amounts of data generated upon short observation times.

### 3.1.3. Summary on physical quantities

This overview of the physical quantities measured for particle resuspension has shown that the level of information differs widely. This ranges from the most basic information on threshold velocities for
incipient motion to detailed statistical information on particle velocities. Yet, the choice of one quantity or another depends on the specific objective which is pursued: studies focused on human exposure to contaminants tend to rely on volume fluxes/rates while investigations on surface contamination prefer to use surface fluxes/rates. Some studies even consider several quantities simultaneously. For instance, in river research about sediment transport, it is frequent to measure both the threshold velocity and a transport rate [63, 64, 65]. Similarly, studies related to airborne particle often characterize the evolution of the resuspended fraction with the friction velocity together with the median critical velocity [21, 48]. In addition, the choice of a given quantity is also driven by the method of measurement, which can be suitable for either in-situ and/or laboratory studies. These techniques together with the corresponding procedure are described in the following Section 3.2.

## 3.2. *Measuring particle resuspension*

Having introduced the physical quantities used to characterize particle resuspension, the next issue is: how can these physical quantities be measured experimentally? The aim of the following paragraphs is to describe the corresponding techniques and protocols designed to access these quantities (see Section 3.2.1) before discussing the limitations of these experimental methodologies (see Section 3.2.2).

### 3.2.1. *Measurement techniques and protocols*

Various methods to set up experimental protocols to measure resuspension factors of interest have been suggested by combining specific procedures and measurement techniques. These methods are briefly recalled in the following (interested readers are referred to detailed reviews dedicated to measurements, e.g., [131, 132]). These techniques are also illustrated in Figure 18 and summarized in Table 1.

<figure><img src="image_26.png" /><figcaption>Figure 18: Sketch of the main available measurement techniques to quantify particle resuspension. It shows how mass balance, collecting traps, particle counting, volume concentration measurements and tracking techniques can be used to monitor resuspension.</figcaption></figure>

* Mass-balance measurement:

*Principle:* Historically, one of the first techniques consisted in measuring variations in the deposit mass before and after exposure to a given flow [36]. This simple procedure provides macroscopic
information on the portion of materials that has been resuspended by applying a given mean flow over the deposit. Yet, it does not provide microscopic information on individual particles nor insights into the resuspension mechanisms.

More recently, the procedure has been extended to provide information not only on the amount of particles but also on their chemical composition. This is achieved by combining weighting techniques with other intrusive methods where the deposits are further analyzed. For instance, when dealing with organic compounds, chemical analysis techniques can be employed such as chromatography (which allows to separate constituents in a complex gas or liquid mixture) and mass spectroscopy. Such combinations have been used to quantify the amount of trace explosive residues resuspended from fibrous surfaces [61].

*Usage:* Due to its simplicity and to its non-intrusiveness, it has been widely employed to quantify resuspension by airflows [37], especially in laboratory measurement. It is still used in combination with other optical counting techniques, especially when dealing with resuspension in complex and confined environments [85].

* Collecting traps or chambers:

*Principle:* The idea here is to collect particles at specific locations by resorting to traps [131]. Compared to simple mass-balance measurements, these techniques can provide some information on the motion of the particles. For instance, particles that are slowly moving along the surface (e.g., creeping motion) can be collected by traps buried within the deposit with the orifice oriented vertically (see Figure 18). Alternatively, traps placed at various heights above the deposit with the orifice oriented horizontally allow to collect particles that are moving on the surface (e.g., through creeping motion) as well as particles moving above the surface (like reptating or saltating particles).

These techniques have been later refined for atmospheric measurements using collection chambers with filters on which particles pile up. These collected particles can then be analyzed with various methods including image analysis (for particle size and shape), gravimetric determination (for mass and density), and chemical speciation (for composition).

*Usage:* Historically, such techniques were used by Bagnold [17] to quantify creeping motion. Such techniques are still widely employed in aeolian transport to measure the amount of creep, reptation, and saltation of sand particles (see the recent review [22]). These techniques are also used to measure the motion of sediments in rivers or in lakes [131, 133]. The popularity of this technique comes from the fact that the procedure can be easily set up for in-situ field measurements since it only requires to install a few traps and to collect the accumulated particles from time to time.

Collection chambers with filters are also widely used to monitor road resuspension of dust and particulate matter due to human activities and traffic (see the recent review on this topic [15]). For instance, such passive sensors are placed at various heights from the ground near roads to quantify traffic-induced resuspension [7]. Another approach consists in placing a sensor behind the tires of a vehicle and to compare the amount of particles collected to the ambient concentration (measured with sensors placed in the front or at the top of the vehicle) [15].

* Particle counting:

*Principle:* Another common technique relies on taking a sequence of images to record the amount of particles deposited on a sampling area. The analysis of these pictures consists in counting the number of particles that were initially deposited on a sample surface and then comparing it to the number of particles remaining on the same sample surface after exposure to a given flow for a certain duration. Hence, this provides direct access to the amount of particles resuspended. The images can be taken using various techniques, including:

- Optical high-resolution cameras:

Cameras placed under/over the sampling area allow taking pictures of the particles deposited on the surface (provided the substrate is transparent to allow the light to go through it). For
instance, it is commonly used to investigate how particle resuspension evolves with time by taking pictures at regular time intervals. Alternatively, this technique is frequently used to measure the evolution of resuspension with respect to the flow rate by taking pictures after exposing a deposit to a given flow rate for a specified time [128].

- **Scanning Electron Microscopy (SEM):**

SEM produces images of a sampling area by scanning the surface with a focused beam of electrons. Compared to traditional optical microscopy, the resolution of an electron microscope is much higher since it allows to visualize objects below the micrometer size. Yet, this higher resolution often implies that the area of the scanned surface is smaller than the one using an optical microscope. Hence, only a limited number of particles is accessible through SEM images, making the technique more prone to statistical errors on the concentration. However, SEM images provide essential information on the particle shape and size [55, 60, 61]. For that reason, this procedure is less frequently used than other optical methods.

***Usage:*** Particle counting techniques from optical images are still widely used to determine the amount of particles remaining on a surface after exposure to a given flow for a certain duration (as in [128]). However, experimental measurements are increasingly relying on SEM images to provide information on the geometry and surface properties of both particles and substrates (see for instance [125]).

- **Measurements of concentrations in the fluid:**

***Principle:*** Techniques based on the measurement of particle concentration within the fluid (and not on the surface) have been quickly adapted to quantify particle resuspension. Such methods are very similar to collecting traps since they consist in sampling the particle concentration at a given location in the domain. For instance, one can measure the concentration at various heights above the surface or in the wake of a deposit. The key difference with collecting traps is that such sensors are non-intrusive in the sense that they do not necessarily modify the flow, thus allowing for in-line measurements. As a result, it becomes possible to monitor how particle resuspension evolves with time as well as to capture how the local instantaneous flow velocity affects the resuspension rate. In fact, the acquisition frequency for particle concentration depends on the modus operandi. Various procedures exist, including:

- **Radioactive materials and activity measurements:**

A first technique consists in marking particles using either radionuclides [133] or by exposing particles to a source of neutron (Neutron Activation Analysis) [54]. By recording the radioactivity (e.g., with X-ray or gamma ray detectors), it is then possible to estimate the local concentration of particles in certain areas of interest. The main advantage of radioactive particles is that such techniques can be applied even in domains with opaque boundaries, whereas other optical techniques (e.g., turbidity) require transparent boundaries near the measurement regions.

- **Fluorescent particles and stereoscopic measurements:**

More recently, researchers have started using fluorescent particles to measure the local concentration of particles in certain areas of interest thanks to fluorescence stereoscopes [117]. This technique is very similar to the one relying on radioactive particles. In fact, it removes the need to handle potentially harmful radioactive particles but requires to work with transparent boundaries in order to measure particle concentrations.

- **Optical turbidity measurements:**

The presence of suspended particles in a fluid affects the reflection, absorption, and scattering properties of a light beam passing through the fluid. The intensity of the scattering effect is actually directly influenced by the particle concentration, the particle size, the particle shape as well as by the wavelength of the light [132]. This effect is caused by the fact that particles have a different refractive index than the surrounding medium. As a result, the intensity of the light transmitted through the suspension is reduced: this is called turbidity [134]. Hence, turbidity measurements provide information on the concentration of particles in a fluid, provided that some
information is already known on their size and shape. In practice, this is achieved by measuring light scattering at different angles, light transmission through the fluid and/or light absorption. This has led to the development of static light scattering and dynamic light scattering methods (interested readers are referred to the recent review [132]).

- Acoustic backscatter measurements:
  Similarly to the optical turbidity, particle concentrations can be measured using acoustic methods. This means that a sound is propagated within the solution and its scattering due to the presence of particles is recorded [4].

*Usage:* Optical sensors have been widely used to measure particle resuspension induced by currents and waves in marine systems (e.g., lakes or shallow water, see [135, 136, 137] and references therein) as well as resuspension in atmospheric environments (see for instance [138, 56]). Due to their ease-of-use for in-situ observations, acoustic backscatter methods are also widely used to measure resuspension in marine systems (see the short review [131]), especially in lakes [137] and in stratified fluids [4] (sometimes in combination with optical methods). These instruments are often coupled to other tools to provide information on the local fluid velocity. This is achieved either by resorting to acoustic Doppler velocimeter (i.e., based on acoustic sensors, see [135, 137] and references therein) or on Laser Doppler Anemometry (i.e., measuring the shift between the incident and scattered light frequencies [139]). Furthermore, these sensors can be coupled to Aerodynamic Particle Sizer (APS) spectrometers [53, 55, 54], which allow to measure the size distribution of particles suspended in airflows (by accelerating particles through a constricted nozzle and recording their velocity as they pass through two laser beams).

