Drop laden flows

O.M.H. Rodriguez, P. Angeli, Dominique Legendre, Eric Climent, A. Soldati

► To cite this version:

O.M.H. Rodriguez, P. Angeli, Dominique Legendre, Eric Climent, A. Soldati. Drop laden flows. International Journal of Multiphase Flow, 2025, 191, pp.105284. 10.1016/j.ijmultiphaseflow.2025.105284. hal-05155906

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Drop laden flows

O.M.H. Rodriguez<sup>a, ID</sup>, P. Angeli<sup>b ID</sup>, D. Legendre<sup>c</sup>, E. Climent<sup>d ID</sup>, A. Soldati<sup>e,f ID,*</sup>

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ARTICLE INFO

Keywords:

Polariton

WGM

BEC

A B S T R A C T

Drops play a pivotal role in both environmental systems and industrial processes, where their physical
properties, dynamics, and interactions have far-reaching implications across diverse domains. In environmental
contexts, water droplets are central to critical phenomena such as carbon dioxide exchange at the ocean-
atmosphere interface, cloud formation, and precipitation—all of which are essential to agriculture, climate
regulation, and ecosystem sustainability. In industrial applications, drops underpin technologies such as
spray systems, coatings, inkjet printing, and fuel injection systems. Their behavior governs key processes
like heat transfer, material deposition, and combustion efficiency, offering opportunities for innovation and
enhanced performance across various sectors. The multiscale nature of drop-laden flows — ranging from
molecular-scale interfacial dynamics to macroscale phenomena — poses significant challenges for numerical
or experimental approaches. This necessitates the development of models capable of bridging these disparate
scales effectively. This paper provides a concise overview of the current state of the art in drop-laden flow
research, reviewing experimental and computational techniques. Additionally, it outlines key challenges and
proposes strategic guidelines to shape future investigations, emphasizing the need for interdisciplinary efforts
to address unresolved questions and drive progress in this multifaceted field.

1. Introduction

A drop is a limited quantity of one liquid immersed in another
fluid, which can be either a liquid or a gas. The existence of a drop
is defined by its interface, a macroscopic manifestation of molecular
properties (Höfling and Dietrich, 2024). Interfaces are not inherent
to the drop or the surrounding fluid; rather, they form the boundary
across which momentum, heat, and mass transfer occur. Droplets play
a crucial role in various natural and industrial processes as illustrated
in Fig. 1. Natural examples include the absorption of carbon dioxide
(CO₂) by oceanic spray droplets, and the formation of rain. In industry,
droplets are involved in processes such as combustion, spray drying,
painting, oil-water transport in pipelines, inkjet printing (Lohse, 2022),
and microfluidic (Fig. 1.d) or nanofluidic applications (Fig. 1.e) for
analysis or complex products formulation. Key parameters influencing
these phenomena include droplet size, surface tension, and the flow

regime-whether it is turbulent or laminar. The viscosity of the drop
is a further important parameter that can influence the dynamics of
a drop (Castro-Bolivar and Rodriguez, 2021), and in a number of
biological, industrial and environmental applications non-Newtonian
effects can be important. However, non-Newtonian effects will not be
covered in this paper (See Chabra, 2019; Zenit and Feng, 2018). The
defining characteristic of drop-laden flows is their inherently multi-
scale nature, from large problem domains down to molecular-scale
interface dynamics, encompassing a vast range of physical phenomena.
The challenge in accurately predicting these flows lies in the intri-
cate interplay between microphysical processes and their macroscopic
manifestations. This coupling creates significant obstacles for direct
investigations through simulations or experiments, as neither approach
currently offers sufficient resolution to capture the full scale range of
the phenomena.

<footnote><sup>a</sup> University of São Paulo (USP), São Carlos School of Engineering (EESC), Mechanical Engineering Department, Industrial Multiphase Flow Laboratory (LEMI), Av. Trab. São Carlense, 400 - Parque Arnold Schimidt, São Carlos-SP, 13566-590, Brazil</footnote>

<footnote><sup>b</sup> UCL, ThAMeS Multiphase, Department of Chemical Engineering, Torrington Place, London, WC1E 7JE, UK</footnote>

<footnote><sup>c</sup> Institut de Mécanique des Fluides de Toulouse (IMFT), CNRS UMR 5502, Institut national polytechnique de Toulouse (INPT), Université Paul Sabatier (UPS), Université de Toulouse, Toulouse, France</footnote>