* Individual particle tracking:

*Principle:* Recent measurements are increasingly relying on instruments to track the motion of individual particles. Such methods provide very detailed information on the time-evolution of the particle position, velocity, and acceleration. Various techniques can be used, including:

- Laser Distance Sensor (LDS):
  LDS corresponds to the tracking of an individual particle placed on a surface using a laser beam focused on the surface of the particle of interest. By measuring the reflected and/or scattered light, it is possible to accurately record the position of an individual particle and to reconstruct the streamwise and/or vertical components of the instantaneous velocity vector. Such methods have been used to measure the initial motion of millimeter-size particles [140, 141].

- Particle Tracking Velocimetry (PTV):
  PTV techniques consist in illuminating particles as they cross laser-light planes and then reconstructing the trajectory of each individual particle by correlating their positions between sequential frames (see the review of PTV techniques in atmospheric sciences [142]). Hence, PTV naturally allows to obtain detailed information on the local position, velocity [143], and acceleration [144] of each tracked particle. In addition, depending on where the cameras are placed, PTV can provide information not only on the streamwise velocities [118] but also on the wall-normal velocities [108]. Yet, as for Particle Image Velocimetry (PIV) which tracks the motion of fluid tracers, PTV methods perform poorly in dense particle suspensions. This is due to the fact that, when the number of particles in suspension increases, it becomes harder to determine the trajectory of each individual particle.

Recent advances in PTV techniques gave rise to the use of transparent millimeter-size particles with dots drawn at specific locations on the surface [129]. By tracking the combined motion of these dots on each particle, it is possible to extract both the translational and rotational motion. Hence, these new procedures pave the way for precise measurements of rolling and sliding motion of particles on surfaces. This method is still restricted to large-enough particles (typically a few millimeters), but with the progress in optical techniques, similar measurements for micrometer-size particles might emerge in the near future.
- Tomographic Particle Tracking Velocimetry (TOMO-PTV) or 3D-PTV:
TOMO-PTV (also called 3D-PTV) is an extension of the PTV technique that allows to illuminate, record, and reconstruct the motion of any particles within a 3D volume. This is achieved using several (at least four) high-speed cameras to record simultaneous views of the whole volume investigated [145]. The key difference with PTV lies in the ability to reconstruct the particle trajectory across a whole volume and not only on preestablished 2D light planes.

- X-ray 3D tomography:
Another recent extension of classical PTV techniques relies on the use of X-ray tomography instead of optical tomography [146]. The main interest of the X-ray tomography is that it allows seeing right through opaque surfaces and through particles. Hence, when dealing with complex multilayer deposits, it provides key information on the deposit morphology that cannot be accessed using optical techniques. It has been successfully used to monitor the motion of a 3D aggregate composed of millimeter-size particles (with close to 1000 particles in the cluster) [147].

***Usage:*** PTV techniques are increasingly used to measure the motion of individual particles on the surface (see the review on creep by sand particles [22]). In addition, PTV is a very practical tool to track motion of particles once they are detached from the surface, especially to investigate the role of near-wall turbulent structures on their subsequent entrainment [148, 144]. More recently, TOMO-PTV has been successfully used to characterize particle entrainment velocities within the near-wall turbulent structures [145] and also in near fixed obstacles [149, 150].

These techniques are often combined with measurement of the local fluid velocities. In fact, most studies rely on a coupling between similar approaches (e.g., PIV+PTV [139] or TOMO-PIV+TOMO-PTV [145]) to track the motion of both tracers and solid particles. Other studies have been carried out using a combination of particle tracking and Laser Doppler Anemometry [139].

**Summary** This overview of experimental protocols illustrates that various quantities can be measured to characterize particle resuspension and that the associated experimental techniques differ depending on the specific objectives and context (see also Table 1). In fact, the level of information that can be extracted from these measurements spans a wide range of scales, from the most basic information on a threshold velocity for incipient motion to more detailed statistical information on particle velocities. In addition, it has been shown that the choice of an experimental procedure is based on a number of criteria including: the quantity to be measured, the properties of the fluid, the particles, and the surfaces involved. One additional piece of information that has not been discussed previously is related to the type of motion used to resuspend particles. In the case of field measurements in aeolian or fluvial research, particles are naturally exposed to the wind or the water flow (see for instance [3, 4]). The flow configuration can either be relatively simple (e.g., sand dunes [17], wave-induced resuspension [151, 4]) or more complex (e.g., flow in vacuum cleaners [59]). Yet, in the case of laboratory measurements, the vast majority of studies rely on particles exposed to a flow within a channel (see for instance [128]), near obstacles in wind tunnels [149, 150], within an impinging-jet, within a nozzle-jet (see for instance [60]), or within a rotating shear flow (see for instance [129]). More recently, an increasing interest in the resuspension of particles due to vibrations has emerged due to their application in granular beds [152], in dust resuspension due to human foot tapping [138], in resuspension from mattresses [56, 153] and from clothing [121].
<table><caption>Table 1: Summary of the main measurement techniques to quantify particle resuspension and of their characteristics (sorted in terms of variable accessible, range of applicability regarding fluid/particle/surface/deposit properties and specificities).</caption><thead><tr><th rowspan="3">Measurement techniques</th><th colspan="12">Characteristics</th><th colspan="2">Particle</th><th colspan="2">Surface</th><th colspan="2">Deposit</th><th colspan="2">Specificities</th></tr><tr><th colspan="2">Fluid</th><th colspan="4">Variable accessible</th><th colspan="2">Motion</th><th colspan="2">Gas</th><th colspan="2">Liquid</th><th colspan="2">Nature</th><th colspan="2">Monolayer</th><th colspan="2">Non-intrusive</th><th colspan="2">Acquisition frequency</th></tr><tr><th>U<sub>p</sub></th><th>U<sub>p</sub>=50%</th><th>u<sub>r</sub>=50%</th><th>f<sub>r</sub></th><th>f<sub>r</sub>=50%</th><th>τ<sub>r</sub></th><th>τ<sub>r</sub>=50%</th><th>R<sub>f</sub></th><th>R<sub>f</sub>=50%</th><th>U<sub>p</sub></th><th>U<sub>p</sub>=50%</th><th>τ<sub>r</sub></th><th>τ<sub>r</sub>=50%</th><th>R<sub>f</sub></th><th>R<sub>f</sub>=50%</th><th>u<sub>r</sub></th><th>u<sub>r</sub>=50%</th><th>τ<sub>r</sub></th><th>τ<sub>r</sub>=50%</th></tr></thead><tbody><tr><td>Mass-balance</td><td>Mass-balance</td><td>Collecting traps</td><td>Optical images</td><td>SEM</td><td>Radio-active</td><td>Fluorescence</td><td>Turbidity or Acoustic</td><td>Concentration measurements</td><td>Radio-active</td><td>Fluorescence</td><td>Turbidity or Acoustic</td><td>Radio-active</td><td>Fluorescence</td><td>Turbidity or Acoustic</td><td>Concentration measurements</td><td>Radio-active</td><td>Fluorescence</td><td>Turbidity or Acoustic</td><td>Concentration measurements</td></tr><tr><td>Individual tracking</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
### 3.2.2. Limitations of experimental methodologies

Despite considerable progress in measurement techniques, there are still a number of limitations and uncertainties that require specific attention to avoid meaningless measurements. The main sources of uncertainties are related to:

* The reverse deposition process:

One of the most frequent issues encountered in measurements of particle resuspension is related to the reverse-process of particle deposition which often occur simultaneously [154]. In fact, this issue was already identified in the early 1980s [155] and it has remained a matter of concern ever since, especially for in-situ observations (where particles are both deposited and in suspension at the start of the measurement). Laboratory measurements of particle resuspension can also be hampered by initially suspended particles. For instance, when simply counting the number of particles before and after exposure to a given flow, no information is available to determine if some (and how many) of the particles present on the surface at the end of the procedure actually come from suspended particles that have deposited on the area of observation.

Various attempts have been made to mitigate this issue. In laboratory studies, a wide range of measurements have been performed using fluids without any suspended particles. This means that the only particles within the domain considered are the particles that are initially deposited on the surface, hence effectively removing any contamination due to the deposition of particles initially in suspension. With the advent of new measurement techniques such as 3D-PTV, laboratory studies or in-situ observations can now explicitly track the motion of each individual particle near the boundary. Hence, these new methods allow to identify particles depositing on the surface and particles resuspending from it. Alternatively, some studies actually rely on the combined effects of deposition and resuspension processes in environments initially devoid of suspended particles. For instance, investigations of resuspension from clothes often characterize the ratio between the amount of particles resuspended from clothing substrates and the portion of these resuspended particles that reach a target surface on the ground due to subsequent deposition (this is referred to as the clothing release fraction) [121, 117]. In such cases, this measure is relevant since it corresponds to the portion of resuspended particles that reach the ground (where they are less prone to be inhaled by an adult person and where they can be collected later in vacuum cleaners).

* The procedure to form the initial deposit:

Even when particle resuspension is studied in environments free of any other sources of particles (meaning that only deposited particles are present in the domain studied), the procedure used for the deposition can have significant consequences on resuspension. This can be due to a variety of reasons including the dry-vs.-wet deposition (e.g., spores from dry deposition are easier to remove than from wet deposition [156, 157]), the flexibility of the particles (which affects the cohesive forces between particles), and even how the particles were deposited. In many wind-tunnel studies, the deposit is usually formed by sedimenting particles within a quiescent flow (i.e., without any airflow) [128]. Once fully formed, the deposit is then exposed to a fluid flow in order to study the resuspension process. In reality, deposits are usually formed by a subtle balance between particle deposition and resuspension since both processes happen simultaneously. This means that the particles loosely adhering to the surface would be removed shortly after deposition. Hence, if the flow velocity is suddenly increased, these particles would not be available for resuspension and the total amount resuspended would be less than the one computed considering that all particles were present initially on the surface (i.e., considering the full adhesion force distribution). This is a matter of concern when trying to distinguish between short-term and long-term resuspension. As mentioned previously, short-term resuspension corresponds to the large amount of particles that leaves the surface over a very short time as soon as the deposit is exposed to a flow [124]. Yet, it appears that this short-term resuspension rate may not be relevant to all practical applications and can even be an artifact from the protocols used to create deposits on a surface.
* Spatial/temporal resolution:

Another limitation is related to the spatial/temporal resolution. In fact, regardless of the measurement technique, particle resuspension is always characterized over a finite surface and finite time: the size of this surface can be the size of the weighting plate, the surface observed in particle counting methods, the surface (resp. volume) considered in PTV (resp. 3D-PTV), or the area behind which collectors are placed. Due to this finite-size, it is not possible to ascertain if some of the particles initially deposited on the surface have actually moved on the surface (resp. detached from the surface) before coming to a halt on the surface outside of the observation area (resp. redeposited outside of the area, like saltating particles). As a result, measurements of particle resuspension always overestimate the number of resuspended particles that would be obtained considering an infinite surface. In fact, this can be even more complicated when relying on counting techniques. In that case, the number of particles still on the surface after being exposed to a flow for a certain duration is simply compared to the initial number of particles. However, some of the particles might have been rolling/sliding on the surface before coming to a stop later on: if they remain within the observation area, these particles are considered as not resuspended (regardless of their motion) while, if they leave the observation area, they are counted as resuspended materials (strictly speaking, they are migrating on the surface).