<footnote><sup>d</sup> Univ Toulouse, Inst Mecan Fluides Toulouse IMFT, CNRS, UMR 5502, F-31400 Toulouse, France</footnote>

<footnote><sup>e</sup> Institute of Fluid Mechanics and Heat Transfer, TU Wien, Vienna, 1060, Austria</footnote>

<footnote><sup>f</sup> Department Engineering and Architecture, University of Udine, Udine, 33100, Italy</footnote>

<footnote>☆ Paper prepared for the 50th anniversary of the International Journal of Multiphase Flow.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: oscarmhr@sc.usp.br (O.M.H. Rodriguez), p.angeli@ucl.ac.uk (P. Angeli), legendre@imft.fr (D. Legendre), eric.climent@imft.fr (E. Climent), alfredo.soldati@tuwien.ac.at (A. Soldati).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** (a) Droplet spray using the SAMES technology based on rotating bell for painting application, (b) Taylor bubble rising in an upward vertical flow of an emulsion of oil in water, a three-phase flow common in oil production, LEMI-USP, (c) Breaking wave contributing to the Ocean spray, (d) Droplet formation in microchannels in the presence of surface active agents, ThAMeS Multiphase, Dept Chemical Engineering, UCL, (e) Laser bio-printer developed by Poietis. A laser pulse produces a cavitating bubble able to eject nano-liter droplets, (f) The DC10 Air Tanker 914 dropping a cloud of retardant on the Central Fire in Arizona on June 2020. Photo by JDH Images, (g) Small scale pulsed Liquid-liquid column developed for solvent extraction to recycle plutonium (Credit photo CEA).</figcaption></figure>

This paper was conceived during the IJMF50 conference held in Vienna in 2023 and represents a collaborative effort by the authors to provide a comprehensive overview of topics within their direct area of expertise. These specific subjects have been the focus of continuous research over the years, remain challenging and elusive for precise simulations and experiments, and offer significant potential for future advancements in the field.

## 1.1. *Drop laden flow applications*

In this section, we explore few applications of drop-laden flows across different fields. We will present the role of sprays in industrial processes, microfluidics for precise fluid manipulation, liquid-liquid extraction for separation techniques, oil transport in pipelines, emulsions in food and pharmaceuticals, and natural phenomena such as rain formation and cloud dynamics. Each of these areas demonstrates the importance of understanding the dynamics of drop-laden flows for both practical and scientific advancements.

### *Sprays*

Spraying systems are widely used in industry for different applications resulting in different technologies and sizes for the injectors (Fig. 1.a). As an example, spray formation by liquid jet atomization in a cross flow has been extensively considered for millimeter-sized jets (Broumand and Birouk, 2016) in the field of combustion, but it also concerns application with liquid jets up to three order of magnitude larger as encountered in Airtanker process (Legendre, 2024) as shown in (Fig. 1.f). The final droplet size distribution in sprays results from primary and secondary atomization mechanisms involving the deformation up to breakup of liquid jets, sheets and large droplets following a similar process where deformation forces (viscous shear,

inertia, gravity) have to overcome cohesion forces such as surface tension (Villermaux, 2020; Ni, 2024).

### *Microfluidics*

Microfluidic channels have been widely used to produce drops with controlled sizes (Fig. 1.d) and narrow size distributions, finding applications in emulsification, ink jet printing, cosmetics and healthcare. Surfactants are commonly added to modify the interfacial properties and enhance the stability of the dispersion. The presence of surfactant molecules significantly affects the non-linear drop formation process and thus the final size (Chen et al., 2015). Recently, the use of colloidal particles as interface stabilizers has emerged as an alternative to surfactants. Unlike surfactant molecules, solid particles can irreversibly adsorb at the interface, and induce viscoelastic effects which modify the drop pinch off dynamics (Bazazi et al., 2023; Chagot et al., 2024) and final drop size.

### *Liquid-liquid extraction*

Although it is a mature technology, solvent extraction is still an important field for investigation. The main challenge is to improve recovery from dilute and multicomponent ores and wastes, while minimizing the required amounts of water, chemicals and organic solvent. For example the nuclear industry is developing future nuclear systems involving repeated recycle of plutonium. For this purpose the efficiency of new solvents are tested in liquid-liquid column as illustrated in Fig. 1.g. In such solvent extraction columns, the contact surface between the two immiscible liquids is enhanced by dispersing (mechanical stirring, pulsation, etc.) one phase in the form of droplets in the second. The efficiency of the system is then directly related to the transfer between the dispersed phase and the continuum phase enhanced by chemical reaction (Rachih et al., 2020; Godé et al., 2024).
## Oil transport and emulsions

In oil transport pipelines, the presence of water droplets dispersed in oil can significantly influence the flow dynamics, leading to phenomena such as slug flow, drop coalescence, and phase separation (Masalova et al., 2003). These processes must be understood in order to optimize the efficiency and safety of oil transport systems. For example, the formation of water-in-oil emulsions can increase friction, cause corrosion, and present operational challenges. Furthermore, controlling the size distribution and stability of droplets in pipelines is crucial for mitigating the risks of flow blockages and minimizing maintenance costs. Both numerical simulations and experimental studies are essential for investigating the complex interactions between droplets and the surrounding oil phase, providing valuable insights that can guide pipeline design and flow management strategies.

While emulsions are particularly relevant in oil transport, they also play a crucial role in various other fields, such as environmental (Masnadi et al., 2021) and industrial applications (Chappat, 1994), including food (Dalgleish, 2001) and pharmaceutical industries (Barkat et al., 2011). These systems consist of closely packed droplets suspended in a continuous liquid phase (Banhart et al., 2008), with volume fractions ranging from less than one percent to as high as 30–35% (Denkov et al., 2009). Despite their apparent similarity to common fluids, emulsions involve complex and dynamic physical behavior. Understanding emulsions requires both experimental and numerical approaches (Bakhuis et al., 2021; Yi et al., 2022) to address several significant challenges. A key limitation is the scarcity of experimental data, partly due to the destabilizing effects of gravity and optical challenges related to turbidity (Banhart et al., 2008). Moreover, the coalescence process remains poorly understood, particularly the phases of film drainage and rupture (Kamp et al., 2017; Soligo et al., 2021; Beckedorff et al., 2025), with various mechanisms still under investigation (Chen et al., 2004; Perumanath et al., 2019). The small-scale physics of coalescence becomes even more complex when surfactants are present, as they strongly influence the interactions at the interface (Langevin, 2020; Yang et al., 2009), further complicating the study of these systems (Zembyla et al., 2020; Velimirov, 1982).

## Natural phenomena

Droplet can be observed in nature from rainfall precipitations to ocean spray aerosols (Fig. 1.c). Water precipitation is a natural process involved in the global water cycle directly impacting terrestrial hydrology. The prediction of precipitation rates remains an open challenge despite it has received a lot of interest from the scientific community over the last decades. During atmospheric rainfall, the cloud evolution is controlled by droplet bouncing, coalescence, or breakup (Testik and Barros, 2007). The canopy storage capacity is the amount of water resulting from the deposition of the droplets on vegetation after rainfall events. It depends on several parameters such as the rainfall intensity, the wind conditions and the types of vegetation (Holder, 2013). The process of droplet impact with leaves presents specific behaviors related to the leaf mechanical response (Gilet and Thoroddsen, 2015) when compared to droplet impact on solid surfaces (Josserand and Thoroddsen, 2009). In high wind speed conditions, sub-millimetric droplets are generated at the surface of the ocean by ligament and bag breakup, and are transported in the atmospheric boundary layer. Droplet are also formed in the marine atmosphere by bursting bubbles resulting from breaking waves. The two mechanisms result in a sea spray process having an important contribution to the air-sea fluxes (Veron, 2015).

## 1.2. Drop-laden flow classification by length scales

The examples described in the preceding section can be classified using important length scales in these flows as illustrated in Fig. 2. One such scale is the droplet diameter, (or its equivalent diameter when the drop is deformed) $d = (6\theta_d/\pi)^{1/3}$ where $\theta_d$ represents the droplet

volume. This diameter can be compared to both the system size, $D$, and the characteristic length scales of the flow field, which govern the dynamics of droplet interactions. In essence, the droplet mean free path is influenced by the system's constraints and the scales of the surrounding flow.

Depending on the relationship between $d$, $D$, and the characteristic flow scales, droplets can exhibit distinct behaviors. For instance, when $D < d$, as in microfluidic applications, droplets are confined by the system size, resulting in restricted movement and interactions primarily dictated by the boundaries. In cases where droplets are free to move within a domain of size $D$ (that also imposes the large-scale flow characteristic $L$), such as in pipe flows, their behavior is shaped by the imposed flow structure. Alternatively, when $d \ll L$, droplets are predominantly influenced by the surrounding flow field, which is often turbulent in practical scenarios. Multiphase turbulent flows are often more complex than the ideal view of homogeneous isotropic turbulence due to new scales related to the presence of the dispersed phase, momentum transfer, and collective effects on the surrounding fluid.