The temporal resolution used when quantifying particle resuspension can have similar effects as the spatial resolution. In fact, in the case where particles undergo small hops over the surface (reptation), the result obtained with simple counting techniques varies with the observation time. In addition, as was mentioned previously, the resuspension rates extracted from such measurements depends both on the observation time $\Delta t$ and on the time $t$ at which the measure is made. Hence, it is paramount to keep this information in mind to avoid drawing inappropriate conclusions by comparing two experimental data (especially if they were made using different observation times).

* Complex surfaces:

Another experimental artifact is related to studies concerning resuspension from vegetation. Experiments in wind tunnels may use an unrealistic unidirectional wind flow compared to the conditions in the actual atmosphere. Resuspension may therefore be underestimated because vegetation like grass can bend in the direction of the flow, thereby protecting the deposited particles. For instance, resuspension from 25-30 cm grass doubled when the grass trays were rotated during the experiment [158].

* Incomplete measurements of surface topology and adhesive properties:

Despite recent advances in measurement techniques which now provide information on near-wall particle velocities, a complete picture of a single particle resuspension event is still missing. This is mostly related to the difficulty in characterizing precisely what is the exact adhesive force between a particle and a surface at a given location. This is due to the chaotic nature of the surface profiles, which usually display roughness at the nanoscale. Such uncertainties severely impact the resuspension of small colloidal particles (which are highly sensitive to adhesive forces). Nowadays, statistical information on roughness profiles and the corresponding adhesive force are available thanks to AFM (atomic force microscopy, see existing reviews [159, 160]), but there are still no procedure allowing to measure both the adhesion force of a single particle located at a certain position and its resuspension as it is exposed to a given flow.

* Lack of uncertainty quantification:

Despite the number of experimental studies performed, there has been no attempt to quantify the sources of uncertainty, at least to the authors' knowledge. Uncertainty quantification in experiments consists in quantifying the sources of uncertainty and errors using probabilistic and statistical tools [161, 162]. To put it differently, “uncertainty quantification offers a rational basis to interpret the scatter on repeated observations, thus providing the means to quantify the measurement inaccuracies and imprecision” [163]. In practice, there are two main sources of uncertainty and errors in experiments: limited/incomplete data and limited accuracy or resolution of sensors. In the present case,
a number of parameters have been shown to result in variations of the measured value for particle resuspension (see the previous paragraphs). The interest of uncertainty quantification is that it allows to sort these various sources of errors with respect to their impact on the measured quantity. As a result, it can provide useful information as to which parameters should be carefully set up (typically those with a high impact on the measured quantity). However, uncertainty quantification requires a thorough examination of available experimental data with a strict procedure (where each parameter is successfully varied over the range of possible values while all the other ones are fixed). For that reason, it has only been seldom carried out (see for instance a recent paper on PIV measurements [163]).

Drawing on these limitations, several suggestions can be made to go beyond the current state-of-the-art measurements. These propositions are detailed in Section 7.

# 4. Modeling particle resuspension

In parallel to experimental measurements, a range of theories have been developed to predict the physical quantities of interest (see the list in Section 3.1). Once terminology issues are clarified, it appears that some of these models are equivalent. There are, however, often structural differences between them due to: (a) the different physical quantities being sought, (b) the various levels of description at which they operate and (c) the range of time/spatial scales considered in each of these theories. It is therefore useful to provide a general framework in which different classes of models can be set.

To that effect, we review here the various types of models developed in the literature, starting with the different types of forces that can either induce or prevent the motion of deposited particles in Section 4.1, along with the existing expressions for these forces. The different levels of description and strategies that can be followed to generate the information required to compute these forces, especially the modeling approaches for the turbulent flow field and for surface roughness, are recalled in Section 4.2. A number of representative resuspension models are then detailed in Section 4.3. This overall description of models paves the way for the more in-depth analysis of resuspension models to be presented in Section 6.

## 4.1. Key forces at play in resuspension

As introduced in Section 2.2.2, particle resuspension results from the rupture of balance between forces inducing motion and forces preventing motion. These forces are often classified in terms of the particle-fluid, particle-particle and particle-surface forces (see previous reviews [10, 11]). Here, we adopt another standpoint related to the 'system' considered and to the nature of the forces involved. More precisely, we consider that the system under consideration is composed of three elements: (a) a fluid, (b) a physical domain with solid boundaries (or walls), and (c) particles immersed in the fluid. From this simple definition, one can then distinguish between external forces (related to interactions with physical objects, or force fields, external to the system) and internal forces (related to interactions between elements within the system). The former include gravity forces or electro-magnetic interactions with external fields while the latter include particle-fluid, particle-wall and particle-particle forces. In the following, we first review external forces (see Section 4.1.1) before listing internal forces within the system, which are classified as: particle-fluid forces (see Section 4.1.2), non-contact forces (see Section 4.1.3) and contact forces (see Section 4.1.4). The main reason for choosing to depict separately non-contact and contact forces is to better identify which forces occur when particles are deposited (i.e., in contact with a substrate) and which forces act on detached particles.

### 4.1.1. Interactions with external fields

**Physical origin:** The system considered here (i.e., composed of a fluid, walls and particles) can interact with external fields. In particular, the dynamics of the fluid as well as the dynamics of particles are affected by their interaction with external fields, such as (see also Fig. 19): gravity forces or electro-magnetic forces.
<figure><img src="image_27.png" /><figcaption>Figure 19: Illustration of the most important forces with external fields at play in particle resuspension.</figcaption></figure>

**Expressions:** Focusing on the dynamics of particles, the interaction with external fields is given by well-known formulas for each contribution.

A. Gravitational forces occur in most applications since particles are deposited on substrates that are often at the surface of a planet. Gravity forces arise due to the weight of the particle and are given by:

$$ \mathbf{F}_{\text{grav}} = \mathcal{V}_{\text{p}} \rho_{\text{p}} \mathbf{g}, \qquad (9) $$

where $\rho_{\text{p}}$ is the particle density, $\mathcal{V}_{\text{p}}$ its volume and $\mathbf{g}$ the gravity acceleration. In many cases, gravity tends to prevent the motion of large particles since it acts to keep particles in place (like rocks or gravels at the bottom of a river).

B. Electro-magnetic forces occur when the two bodies display electro-magnetic properties (such as ferro-magnetic particles) [164]. In that case, the force on each particle is actually given by Lorentz force:

$$ \mathbf{F}_{\text{magn}}(h) = q (\mathbf{E} + \mathbf{U}_{\text{p}} \times \mathbf{B}) , \qquad (10) $$

where $q$ is the particle charge, $\mathbf{E}$ the ambient electric field, $\mathbf{B}$ the ambient magnetic field and $\mathbf{U}_{\text{p}}$ the particle velocity. Although these effects are rarely present in the applications mentioned until now, their contribution to particle resuspension can yield rich and complex collective phenomena due to the induced magnetic dipoles and their effects on the local magnetic fields.

**Role in resuspension:** External forces can impact particle resuspension in different ways. First, gravity forces tend to prevent the motion of particles that have sedimented on a surface (like gravels on river beds). On some rare occasions where particles have accumulated on the upper regions within pipes, gravity can favor resuspension. Second, most electromagnetic forces induce a motion of both the fluid and the particles. Hence, they rather tend to favor the onset of particle motion.

### 4.1.2. Fluid forces on particles

**Physical origin:** Any body placed in a fluid flow undergoes aero- or hydro-dynamic forces related to the interactions between the fluid and the particle. Using a continuum description of fluids (which will resurface in Section 4.2.1), these forces are either functions of the fluid pressure or of the relative velocity between the fluid and the object.

**Expressions:** As depicted in Fig. 20, particle-fluid interactions have two origins:

$$ \mathbf{F}_{f \to p} = \underbrace{\mathbf{F}_{\text{PG}} + \mathbf{F}_{\text{buoy}}}_{\mathbf{F}_{\text{pressure-driven}}} + \underbrace{\mathbf{F}_{\text{drag}} + \mathbf{F}_{\text{lift}} + \mathbf{F}_{\text{AM}} + \mathbf{F}_{\text{hist}}}_{\mathbf{F}_{\text{relative-motion}}}. \qquad (11) $$

A. Pressure-driven forces are actually decomposed in two contributions:
<figure><img src="image_28.png" /><figcaption>Figure 20: Illustration of the approaches to evaluate the hydrodynamic force exerted on object immersed in a fluid.</figcaption></figure>

i - Buoyancy is the force exerted by the fluid that opposes the weight of an object immersed in a fluid. Strictly speaking, it comes from the pressure difference in a fluid: since pressure increases with depth (due to the weight of the fluid in the upper layers), the pressure at the bottom of a particle is slightly greater than at its top. This pressure difference results in a net upward force on the object. Following Archimedes' principle, this force is equivalent to the weight of the displaced fluid (i.e., the fluid that would otherwise occupy the submerged volume of the object). It is simply given by:

$$ \mathbf{F}_{\text{buoy}} = -\nabla_{\mathbf{p}} \rho_{\mathbf{f}} \mathbf{g}, \qquad (12) $$

where $\rho_f$ is the fluid density. If we combine buoyancy with the gravity force, Eq. (9), we retrieve the well-known expression of the apparent weight of an immersed body:

$$ \mathbf{F}_{\text{grav}} + \mathbf{F}_{\text{buoy}} = \nabla_{\mathbf{p}} (\rho_{\mathbf{p}} - \rho_{\mathbf{f}}) \mathbf{g}. \qquad (13) $$

ii - The pressure-gradient force $\mathbf{F}_{\text{PG}}$ corresponds to the force exerted by the velocity field undisturbed by the presence of the particle [165]. It is usually expressed as:

$$ \mathbf{F}_{\text{PG}} = \nabla_{\mathbf{p}} \rho_{\mathbf{f}} \frac{D\mathbf{U}_{\mathbf{f}}}{Dt}, \qquad (14) $$

where $D/Dt = \partial/\partial t + \mathbf{U} \cdot \nabla$ is the material derivative.

B. Hydrodynamic forces related to velocity differences between particles and fluid can be evaluated using two approaches:

i - Integration on a fully-resolved flow field: When the instantaneous flow field around the object is fully resolved, one can directly estimate the strength of these hydrodynamic forces by integrating the extra fluid stress tensor $\delta\mathcal{T}_f$ due to the perturbation induced by the particle along the entire surface of the object $S_p$. This gives:

$$ \mathbf{F}_{\text{relative-motion}} = \int_{S_p} \delta\mathcal{T}_f \cdot \mathbf{n}_S \, dA \qquad (15) $$

where $\mathbf{n}_S$ is the outgoing normal to the local surface. As one can expect, such integrations require very detailed information on the flow field around each individual particle. This can be obtained either with prescribed flow fields in simple configurations or with fine numerical simulations of turbulent flows (see Section 4.2 for more details).
## ii - Approximate formulas:

When direct integration of the hydrodynamic force is not possible, one can resort to approximate expressions to estimate the hydrodynamic forces acting on a particle. This usually occurs when particles are much smaller than the grid size used in the computation of the flow field, leading to the so-called point-particle approximation. The interest of this point-wise particle approximation is that the hydrodynamic forces are expressed using information only on the fluid velocity at the particle position, which defines the velocity of the fluid seen $U_s(t) = U_f(t, X_p(t))$.

Despite decades of research on the subject (see the work of Stokes in the mid-19th century and those of Basset-Boussinesq-Oseen between 1888 and 1927), the exact expression of these hydrodynamic forces is still a subject of research today. In the case of point particles, the hydrodynamic forces are generally given by the Maxey-Riley-Gatignol equation [166, 167] (also referred to as the BBO equation). This equation takes into account several contributions [8, 165, 168]:

$$ \mathbf{F}_{\text{relative-motion}} = \mathbf{F}_{\text{drag}} + \mathbf{F}_{\text{lift}} + \mathbf{F}_{\text{AM}} + \mathbf{F}_{\text{hist}}. \qquad (16) $$

a. Drag forces $\mathbf{F}_{\text{drag}}$ arise from the friction due to the relative velocity between the fluid and particle velocities. Drag forces act along the direction of the relative velocity and tend to move particles in the same direction as the flow (such that the relative velocity becomes zero). Hence, these forces are directly expressed in terms of the relative velocity $\mathbf{U}_r = \mathbf{U}_p - \mathbf{U}_s$ [169, 170]:

$$ \mathbf{F}_{\text{drag}} = \frac{1}{2} \rho_f C_D A_p |\mathbf{U}_r| \mathbf{U}_r, \qquad (17) $$

with $A_p$ the particle cross-sectional area exposed to the flow and $C_D$ the drag coefficient. The particle cross-sectional area $A_p$ is straightforward to compute for simple mathematical shapes (like spheres, spheroids) but can be more cumbersome when dealing with irregular particles. The difficulty comes from the expression of the drag coefficient, which is a non-linear function that depends on the particle shape and on the particle Reynolds number $Re_p = \|\mathbf{U}_r\|d_p/\nu_f$ ($\nu_f$ being the fluid kinematic viscosity and $d_p$ the particle size). Nevertheless, semi-empirical expressions for $C_D$ can be found in simple cases. For instance, for spherical particles, the Stokes drag coefficient is rigorously given by $C_D = 24/Re_p$ (derived in the Stokes regime where $Re_p \ll 1$), leading to the exact formula for the Stokes drag force [169, 170]:

$$ \mathbf{F}_{\text{drag}} = 3 \pi d_p \rho_f \nu_f (\mathbf{U}_s - \mathbf{U}_p). \qquad (18) $$

Extended formulations of the Stokes drag coefficient have been proposed by adding a correction factor $C_D = 24/Re_p f(Re_p)$ which extends the range of validity of the formula. For example, a frequently used correction factor is $f(Re_p) = 1 + 0.15 Re_p^{0.687}$ (valid up to $Re_p \le 1000$, at which point the drag coefficient is usually fixed to a constant equal to 0.44) [10, 169]. More recently, general approximations of the drag coefficient have been pursued using machine-learning techniques (as in [171] for particles with arbitrary shapes or in [172] for dense suspensions). In the context of particle resuspension, the Stokes drag formula should be also corrected to account for the presence of the wall, giving:

$$ \mathbf{F}_{\text{drag}} = 3 \pi d_p \rho_f \nu_f (\mathbf{U}_s - \mathbf{U}_p) f_{\text{wall}}, \qquad (19) $$

with $f_{\text{wall}}$ a correction factor (equal to 1.7009 according to [173]). Another consequence of the fluid motion anisotropy in the near-wall region is that it produces a non-zero torque (since the velocity is smaller near the surface). In the Stokes regime, the torque is usually approximated with the O'Neill formula [173]:

$$ \mathbf{M}_{\text{drag}} = 1.4 r_p \mathbf{F}_{\text{drag}}, \qquad (20) $$

with $r_p$ the particle radius.
b. Lift forces $\mathbf{F}_{\text{lift}}$ correspond to the force exerted by the fluid flow in the direction perpendicular to the oncoming flow direction (contrary to the drag forces that act along the direction of the relative velocity). The lift force depends on the relative velocity between the particle and the fluid (termed Saffman lift) and on the particle rotation rate (labelled Magnus lift) [170]. As for drag forces, several analytic or empirical formulas have been presented in the literature. Yet, there is no consensus on a valid formulation (or taken as such) regardless of the particle Reynolds number $\mathcal{R}e_p$. For instance, empirical formulas have been written for spherical particles in a fully developed turbulent boundary layer [174, 175], which reads:

$$ \mathbf{F}_{\text{lift}} = \alpha \nu_f^2 \rho_f (\mathcal{R}e_p)^\beta, \qquad (21) $$

with $\alpha$ and $\beta$ two constants. Hall proposed to use $\alpha = 20 \pm 1.57$ and $\beta = 2.31 \pm 0.02$ for large particles in the range $3.6 \le \mathcal{R}e_p \le 140$ [174], while Mollinger et al. suggested $\alpha = 56.9 \pm 1.1$ and $\beta = 1.87 \pm 0.04$ for small particles such that $0.6 \le \mathcal{R}e_p \le 4$ [175].

Yet, these formulas should be carefully chosen and assessed, especially since discrepancies have been found between theoretical and experimental works leading to estimates that differ by orders of magnitude [176]. It should be highlighted that the approach to determine the drag and lift forces has to be consistent and based on the same assumptions. For instance, if the O'Neill formulation for drag is chosen, the lift force is zero and cannot be replaced with just any other equation. In addition, as for the drag force, the length scale in the lift force computation may depend on the particle orientation for non-spherical particles.

c. Added-mass forces $\mathbf{F}_{\text{AM}}$ are associated with the additional force that is needed to accelerate/decelerate a body immersed in a dense fluid due to the fact that the body also moves some of the nearby surrounding fluid as it moves through it. Hence, the added-mass force is directly proportional to the difference between the particle acceleration and the fluid one at the particle location. It is given by:

$$ \mathbf{F}_{\text{AM}} = \frac{1}{2} \rho_f \mathcal{V}_p \left[ \left( \frac{D\mathbf{U}_f}{Dt} \right) (t, \mathbf{X}_p(t)) - \frac{d\mathbf{U}_p}{dt} \right]. \qquad (22) $$

While added-mass forces can severely impact the dynamics of small particles with a density close to the fluid density, it is often negligible compared to drag forces for particles much heavier than the fluid, i.e. $\rho_f \ll \rho_p$ (like sand in air).

d. Basset history forces are related to the response of the fluid boundary layer to an acceleration of the relative velocity. Due to viscous effects, there is indeed a temporal delay in the development of the boundary layer around a particle when the relative velocity changes, resulting in a 'history' effect [177]. Hence, the Basset force is usually expressed by an integration of the relative velocity over the trajectory of the particle:

$$ \mathbf{F}_{\text{hist}} = \frac{3}{2} d_p^2 \rho_f \sqrt{\pi \nu_f} \int_{-\infty}^{t} \frac{d}{ds} (\mathbf{U}_s(s) - \mathbf{U}_p(s)) \frac{ds}{\sqrt{t-s}}. \qquad (23) $$

The Basset force is negligible in stationary flows [177] but not in unsteady flows. Yet, in practice, it is often neglected due to its cumbersome evaluation (with a singularity in the term $1/\sqrt{t-s}$).