This classification highlights the multiscale complexity of drop-laden flows, where the interplay between droplet size, system constraints, and flow characteristics dictates the governing dynamics. Understanding these relationships is essential for designing efficient systems and accurately modeling droplet behaviors across a variety of applications, from microfluidics to turbulent multiphase flows.

## 2. Experimental techniques and measurements

Several techniques have been used to obtain drop size distributions in liquid dispersions. These can be local sensors or tomographic systems employing a suite of sensors. They are based on the differences of electrical (conductance or capacitance) or optical properties between the phases or on local heat transfer as in hot film anemometers. Often these sensors are intrusive and can disturb the flow field, thus affecting the accuracy of the measurements. Techniques based on radiation such as magnetic resonance imaging (MRI), and gamma-ray or X-ray attenuation techniques are relying on the density differences of the materials; for liquid-liquid flows where the density differences between the two phases are small, their application is limited (Iannello et al., 2022).

Single and multi-tip needle and optical fiber probes (Kim et al., 2000; Descamps et al., 2007; Hu and Angeli, 2008; Manera et al., 2009) are examples of well consolidated instruments for time-averaged local electrical-impedance measurements of bubble size. Electrical resistance (ERT) or capacitance (ECT) tomography (Abdulkareem, 2010; Matusiak et al., 2010; Abdulkadir et al., 2011, 2013; Olerni et al., 2013; Guo et al., 2019) have been used to measure and reconstruct bubble and drop size, but spatial resolution, which is related to the inverse problem of image reconstruction, and dependence on the flow pattern are still evident drawbacks. Conductivity- or capacitive-based wire-mesh technology (WMS) has been used for bubble and drop size measurement and distribution with spatial resolution ranging from 5 to 2 mm (Prasser et al., 2001, 2005; Pietruske and Prasser, 2007; Da Silva et al., 2011; Parsi et al., 2015; Hernández Cely et al., 2018; Zhang et al., 2020; Lee et al., 2021; Velasco-Peña et al., 2022; Zhai et al., 2023).

Quantitative measurements of liquid dispersions based on imaging have followed the evolution of digital cameras, image processing capacity, high energy and frequency lasers (Dong et al., 2019). These approaches require transparent test sections. Imaging is limited to low volume fractions of the dispersed phase, usually of the order of a few percent. Some of the consolidated techniques that require the use of laser are Phase Doppler Anemometry (PDA) and Planar Laser Inducer Fluorescence (PLIF). In the latter a fluorescent dye is added in one of the phases while a laser sheet illuminates the test sections (Voulgaropoulos et al., 2019; Charogiannis et al., 2019; Miranda-Lugo, 2024). A recent and promising technique to improve the understanding and modeling of the phenomenon of bubble and droplet deformation and
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Typical droplet flow configurations depending on the droplet diameter *d* with respect to the other characteristic length scales in the flow. (a) Microfluidics: When the droplet diameter *d* is comparable to or larger than the system size *D*, the droplet is confined, leading to constrained motion and interactions typically observed in microfluidic systems. (b) Pipe flow: In this regime, the droplet size *d* is small relative to the system size *D~L* (the characteristic length of the flow domain that controls the largest flow scale *L*), but the droplet dynamics are influenced by large-scale flow features such as turbulence and shear. (c) Dispersed flow: Here, the flow is unbounded and droplets are influenced by the surrounding turbulent flow, with *d* ≪ *L*, as seen in many multiphase flow systems. Key length scales include *D* (system size), the smallest scale of the carrying flow η (this will be close to the Kolmogorov micro-scale in the dilute regime), λ (Taylor scale), and *L* (integral scale), which govern the interactions and behavior of the droplets at different scales within the flow field.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Spatial and temporal resolutions, and estimated cost of available experimental techniques for drop size measurement. The resolution bands were chosen according to the most common and representative applications, and the cost is an estimation based on standard equipment. Advanced systems, such as multiple-camera setups or micro-PIV/PLIF systems, can achieve exceptionally high spatial resolution. The dashed box stands for recent ultra-fast and -resolution X-ray techniques.</figcaption></figure>

break up due to turbulence is the use of multiple-camera setups and 3-D reconstruction algorithms that use multiple images (Qi et al., 2020; Moustafa et al., 2022; Ni, 2024).