At this stage, it is worth noting that the Maxey-Riley equation is, strictly speaking, only valid for particles such that $\mathcal{R}e_p \ll 1$. However, they are also used when $\mathcal{R}e_p \gg 1$ (but resorting to generalized formulas for the drag and lift forces). The particle equation of motion can be further extended to include effects related to Brownian motion. This corresponds to the random motion of particles suspended in a fluid due to their collision with molecules of the fluid. While Brownian motion has been shown to be relevant for colloidal particles (i.e., with a size smaller than a few micrometers), it quickly becomes negligible when dealing with larger particles (which are more sensitive to gravity forces) [10]. A usual and straightforward way to include Brownian motion in a particle equation of motion is to adopt a stochastic description of the particle dynamics and to add a white noise term (to account for the random Brownian
motion). Neglecting other contributions, this gives the following Langevin equation (more details in [9]):

$$d\mathbf{X}_p = \mathbf{U}_p dt \qquad (24a)$$

$$d\mathbf{U}_p = (\text{other accelerations}) + K_{\text{Br}} d\mathbf{W} \qquad (24b)$$

where $K_{\text{Br}}$ is the diffusion coefficient for Brownian motion and $\mathbf{W}$ is a vector of independent Wiener process (independent noise terms).

**Role in resuspension:** Hydrodynamic drag and lift forces are key ingredients in particle resuspension. They are directly involved in particle detachment, since their action tends to set in motion particles initially at rest on a surface. In addition, they play a role in the dynamics of particles once they are detached from the surface (i.e., the re-entrainment phase). Due to turbulence effects (see Section 4.2.1), these forces exhibit fluctuations and are therefore characterized by a distribution of values. This distribution is responsible for the range of values at which resuspension occurs for a given type of particle and flow conditions.

### 4.1.3. Non-contact forces

**Physical origin:** Having described particle-fluid forces, we now turn our attention to particle-wall and particle-particle interactions. These forces are usually referred to as inter-surface forces (or simply surface forces), since they act between two macroscopic bodies (more details in [178]). From a physical point of view, they arise due to the intermolecular (weak) forces acting between every atoms/molecules composing each body. Hence, surface forces are short-ranged and important when dealing with small particles (typically colloids). We focus specifically on forces when there is no contact between surfaces, which means that these forces affect the dynamics of particles detached from the surface (contact forces are described in Section 4.1.4).

<figure><img src="image_29.png" /><figcaption>Figure 21: Illustration of some of the non-contact forces at play in particle resuspension.</figcaption></figure>

**Expressions:** Non-contact surface forces encompass various contributions, including (see also Fig. 21):

A. van der Waals (VDW) interactions result from the correlations in the fluctuating polarization of nearby dipoles [178]. This includes contributions from two permanent dipoles (Keesom force), a permanent dipole and an induced one (Debye force), as well as two induced instantaneous dipoles (London dispersion forces). In most cases, VDW forces are attractive and responsible for adhesion between macroscopic bodies. Yet, they quickly vanish when distances exceed a few nanometers/micrometers. As a result, VDW interactions are playing a significant role in the adhesion of small particles, with a size ranging
from a few nanometers up to a few hundreds of micrometers. Above this threshold, VDW forces are still present but their magnitude becomes negligible compared to other macroscopic forces (such as gravity). Methodologies to evaluate VDW forces are based on three different levels of description:

* First, direct numerical integration are based on Surface Element Integration methods (SEI). As depicted in Fig. 21, these approaches describe each body as an ensemble of molecules (i.e., using a molecular level of description). Then, the potential energy $E_{VDW}$ is simply obtained by integrating the elementary molecule-molecule interactions $E_{vdw}$ over the exact topography of the two bodies involved [179]. This gives:

$$E_{VDW} = \int_{V_1} \int_{V_2} E_{vdw}(\text{mol 1} - \text{mol 2}) dV_1 dV_2 \quad (25)$$

Note that we use capital letters for interactions between macroscopic bodies and lowercase letters for molecular interactions. The resulting VDW force is then simply obtained by deriving the potential energy with respect to the separation distance $h$:

$$F_{VDW} = -\frac{dE_{VDW}}{dh} \quad (26)$$

The main interest of this approach is that it can be used regardless of the particle shape and topography. It only requires to define the inter-molecular potential energy: this is often taken as a Lennard-Jones potential $E_{L-J} = C_1/h^{12} - C_2/h^6$ (with $C_1$ and $C_2$ two constants), which combines a van der Waals attractive term ($1/h^6$ law) and a repulsive term due to Pauli exclusion principle [178]. Yet, the computational costs are very high since VDW forces are obtained from a molecular integration (similar to Molecular Dynamics in terms of the level of description).

* Second, approximate expressions can also be obtained using the Hamaker approach. This approach still adopts a microscopic point of view, where each macroscopic body is described in terms of its molecules. However, instead of performing direct numerical integration, simplified formulas are obtained with the following assumptions: (a) dispersion forces between molecules inside the same body are neglected; (b) only simple geometries are considered (where mathematical integration is possible). For instance, assuming that the basic molecule-molecule interactions is given by the unretarded energy $E_{vdw} = -C/h^6$ (with $C$ a constant), one obtains the unretarded VDW interactions between two spheres [180]:

$$E_{VDW,Sph1-Sph2} = -\frac{A_{Ham}}{6} \left[ \frac{2 r_{p,1} r_{p,2}}{(2 r_{p,1} + 2 r_{p,2} + h) h} + \frac{2 r_{p,1} r_{p,2}}{(2 r_{p,1} + h) (2 r_{p,2} + h)} + \ln \frac{(2 r_{p,1} + 2 r_{p,2} + h) h}{(2 r_{p,1} + h) (2 r_{p,2} + h)} \right], \quad (27)$$

with $A_{Ham} = \pi^2 C \rho_1 \rho_2$ the Hamaker constant (related to the constant $C$ for molecular vdw interactions and to the number of atoms per unit volume $\rho$).

In general, these approximate formulas are written in the form of a Hamaker constant $A_{Ham}$ multiplied by a geometrical term (accounting for the separation distance $h$ between bodies and for their shape). For instance, using the Derjaguin approximation (which assumes that the separation distance is much smaller than the radius of both bodies), a simplified formula for the unretarded VDW forces between spheres is obtained [134]:

$$E_{VDW,Sph1-Sph2} = -\frac{A_{Ham}}{6h} \frac{r_{p,1} r_{p,2}}{r_{p,1} + r_{p,2}} \quad (28)$$

The formulas can be more complex depending on whether retardation effects are accounted for (more details in [134, 181]). These retardation effects result from the finite propagation velocity of electromagnetic waves between dipoles, which induces vdw energies that decay as $1/h^7$ at larger
separation distances between molecules. Such effects are often introduced into the Hamaker expressions by adding a correction term proportional to a characteristic retardation wavelength $\lambda$ (that measures the distance above which retardation effects play a role).

* Third, a macroscopic point of view can be adopted using the Lifshitz approach [181]: it consists in treating each body as a continuous media instead of considering atomic details. As a result, VDW forces between macroscopic bodies are expressed with the electromagnetic properties of the interacting media (such as dielectric constants and refractive indices). In fact, the formulas are often very similar to those obtained with a Hamaker approach, except that Hamaker constants are here calculated considering the whole dielectric spectra of each material (including retardation effects).

B. Electrostatic forces occur when particles are electrically charged (e.g., soot particles from combustion engines [182]). Expressions for these forces depend on the medium in which the particles are immersed.

i. When particles are immersed in the air (or a gas), electrostatic forces occur between each charged body. In the case of punctual spherical particles, this force is inversely proportional to the square of the distance between the objects $h$:

$$F_{\text{elect}}(h) = \frac{k_0 q_{p,1} q_{p,2} e^2}{h^2} \mathbf{x}_{1-2} \qquad (29)$$

where $q_{p,1}$ and $q_{p,2}$ are the number of elementary charges for each particle, $e = 1.602 \times 176 \times 634 \times 10^{-19}$ C the elementary charge, $k_0 = 8.987 \times 551 \times 792 \times 10^9 \text{ kg m}^3/\text{s}^2/\text{C}^2$ the Coulomb constant and $\mathbf{x}_{1-2}$ the unit vector connecting the two particles.

ii. Electrostatic double-layer forces arise between charged bodies immersed in a liquid. In that case, the surface charge of each particle is balanced by an excess of oppositely-charged ions present in the solution near the surface [178]: this results in the formation of a so-called electrostatic double-layer (i.e., a volume of fluid in close proximity of the body where counter-ions are more abundant than co-ions). When two bodies are in close proximity, the electrical double-layer of each body can overlap with each other. In turn, this overlap induces either an attractive force (when the two bodies are oppositely charged) or a repulsive interaction (for bodies with the same sign of charges). This force is called the electrostatic double-layer (EDL) interaction. Like VDW forces, EDL forces are relatively short-ranged (with a range that depends on the concentration of ions in the solution, typically from a few tenths of nanometers up to a few hundreds of micrometers). Hence, EDL forces are not acting at distances as large as electrostatic forces in air.