The application of ultrasound has been extended to obtain particle size distributions in dispersed flows (Hossein et al., 2021). This technique relies on measuring the attenuation of an ultrasound wave as it propagates from a transmitter to a receiver through the dispersed medium (Hossein et al., 2023). The minimum detectable drop size is influenced by the frequency of the ultrasound wave. Studies have shown that ultrasound waves in the frequency range of 0.5 to 4 MHz are capable of detecting droplets as small as 0.2 mm in dispersions formed within stirred vessels.

Measurement techniques utilizing ionizing radiation offer several advantages, including non-intrusiveness, the ability to operate without

transparent test sections or fluids, and relatively high spatial resolution achievable through specially designed collimators. These techniques have been employed not only to determine in-situ volumetric fractions (Rodriguez and Oliemans, 2006) but also to map dispersed phase fraction distributions across the cross-sections of pipes (Kvandal et al., 2000; Elseth, 2001; Amundsen-Lene, 2011; Shmueli et al., 2015; Vorobieff and Brebbia, 2018; Alvarez-Pacheco et al., 2024). Chordal distributions derived from these measurements can be used to calculate entrainment fractions of droplets in a turbulent continuous gas phase or bubbles in a turbulent continuous liquid phase. Ultra-fast and -resolution X-ray techniques, such as ultra-small angle X-ray scattering, stand out as capable of delivering exceptionally high temporal and spatial resolution, but they can be considered as recent and not yet sufficiently widespread among the community (Kastengren et al., 2017;
<figure><img src="image_7.png" /><figcaption>Fig. 4. Numerical methods for the description of flow field and interfacial topology in multiphase turbulence. The top part shows the available approaches for the description of the flow field: (i) one-fluid approach; (ii) ghost-fluid approach; (iii) lattice-Boltzmann methods (LBM); (iv) smoothed-particle hydrodynamics (SPH). The bottom part shows the available methodologies to describe interface topology: (i) front-tracking; (ii) volume of fluid; (iii) level-set; (iv) phase-field; (v) fluid-particles. Arrows linking the top and bottom parts (flow and phase description) of the figure identify how these methods are commonly coupled.</figcaption></figure>

Chen and Llavsky, 2024) A detailed review on X-ray visualization techniques can be found in Heindel (2024).

In Fig. 3 the positioning of state-of-the-art techniques applied to characterize dispersed flows is illustrated, considering their spatial and temporal resolution as well as cost.

### 3. Interface-resolved simulations of multiphase turbulent flows

Simulation of multiphase turbulent flows requires numerical methods suitable to describe the different physics aspects of the system. Characteristics of these methods are schematically depicted in Fig. 4. The methods must describe first flow field in each phase, and second the coupling of the phases at the interface, so that each method must handle abrupt changes of density and viscosity at the interface and must also include the action of surface tension (Prosperetti and Tryggvason, 2009; Soligo et al., 2021), while resolving all the scales of both the velocity field and the interface deformation. The oldest approach is the one-fluid approach, which relies on the solution of a single set of Navier-Stokes equations in the entire domain (usually performed on a structured grid). An alternative approach is the two-fluid approach (Prosperetti and Tryggvason, 2009), which relies on the solution of multiple sets of Navier-Stokes equations, one for each phase, coupled at the interface (Fedkiw et al., 1999; Glimm, 1982; Glimm and Mcbryan, 1985; Udaykumar et al., 1997). Lattice-Boltzmann methods (LBM) can also be used — where Navier-Stokes equations are not solved but are approximated via streaming and collision operators (Nourgaliev et al., 2003) — and also smoothed-particle hydrodynamics (SPH)—where Navier-Stokes equations are approximated using discrete Lagrangian particles (Liu and Liu, 2010; Monaghan, 2005, 2012).

Each of these methods must be coupled with a methodology for the description of the interface topology, and available methods can be broadly classified in two families: interface tracking and interface capturing methods. The fundamental difference resides in the definition