Expressions of EDL forces between two charged bodies can only be obtained by solving the Poisson-Boltzmann equation with appropriate boundary conditions at the surfaces. For instance, assuming that the surface potential remains constant when the two bodies come closer and that the potentials are small enough to use a linearized Poisson-Boltzmann equation, one obtains the following equation for EDL potential energy between two spheres [134]:

$$U_{\text{EDL,Sph1-Sph2}} = 64\pi\varepsilon_0\varepsilon_R \left(\frac{k_B T_f}{z e}\right)^2 \tanh^2\left(\frac{z e \Psi_0}{4 k_B T_f}\right) \left(\frac{r_{p,1} r_{p,2}}{r_{p,1} + r_{p,2}}\right) e^{-\kappa h}, \qquad (30)$$

with $\Psi_0$ the surface potential, $k_B = 1.38 \times 10^{-23} \text{ J K}^{-1}$ the Boltzmann constant, $T_f$ the fluid temperature, $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C V}^{-1} \text{ m}^{-1}$ the dielectric permittivity of vacuum, $\varepsilon_R$ the dielectric constant of the media (e.g., $\varepsilon_R = 78.5$ for water), and $z$ the valence of ions present in the solution. Equation (30) involves the inverse Debye length $\kappa$, defined by:

$$\kappa = \sqrt{\frac{e^2 \sum_i n_i^0 z_i^2}{\epsilon_0 \epsilon_r k_B T_f}} = \sqrt{\frac{2e^2 I}{\epsilon_0 \epsilon_r k_B T_f}}, \qquad (31)$$

$I$ being the solution ionic strength.
C. For particles immersed in a viscous fluid, an additional force comes from the extra pressure generated by the interstitial fluid that is being squeezed between the two surfaces approaching each other. Due to viscous effects at such small scales, an extra repulsive force is induced, called the lubrication force in the multiphase flow community [183, 184]. When the flow velocity field is known even at scales much smaller than the separation distance between particles $h$, this force is fully determined. Yet, in simulations of particle resuspension, the fluid flow is not resolved at such microscale distances (see Section 4.2.1). Hence, additional models for lubrication forces are usually introduced. For instance, in the case of two spherical particles nearing each other, lubrication forces are often expressed by [185]:

$$ \mathbf{F}_{\text{lub}} = \begin{cases} -\frac{6\pi\nu_f\rho_f\mathbf{U}_r \cdot \mathbf{n}_{\perp}}{h} \frac{r_{p,1} r_{p,2}}{r_{p,1} + r_{p,2}} \mathbf{n}_{\perp}, & \text{if } h_{\text{min}} \le h \le h_{\text{max}} \\ 0, & \text{elsewhere} \end{cases} \quad (32) $$

where $\mathbf{U}_r \cdot \mathbf{n}_{\perp}$ is the projection of the relative velocity between the two particles along the direction connecting their center of gravity $\mathbf{n}_{\perp}$. The force is set to 0 when the separation distance is larger than $h_{\text{max}}$, which is usually taken as twice the smallest scale solved by the solver for the flow field (see Section 4.2.1): in fact, at higher distances, such effects would already be accounted for and do not need to be modeled. Similarly, another minimum cutoff distance $h_{\text{min}}$ is introduced to avoid the singularity at $h = 0$. The main reason for this cutoff distance $h_{\text{min}}$ is that real surfaces are rough (see Section 4.2.2), meaning that, at the macroscopic scale, contact happens at a short (but non-zero) distance. Another justification for this cutoff distance could arise even without accounting for surface roughness by bridging the fields of multiphase flow and interface science. In fact, in the latter field, specific forces account for the repulsion arising from the ordering of solvent molecules when only a few layers are left in the small interstitial space between the two surfaces (see Fig. 21). These forces are referred to as solvation or structural forces (or even hydration forces when the medium is water). When dealing with perfectly smooth surfaces, the solvation potential is usually approximated by an exponentially decaying cos-function of the form [178]:

$$ E_{\text{solv}} \approx -2 \Delta\gamma \cos\left(\frac{2\pi h}{\sigma}\right) e^{-h/\sigma}, \quad (33) $$

where $\sigma$ is the size of solvent molecules, and $\Delta\gamma$ is the interfacial energy between the two surfaces (which is a function of the interfacial energy of each surface taken alone and of the medium). This gives rise to a number of potential wells at different separation distances at the molecular scale (one of which could be related to the cutoff distance used in more macroscopic descriptions for lubrication forces).

D. Capillary forces develop when a capillary bridge forms between two bodies [178]. This effect induces higher adhesion between the two bodies, since separating them requires breaking the capillary bridge. Such capillary adhesion is especially important when dealing with particles in the air with a high relative humidity (as for bacteria in indoor surfaces [44] or sand dunes [186]). Alternatively, it can also occur for particles immersed in a liquid but with a high relative saturation (meaning that air bubbles may spontaneously form). The strength of the capillary bridge also depends on the surface properties of the two bodies: due to their affinity with water, it is easier to form capillary bridges with hydrophilic surfaces (e.g., dust mite) than with hydrophobic particles (such as cat fur, dog fur) [44].

In general, the force due to a capillary bridge between a sphere and a smooth plate can be expressed using the Laplace pressure in the liquid, which acts on a small area between the two surfaces [178]:

$$ \mathbf{F}_{\text{capil}} = -4\pi r_{\text{p}} \gamma_{\text{L}} \cos\theta \left(1 - \frac{h}{2r_{\text{menisc}}\cos\theta}\right) \mathbf{n}_{\perp}, \quad (34) $$

where $\gamma_{\text{L}}$ is the liquid interfacial surface tension (i.e. between the air and the liquid), $\theta$ the liquid contact angle and $r_{\text{menisc}}$ the radius of curvature of the liquid meniscus formed between the two surfaces.

E. Specific chemical bindings arise due to strong intermolecular bonds between atoms, ions or molecules (such as covalent, ionic or metallic bonds) [178]. Other specific surface forces can arise between inorganic
bodies due to the presence of polymers or surfactants in the solution. In fact, the presence of polymers attached to each body can result in more complex polymer-mediated interactions between the two bodies. This includes both repulsive steric forces (due to Pauli exclusion principle between polymer-covered surfaces) as well as attractive intersegment and bridging forces (when the polymers form bonds like hook-and-loop fasteners). The expression of such forces is not straightforward since it depends on how polymers are arranged near the surface. In general, the interaction potential is often considered to be an exponentially decaying function of the separation distance *h* (more details in [178]).

F. Singular biological forces can take place when dealing with organic or biological bodies (such as pollen or bacteria). In addition to the forces mentioned previously, selective forces can significantly increase the adhesion between two bodies (e.g., due to ligand-receptor interactions or due to bridging effects when polymers get bounded to both surfaces) [178]. Another specificity of the biological forces is related to their non-equilibrium properties, mostly due to the fact that organic surfaces continuously change according to their interactions with the local environment. For instance, organisms like bacteria can grow on the surface of water distribution systems [2] (provided they find sources of proteins, lipids or other compounds on the surface). Hence, while the previously mentioned forces can be viewed as ‘static’, biological interactions are by essence ‘dynamic’. For the sake of simplicity, biological particles are not specifically addressed in the rest of this review (unless specifically mentioned to insist on the similarities with non-organic materials).

**Role in resuspension:** Non-contact forces play a double role in particle resuspension. On the one hand, many of these forces act on particles deposited on the surface (together with contact surfaces described in the next Section 4.1.4). On the other hand, these forces affect the dynamics of particles once they are detached from the surface. Hence, like hydrodynamic forces, they play a key role in the re-entrainment phase. At this stage, it is important to realize that short-ranged surface forces are often characterized by a distribution of possible values (due to effects of surface roughness, see Section 4.2.2), which is responsible for the range of values at which resuspension occurs for a given type of particles and flow condition.

### 4.1.4. Contact forces

**Physical origin:** Various physico-chemical effects play a role in the forces acting between two objects in contact (see also Fig. 22). This includes contributions from: (a) adhesion forces, which belong to the category of surface forces and are responsible for the adhesion between two surfaces (with or without surface deformations); (b) friction forces, which resist the lateral motion between two surfaces in contact due to the non-trivial contact between them; (c) momentum/energy exchanges during inter-particle collisions (e.g., due to action-reaction principles and conservation laws); (d) vibrations, which can induce extra mechanical effects that can favor particle resuspension.

<figure><img src="image_30.png" /><figcaption>Figure 22: Illustration of contact forces at play in particle resuspension.</figcaption></figure>

**Expressions:** Contact forces can be evaluated using a range of descriptions.

A. Adhesion of solids arise from short-ranged intermolecular interactions between the molecules composing each solid. These forces can be calculated using various models, which can be classified in two main
categories sketched in Fig. 23 (see [178, 10, 187] for more details): those based on Hamaker’s approach for van der Waals forces [180] and those based on contact mechanics theories [188].

i - The Hamaker model describes adhesion as a result of the short-range attractive force between surfaces at very close separation distances. For instance, many models are based on a Lennard-Jones potential for inter-molecular interactions, which includes two contributions: an attractive term that scales as $1/h^6$ (due to the London dispersion force) and a repulsive term that scales as $1/h^{12}$ (due to Pauli repulsion effect that prevents the overlap of electron orbitals). The interest of such descriptions is that it allows to grasp what is meant by the notion of ‘contact’. In fact, when two macroscopic bodies are in contact, this means that the two surfaces are separated by a distance typically smaller than or within the nanometer range (since the molecules from each body cannot overlap due to Pauli exclusion principle).

Using the same Hamaker approach introduced previously for VDW forces (see Section 4.1.3), formulas between simple geometrical objects have been obtained by integrating inter-molecular forces over the volume of both objects. In fact, the adhesion force obtained is usually very similar to the VDW force obtained for a separation distance equal to the “contact distance” $z_0$ (since repulsive forces decay extremely rapidly with the separation distance). At this stage, it is worth noting that other inter-surface contributions (such as electric double-layer forces, solvation forces, steric forces) can sometimes be non-negligible and are easy to incorporate in such descriptions of adhesion forces based on a Hamaker model.

<figure><img src="image_31.png" /><figcaption>Figure 23: Illustration of the various modeling approaches to evaluate the adhesive force between surfaces. Models can be either based on descriptions using van der Waals forces or contact mechanics theories (with models based on Hertz theory for elastic deformation, such as JKR or DMT models).</figcaption></figure>

ii - Contact-mechanics models are concerned with surface deformation of the two bodies in contact [188] (see also Fig. 23). In practice, various models have been suggested in the literature to account for the elastic deformations of surfaces upon contact. This includes the JKR theory (named after the work of Johnson Kendall and Roberts [189]) and the DMT theory (named after the work of Derjaguin, Muller, and Toporov [190]).

* The JKR theory is based on the assumption that surfaces deform elastically upon contact according to the Hertz theory (i.e., for elastic deformations) and that adhesion occurs only within the contact area. As a result, the pull-off force (i.e. the force required to separate two surfaces) predicted by the JKR theory between a sphere and a plate is given by:

$$F_{JKR} = 3\pi \Delta\gamma r_p, \qquad (35)$$

with $\Delta\gamma$ the surface energy between the two objects. The JKR theory is valid only for large and ‘soft’ particles (i.e., highly deformable).
* The DMT theory is based on the assumption that surfaces deform elastically upon contact according to the Hertz theory but that adhesion includes also non-contact forces (van der Waals contribution) which act across the gap between the two surfaces. As a result, the pull-off force predicted by the DMT theory between a sphere and a plate is given by:

$$F_{\text{DMT}} = 4\pi \Delta\gamma r_p . \qquad (36)$$

The value obtained is larger than the one predicted with the JKR theory (due to the additional contribution from VDW forces outside the contact area). The DMT theory is valid for small or ‘hard’ particles (i.e., slightly deformable).

More recently, extended theories for elastic deformations have been proposed to encompass both JKR and DMT theories, while naturally recovering both theories in the limit cases of soft or hard particles. Other formulations include the role of plastic deformations in adhesion forces (e.g. the Maugis-Pollock model [191]). In fact, all materials can deform to a certain extent under the action of an external load/force [192, 193]. When the load is small, the surface on which this load is applied undergoes an elastic deformation. During this phase, the induced deformation (also called strain in material sciences) depends linearly on the magnitude of the force applied (called stress in material sciences). This relation is actually given by Hooke’s law and the constant of proportionality is referred to as the modulus of elasticity (also called Young’s modulus). Such elastic deformations are reversible (meaning that the surface returns to its original shape once the stress is removed) until the yield point. When the force is so high that the resulting stress becomes higher than the yield stress, plastic deformation happens. At that point, some of the deformations induced in the structure become permanent and non-reversible. During plastic deformations, the relationship between the stress and the deformation does not follow a linear trend anymore. At even higher values of the stress, fracture occurs resulting in permanent damage to the surface structure (see Fig. 24).

<figure><img src="image_32.png" /><figcaption>Figure 24: Illustration of the stress-strain curve showing the deformation of a particle in contact with a smooth surface (all the way from elastic deformation to plastic deformation and eventually fracture).</figcaption></figure>

B. Friction forces correspond to the resistance to lateral motion when two bodies are in contact. Unlike previously mentioned forces, these forces do not follow a force-versus-distance law but rather arise only as a reaction to motion or to another force (interested readers can find more details in the literature related to tribology, see [178, 194, 195, 196]). From a microscopic point of view, friction forces emerge from the contact between two irregular surfaces, i.e., surfaces that are not smooth (which is always the case at the molecular level). As sketched in Fig. 22, the presence of ‘protruding’ elements on the surface of each body leads to a sort of entrapment which resists lateral motion. To put it differently, the two irregular surfaces can only move freely in the lateral direction once they are separated by a distance larger than the size of the irregularities present on the surfaces. This means that lateral motion is preceded by a short motion along the perpendicular direction. Yet, this perpendicular motion is prevented by adhesive forces (such as VDW contributions) and/or load forces (such as gravity). Hence, lateral motion is only possible provided that the lateral force is high enough to overcome the forces that oppose this short
separation along the perpendicular direction. From this simplified picture of friction forces, it becomes evident that the stronger the adhesion between two bodies, the stronger the resistance to motion and hence the higher is the friction force. In practice, friction forces are expressed with the ratio between lateral and perpendicular forces: this is the so-called friction coefficient $\mu_{frict}$. This gives:

$$F_{\text{resist,||}} = \mu_{\text{frict}} F_{\perp}. \qquad (37)$$

A distinction is usually introduced between the static friction coefficient (i.e., when the two surfaces are not moving yet) and the dynamic friction coefficient (i.e., when the two surfaces are already moving). The values of these coefficients are often tabulated [197].

### C. Exchanges during inter-particle collisions:

Deposited particles can also be set into motion when they are hit by other moving particles (either in suspension or migrating on the surface) [111, 110]. This brings out two important questions (see Fig. 25):

#### i - How to detect collisions?

Detecting particle collisions implies to take into account that particles have a finite size. This means that specific algorithms are used to detect when and where two particles collide, such as:

* When detailed information is available on the position and velocity of each particle, one can resort to deterministic algorithms. For instance, assuming particles to be non-deformable ‘hard’ spheres, a collision happens if two particles overlap, i.e. if the inter-particle distance $h$ becomes smaller than the sum of the two radii at any time $h(t) \le r_{p,1} + r_{p,2}$ [198]. Otherwise, assuming a fixed velocity during each small time interval, one can also use geometric criteria to detect if two particles collide during this time interval [199].

* Alternatively, when the exact trajectory of each particle is not known exactly during an observation time, one can resort to probabilistic algorithms to detect collisions. These algorithm use a-priori information on the relative motion between each pair of particle to estimate the probability that a pair of particle does collide during a certain time (as in [200, 201, 202]).

<figure><img src="image_33.png" /><figcaption>Figure 25: Illustration of the issues related to inter-particle collisions: the detection of such collisions (left) and the treatment of each collision outcome (right).</figcaption></figure>

#### ii - How to treat collisions?

Once a collision is detected, the outcome of each event has to be determined. In practice, the outcome of a collision depends on the conservation laws that apply to the system made up by the two particles and the surface. In a closed system, momentum is conserved but not necessarily kinetic energy. For instance, elastic collisions correspond to collisions where kinetic energy is conserved during the collision while inelastic collisions occur when a fraction of the energy is dissipated (e.g., due to plastic deformations). These phenomena have been extensively studied in the field of impact mechanics, which is concerned with the “reaction forces that develop during a collision and the dynamic response of structures to these reaction forces” [203]. In the context of particle collisions away from boundaries, they are traditionally modeled by a restitution coefficient $e_{coll}$, which measures the ratio between the kinetic energy just before impact $E_{kin}^{-}$ and the kinetic energy right after it $E_{kin}^{+}$. For particles impacting a surface, the same ideas are used although one makes
a difference between the normal and tangential components [112]:

$$E_{\text{kin},\perp}^{+} = e_{\text{coll},\perp} E_{\text{kin},\perp}^{+} \qquad (38a)$$

$$E_{\text{kin},||}^{+} = e_{\text{coll},||} E_{\text{kin},||}^{+} \qquad (38b)$$

Hence, such formulations can be used to determine the motion of each particle after the collision, thereby allowing to capture successive collisions and avalanche phenomena.

D. Mechanical forces (like those induced by vibration) can be relevant in some applications [152, 153]. For instance, in walking-induced resuspension, vibration is induced when a foot hits the ground. Part of the vibrating energy can be transferred to the particle, thanks to the contact between surfaces. Yet, when surfaces are deformable, vibration can facilitate/hinder the breakage of the contact depending on the vibrating frequency. Vibration can also be important in the context of outdoor resuspension from vegetation, where the wind can shake grass blades and tree branches.

**Role in resuspension:** Contact forces are important only for particle detachment since they vanish when the particle/substrate contact is ruptured. They actually play a key role in the detachment of small colloidal particles, since adhesion forces are the most significant ones preventing particle motion.

### 4.1.5. Summary of forces

From this overview of the expressions for the forces/torques acting on particles, it appears that a closed set of equations can be defined to describe particle dynamics. However, this requires to carry out a careful analysis of the input variables (available from experiments) and the output ones (those that should be predicted). This partially explains the profusion of models available in the literature, since each model can introduce slight changes either in the equations of motion or in the expressions of the forces/torques involved.

This brief catalog illustrates not only the diversity of fields involved in particle resuspension but also the complexity of the forces at play. In fact, it appears that the forces listed previously can sometimes act to hinder particle motion or favor it, depending on the conditions. Table 2 summarizes how each of these forces contribute to the resuspension of particles.

### 4.2. Accounting for turbulence and surface roughness

The hydrodynamical forces detailed in Section 4.1 appear as deterministic expressions but require to have access to the exact velocity near/at particle locations. The same is true for adhesion forces which require to know the exact topographies of the surfaces involved. In practice, such information is not always available or is provided in a statistical sense. For instance, in turbulent flows, the velocity at a given point is often described only in a probabilistic manner. Similarly, surface profiles can be depicted as ‘rough’ and characterized only by some statistical measures. This implies that hydrodynamical and adhesion forces are to be addressed also in a statistical sense. To understand the impact of turbulence and of surface roughness on the key forces at play in particle resuspension, it is therefore useful to recall how turbulent flows and rough surfaces are modeled in practice. This is done here by considering, first, turbulence models in Section 4.2.1 and, second, how surface roughness is represented in Section 4.2.2.