of the interface: interface tracking approaches explicitly follow the position of the interface with Lagrangian markers, while interface capturing methods define the interface position as a prescribed value of a color function/concentration field. While in principle any combination of a method for the flow description and interface topology is possible, some combinations are more suitable than others, as shown in Fig. 4. Considering interface tracking methods, the front-tracking has emerged as one of the most popular: it is based on the advection of a set of Lagrangian markers that define the instantaneous position of the interface and it has been mainly used in combination with the one-fluid approach (Tryggvason et al., 2001; Unverdi and Tryggvason, 1992). Moving to interface capturing methods, we have three main approaches: (i) volume-of-fluid, which is based on the advection of a phase indicator that represents the local concentration of each phase (Scardovelli and Zaleski, 1999; Hirt and Nichols, 1981); (ii) level-set methods (Osher and Sethian, 1988, 1994; Osher and Fedkiw, 2001), which relies on a smooth color function characterized by the property $|∇φ| = 1$; (iii) phase-field approaches (Badalassi et al., 2003; Jacqmin, 1999; Kim and Lowengrub, 2005; Roccon et al., 2023), which employ a concentration field that is constant in the bulk of each phase and undergoes a smooth transition across the interface to describe the interface shape. All these interface capturing approaches have been successfully used in combination with the one-fluid approach and also combined, e.g. CLS-VoF. When ghost-Fluid method are employed (Fedkiw et al., 1999; Glimm, 1982; Glimm and Mcbryan, 1985; Udaykumar et al., 1997), level-set is the most common choice, although in principle any interface tracking or capturing method can be also used. For LBM, the available approaches (Zhang et al., 2001; Chin et al., 2002; Inamuro et al., 2004; Lamorgese et al., 2017) roughly resemble the phase-field approach, with a color function that is used to define the interface topology. Finally, when SPH techniques are used, colored fluid particles are used to distinguish between the two phases; this resembles the idea behind interface capturing method, even though is important to note that is based on Lagrangian particles (Pozorski and Olejnik, 2023).
Please refer to the following works for an overview of the weaknesses and strengths of the different methods (Bilger et al., 2017; Mirjalili et al., 2019; Rossi et al., 2025).

The numerical challenges currently addressed when investigating flows with droplets are of two types:

**Mesh refinement:**

Making more efficient resolution of the interface and fluid scales, in particular going beyond actual computer capacity when using uniform grids, with the development of Adaptive Mesh Refinement (AMR) methods (Pairetti et al., 2020) in order to avoid the development of sub-grid modeling requiring a coupling between interface and turbulent fluctuations (Liovic and Lakehal, 2012). Such improved resolutions are in particular well adapted to provide converged evolution of interfaces coalescence or breakup (Ni, 2024).

**Interface phenomena:**

Considering more complex physical, thermal and chemical phenomena involved at interfaces such as evaporation or condensation, solutal or thermal Marangoni flows as well as heat and Mass transfer. These mechanisms can be potentially all coupled as illustrated by the Leidenfrost effect occurring when a droplet is levitating on its own vapor over a hot plate (Mialhe et al., 2023).

# 4. Drop diameter distribution: A multiscale challenge for experiments and simulations

Referring to Fig. 2, we will examine the key challenges encountered in determining the drop diameter distribution under the conditions being studied. Specifically, we will discuss the limitations and complexities associated with the currently available methods for predicting drop size distribution in three distinct scenarios: microchannels, oil transport applications, and open turbulent flows.

## 4.1. Drop diameter in microchannels

Different channel inlet configurations have been implemented to generate drops including T- and Y-junctions, cross and co-flow inlets, while membranes and step emulsification have been used to generate large numbers of drops (Charcosset et al., 2004). During drop formation in microchannels different regimes can occur, namely squeezing, dripping, jetting and threading (Kalli and Angeli, 2022). The squeezing regime appears at low flowrates of both liquids, where the dispersed phase has time to grow and obstruct the channel, with drop detachment caused by the build up of pressure behind the drop. The dripping regime appears for a wide range of continuous phase flowrates and medium dispersed phase flowrates; the drop detaches when the shear forces from the continuous phase overcome the interfacial tension ones. The jetting regime appears at similar continuous phase flowrates to the dripping regime, but at high dispersed phase flowrates. In this case, the increased continuous phase flowrate pushes the dispersed phase inside the main channel where pinch-off takes place. The critical jet length before pinch-off is proportional to the visco-capillary time scale, $t = D\mu/\sigma$, where $\mu$ is the viscosity of the dispersed phase. At low continuous phase flowrates and for increasing dispersed phase ones, the threading regime occurs, and no drops form.

The presence of surfactants reduces the areas occupied by the squeezing and dripping regimes in maps with the flowrates of the two liquids as coordinates, while a tip streaming regime is also seen. Flow pattern maps are often presented with the phase flowrates as coordinates, while the use of dimensionless numbers such as Ca, We and Re or their combinations has been proposed to generalize the applicability of the maps.