#### 4.2.1. Capturing the impact of turbulence on resuspension

**Physical origin and characteristics:** Whereas molecular viscosity is a property of a fluid, turbulence is a property of a flow and is manifested by the fact that characteristic fields, such as the velocity, pressure or temperature fields, exhibit variations in space and in time which cover continuously a wide range of scales (interested readers are referred to books dedicated to turbulence [204, 205]). This explains that turbulent flows are traditionally treated as random fields (thus the terminology of ‘fluctuations’ to describe apparently chaotic, or rather random, behavior). Classical descriptions of turbulent flows have essentially relied on statistical, or probabilistic, formulations but, in recent decades, experimental and numerical studies have revealed that these random fluctuations can be consistent with the existence of some specific geometrical
<table><caption>Table 2: Summary of the forces and their effect on resuspension (especially whether each force favors or hinders resuspension).</caption><thead><tr><th rowspan="2">Type of force</th><th rowspan="2"></th><th colspan="2">Role in resuspension</th></tr><tr><th>Inducing motion</th><th>Resisting motion</th></tr></thead><tbody><tr><td rowspan="2">Interactions with<br>external fields</td><td>Gravity</td><td>✓<br>(ceiling)</td><td>✓<br>(floor)</td></tr><tr><td>Electro-magnetic</td><td>✓</td><td>—</td></tr><tr><td rowspan="6">Hydrodynamic forces</td><td>Pressure gradient</td><td>—</td><td>—</td></tr><tr><td>Buoyancy</td><td>✓<br>(lighter particles)</td><td>✓<br>(heavier particles)</td></tr><tr><td>Drag</td><td>✓</td><td>✗</td></tr><tr><td>Lift</td><td>✓</td><td>✗</td></tr><tr><td>Added-mass</td><td>✓</td><td>✗</td></tr><tr><td>Basset history</td><td>✓</td><td>✗</td></tr><tr><td>Brownian motion</td><td>✓</td><td>—</td></tr><tr><td rowspan="5">Non-contact forces<br>(incl. particle-substrate &amp;<br>particle-particle interactions)</td><td>van der Waals</td><td>✗</td><td>✓</td></tr><tr><td>Electrostatic (in air/liquid)</td><td>✓<br>(Opposite charges)</td><td>✓<br>(Similar charges)</td></tr><tr><td>Lubrication / solvation</td><td>—</td><td>—</td></tr><tr><td>Chemical bindings</td><td>✓<br>(repulsive)</td><td>✓<br>(attractive)</td></tr><tr><td>Biological</td><td>✓<br>(repulsive)</td><td>✓<br>(attractive)</td></tr><tr><td rowspan="3">Contact forces<br>(incl. particle-substrate &amp;<br>particle-particle interactions)</td><td>Capillary</td><td>✗</td><td>✓</td></tr><tr><td>Adhesion</td><td>✗</td><td>✓</td></tr><tr><td>Friction</td><td>✗</td><td>✓</td></tr><tr><td></td><td>Collision exchange</td><td>✓</td><td>✗</td></tr><tr><td></td><td>Mechanical (vibration)</td><td>✓</td><td>✗</td></tr></tbody></table>

features, referred to as coherent structures. This is especially the case in the near-wall region of turbulent flows, where the presence of a wall surface induces a highly anisotropic flow, and in which the dynamical role played by long-lasting and intense vortices often named hairpin-vortices (see also Fig. 26) has been studied in depth. This is an illustration that the distinction between mean values and fluctuations does not correspond to a clear-cut separation between order and disorder and one of the intricate characteristics of turbulent flows is, precisely, that these fluctuations have non-zero space and time correlations which makes them inherently different from ‘heat bathes’ as in classical thermodynamics.

**Role in resuspension:** These hairpin vortices induce near-wall velocity fluctuations, which are especially strong in the logarithmic layer but are also found within the viscous sublayer (their intensity being smaller than further away from the wall). Therefore, particles large enough to interact with the regions of strong fluctuations associated to these near-wall coherent structures are more easily set in motion (see for instance [122] and references therein). This is typically the case for particles larger than a few tens of micrometers while colloidal particles (i.e., with sizes smaller than a few micrometers) are rather deeply embedded within the viscous sublayer so that their resuspension is much less dependent on near-wall structures. Such phenomena correspond to the so-called episodic bursts in resuspension events [207, 109, 208, 209, 210].

Another aspect related to turbulence is that it can be induced by thermodynamic (essentially heat) effects, i.e., when surfaces are warmer/cooler than the fluid. For instance, in wildfires, the creation of an updraft at the flame location induces a convergent wind field, resulting in winds close to the surface that differ from the undisturbed ones if the fire was not present. Turbulence is also more intense at the fire front [211]. The net result is that prescribed fires and wildfires promote resuspension, especially of soil and dust particles, due to the increased wind speeds and turbulence as well as because of the presence of a vertical updraft.

**Modeling turbulent flows:** The basic equations governing fluid flows are well-established: these are the Navier-Stokes equations. In turbulence modeling, the issue is therefore to come up with reduced
<figure><img src="image_34.png" /><figcaption>Figure 26: Sequence of two images showing the instantaneous vortical structures in the vicinity of a hydrogel sphere (labelled $S_4$, in yellow with black markers to indicate its change in orientation). The two images are separated by a time $\Delta t^+ = 16.7$. Vortical structures are visualized by iso-surfaces of the second invariant of the velocity gradient tensor (Q-criterion) that were overlaid by the fluid vorticity $\omega_1^+$. Reprinted with permission from [206]. Copyright 2022, Cambridge University Press.</figcaption></figure>

statistical descriptions, formulated in terms of a tractable number of degrees of freedom, still able to capture relevant flow characteristics. The conventional approach consists in applying a statistical operator to the Navier-Stokes equations to derive exact but open equations and, then, to introduce closure proposals. This leads naturally to field-based methods which are solved by using mesh-based numerical methods. In recent years, particle-based models, such as probability density function (PDF) or smooth particle hydrodynamics (SPH) approaches, have also been developed. Their similarities and differences with respect to classical ones have been analyzed in a recent review [9] and are not repeated here. Indeed, to provide the necessary background for the introduction of particle resuspension models in Section 4.3, it is sufficient to limit ourselves to the most usual turbulence models, which are illustrated in Fig. 27. Interested readers are referred to other reviews [9], as well as existing books on the topic among which [205, 212, 204, 213].

<figure><img src="image_35.png" /><figcaption>Figure 27: Summary of the main various levels of description for the computation of a turbulent flow.</figcaption></figure>

The classical turbulence models pictured in Fig. 27 can be outlined as follows:

A. In single-phase turbulence, the most detailed level of description consists in solving numerically the Navier-Stokes equations without any turbulence model. This implies that all the degrees of freedom of a turbulent flow must be explicitly captured [214] and this approach is referred to as direct numerical simulations (DNS). Given initial and boundary conditions, each DNS computation corresponds to a single realization of the flow and one can extract statistics either by averaging over several realizations or by using time-averages for locally stationary flows or space-averages for homogeneous flows. In the
case of an incompressible flow, where the fluid density $\rho_f$ is constant, the Navier-Stokes equations read:

$$ \nabla \cdot \mathbf{U}_f = 0, \qquad (39a) $$

$$ \frac{\partial \mathbf{U}_f}{\partial t} + \mathbf{U}_f \cdot \nabla \mathbf{U}_f = -\frac{1}{\rho_f} \nabla P_f + \nu_f \nabla^2 \mathbf{U}_f + \mathbf{F}, \qquad (39b) $$

where $P_f$ is the fluid pressure field and $\mathbf{F}$ possible external forces. The first equation, Eq. (39a), corresponds to the conservation of mass written for an incompressible fluid. The second equation, Eq. (39b), corresponds to the conservation of momentum and relates the fluid acceleration to all forces acting on the fluid (including pressure, viscous constraints and external forces, like gravity). Additional boundary conditions are applied when dealing with wall-bounded flows (e.g., by imposing a no-slip velocity condition on wall surfaces).

In dispersed two-phase flow, an additional complexity comes from the treatment of solid particles and two different levels of descriptions can be found:

i - When the particle size is larger than the Kolmogorov scales, finite-sized particles can be explicitly considered as such in the simulation. This means that the grid size used in the computation of the fluid flow is smaller than the characteristic particle size. Hence, additional boundary conditions are imposed along the grid points where particles are present (to avoid penetration of the fluid inside the particle). As a result, the flow around each particle is resolved (as shown in the left panel of Fig. 20): such methods are called PR-DNS (for “Particle-Resolved Direct Numerical Simulation”). This approach is consistent with fine descriptions of the particle dynamics, where the hydrodynamic forces due to the relative motion between the fluid and the particle are explicitly calculated by integrating over all the grid points constituting the particle surface (i.e., without resorting to approximate formulas for these hydrodynamic forces) [183, 184]. The only force related to the fluid that has to be modeled is the lubrication force, which occurs at scales smaller than the grid size (thus not resolved here, see [183, 215, 216]). Other non-contact and contact forces can also be added to these descriptions (such as hard-sphere models to avoid the inter-penetration of particles).

ii - The point-particle approximation is widely used for particles smaller than the Kolmogorov scales, leading to the so-called PP-DNS [165] (see also the right panel of Fig. 20). In that case, the flow around each individual particle is not available. Nevertheless, it is possible to calculate two-phase turbulent flows, including modifications due to the presence of a large number of particles in the flow. This is achieved by including in the term $\mathbf{F}$ in the Navier-Stokes equations, cf. Eq. (39b), the forces exerted by the particles on the fluid which can be obtained by averaging the individual force exerted by a single particle on the fluid (which, according to the action-reaction principle, is simply the opposite of the hydrodynamic forces) over all particles present in each grid cell. In that case, the flow is said to include particle feedback on the fluid or two-way coupling. Such PP-DNS are consistent with approximate expressions for the hydrodynamic forces (i.e. those due to the relative motion between the particle and the fluid). In fact, a PP-DNS provides the exact instantaneous fluid velocity at any point in space (including at the position of each individual particle) and no additional model is required to evaluate the hydrodynamic forces acting on particles.

B. At a less detailed level, Large Eddy-Simulation (LES) consists in solving the Navier-Stokes equations but only down to a certain cut-off length. This cut-off length correspond to a spatial filtering operation (the cut-off length happens to be the size of the grid used). This means that “large-scales” motions of the fluid are explicitly computed while residual “small-scales” motions are modeled. Hence, each variable is decomposed into a filtered value $\langle \cdot \rangle_{ls}$ and a residual one $'$, which gives for the velocity field:

$$ \mathbf{U}_f(t, \mathbf{x}) = \langle \mathbf{U}_f \rangle_{ls}(t, \mathbf{x}) + \mathbf{u}'_f(t, \mathbf{x}). \qquad (40) $$

Applying this filtering operation, one obtains the spatially filtered Navier-Stokes equation (neglecting particle feedback on the flow):

$$ \frac{\partial \langle \mathbf{U}_f \rangle_{ls}}{\partial t} + \nabla \cdot (\langle \mathbf{U}_f \mathbf{U}_f \rangle_{ls}) = -\frac{1}{\rho_f} \nabla \langle P_f \rangle_{ls} + \nu_f \nabla^2 \langle \mathbf{U}_f \rangle_{ls}. \qquad (41) $$