During drop formation in a liquid in microchannels, a fresh interface is continuously formed. When surfactant molecules are present in either

of the two phases, equilibrium interfacial tension values will only establish if the mass transport and adsorption of surfactants to the interface is fast compared to the drop formation time or the rate at which the new interface forms. In microfluidic channels, drop formation times are often short, leading to interfacial tension values different from the equilibrium ones, which are often used in the calculation of capillary number in predictive equations for the droplet size (Kalli et al., 2022). It has been shown that when dynamic interfacial tension values are used instead of the equilibrium ones, the boundaries between the dripping and jetting regimes of drop formation in systems with different surfactants, collapsed (Kalli et al., 2022). In addition, shear stresses from the continuous phase can redistribute surfactant molecules at the interface and induce interfacial tension gradients and Marangoni stresses (Kovalchuk and Simmons, 2021; Roumpea et al., 2019). It is important, therefore, to characterize the mass transfer and kinetics of surfactants during drop formation.

## 4.2. Drop diameter prediction in pipes

The experimental techniques and measurements discussed in Section 2 are applied having in mind the prime objective of providing data to allow the development of phenomenological models or empirical correlations for drop diameter prediction. Innumerable applications require drop size information, and in as many industries and sectors as one can imagine, as pharmaceutical, automotive, food, chemical, energy, medical, even art, to mention just a few. In Appendix one can see a compilation of phenomenological models and empirical correlations for drop size calculation in pipe flow. The former is usually based on the seminal model of Hinze (1955).

The dimensionless drop diameter, as predicted by several models from the literature, is normalized by the pipe diameter and plotted as a function of the product of the Reynolds and Weber numbers in Fig. 5. The models by Acrivos and Lo (1978) and Razzaghi and Shahraki (2009) are limited to continuous laminar flow, while the others apply to continuous turbulent flows. Please note that all cases consider liquid-liquid flows in pipes, with the exception of Castro-Bolivar and Rodriguez (2021), which involves a liquid-liquid-gas flow. By inspecting Fig. 5, the first conclusion is that the predicted dimensionless drop diameter varies significantly depending on the expression taken to calculate it. Considering only turbulent continuous flow, relatively good agreement is observed in the range $10^6 < \text{ReWe} < 10^{10}$, but considering the models of Hinze (1955), Kubie and Gardner (1977), Karabelas (1978), Brauner (2001), Kouba (2003), Torres et al. (2016), and Castro-Bolivar and Rodriguez (2021). The models of Sleicher (1962), Davies (1987), and Angeli and Hewitt (2000) also present good agreement among themselves in the range $10^4 < \text{Re We} < 10^8$, but they disagree with the former team. The model of Ward and Knudsen (1967) strongly disagrees in comparison to all others.

The Ward and Knudsen (1967) model uses Einstein's model to calculate the mixture viscosity, which works up to 5% of dispersed liquid volumetric fraction, the two-phase mixture though, was circulated with the help of a single pump, i.e., the additional shear of the pump blades causes intense droplet breakup, generating smaller droplets than would be expected if the phases were injected separately. The Sleicher (1962), Angeli and Hewitt (2000), and Davies (1987) models are adjusted for viscosity ratios near 1 and volumetric fractions lower than 30%. Models that include the effect of the dispersed droplet viscosity in some way notably provide larger droplet diameters than those that do not. In the inviscid-droplet models, interfacial tension is the only one responsible for the droplet stability. However, the viscous-droplet models implicitly include the effect of the droplet's internal viscous stresses, which contribute to droplet stability.

Even considering the droplet diameter predictions of only the viscous-droplet models of Fig. 5, one may argue that the disagreement is still significant. Notice that each type of dispersed turbulent flow encountered in practice, such as in a pipeline or mixing tank, should have a semi-empirical expression for predicting droplet diameter, i.e., there are coefficients to be adjusted based on experimental data for each specific condition.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Drop diameter normalized by the pipe's internal diameter in dispersed pipe flow as a function of the product of Reynolds and Weber numbers according to different models from the literature. Two examples of water-in-oil dispersions for which the continuous oil flows were laminar (left), and several oil-in-water flows for which the continuous water flows were turbulent (right). The phase inversion model of Castro B. and Rodriguez (2021) was applied to a typical case (dashed vertical line), but the phase inversion range of all tested cases is $10^2 < ReWe < 10^4$.</figcaption></figure>

### 4.3. Drop diameter and drop size distribution in turbulence

There have also been numerous studies of the fragmentation of liquid masses in various physical configurations (Villermaux, 2007; Gorokhowski and Herrmann, 2008). One configuration that attracts attention is droplets fragmentation by turbulence. When a droplet is exposed to a strong deformation due to gravity, a shear flow or an intense turbulent environment, breakage can occur. Different models have been proposed to describe droplet breakup and we refer the reader to Risso (2000) and Ni (2024) for detailed reviews on the subject. The simplest approaches consist in proposing limit values for the Bond, Capillary and Weber numbers to describe limit deformation induced by gravity, viscosity or inertia, respectively (Risso, 2000). For turbulent flows, according to the Hinze–Kolmogorov theory (Hinze, 1955), droplet breakup is induced by turbulent eddies of size comparable to droplet size, resulting in a turbulent Weber number as

$$W_e t = \frac{2\rho\epsilon^{2/3}d^{5/3}}{\sigma} \qquad (1)$$

where the corresponding difference of the velocity fluctuations $\delta u^2(d, x, t)$ at the droplet scale is related to the dissipation rate $\epsilon$ assuming that turbulence is isotropic and within the inertial turbulent subrange, so that $\delta u^2(d, x, t) = \|u'(x + d, t) - u'(x, t)\|^2 \approx 2(\epsilon d)^{2/3}$. The droplet is expected to break if $W_e t$ exceeds a critical Weber number $W_{eC}$ corresponding to a droplet size larger than the Hinze scale $d_H$:

$$d_H = W_{eC}^{3/5} \left( \frac{\sigma}{2\rho\epsilon^{2/3}} \right)^{3/5} \qquad (2)$$

A dynamic description of droplet breakup consists in considering the droplet as a linear oscillator where the mass of the oscillator is the droplet mass plus the added mass of the surrounding liquid, the stiffness is the surface tension, the dissipation comes from the liquid viscosities and the external forcing is imposed by the instantaneous local turbulent fluctuations (Lalanne et al., 2019; Ni, 2024). The normalized amplitude of droplet oscillation $\xi^* = \xi/d$ is then given as a function of the

normalized time $t^* = t\omega_2$ as:

$$\frac{d^2\xi^*}{dt^*2} + 2\frac{D_2}{\omega_2}\frac{d\xi^*}{dt^*} + \xi^* = K\frac{W_e t}{W_e 2} \qquad (3)$$

where $\omega_2$, $D_2$ and $W_e 2 = \rho\omega_2^2 d_b^3/\sigma$ are frequency, damping rate and Weber number related to the free droplet mode 2 oscillation. A critical deformation $\xi^* = 2.5K/W_e 2$ is proposed for the breakup criteria (Lalanne et al., 2019).

This dynamic framework illustrates the complexities involved in predicting droplet size distributions, which serve as a prime example of the challenges faced by current numerical methods in computational fluid dynamics. The first challenge pertains to grid resolution for accurately resolving coalescence and breakup events: what is the minimum resolution required to capture the essential physics? The second challenge involves the interaction between resolved interfaces and turbulent fields. This raises a critical trade-off between computationally expensive and often inaccessible Direct Numerical Simulations (DNS), which resolve all relevant scales, and the development of accurate Large Eddy Simulations (LES) that rely on sub-grid modeling to account for small-scale interface-turbulence interactions.

In this context, the fragmentation problem seems to bear a universal character as it is depicted in Fig. 6, where a comparative analysis of drop size distributions obtained from several interface resolved simulation methodologies — VOF and Phase Field — and experimental data. The figure displays distributions for drop-laden turbulent flows, specifically from homogeneous isotropic turbulence (HIT) and turbulent channel flow (TCF) which are compared with experimental and direct numerical simulation (DNS) studies of breaking waves. Additionally, analytical scaling laws for the coalescence- and breakage-dominated regimes — represented by $d^{-3/2}$ and $d^{-10/3}$, respectively — are included to provide a reference framework. This comparison highlights the effectiveness of different approaches in capturing the size distribution trends across regimes and the critical role of the Kolmogorov–Hinze scale in determining the primary mechanisms of drop size variation.

In examining Fig. 6, we can first observe a good alignment across studies, suggesting that the turbulent break-up cascade model is reasonably applicable across a wide range of drop- and bubble-laden
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Comparison of Size Distributions from Experimental and Numerical Investigations of Breaking Waves and Drop-Laden Turbulent Flows: This analysis includes homogeneous isotropic turbulence (HIT) and turbulent channel flow (TCF). The drop diameter is normalized using the Kolmogorov–Hinze scale for each case (estimated where information is lacking), while the distributions are presented in arbitrary units due to varying normalization methods. Data sources include: (Deane and Stokes, 2002) (field observations on breaking waves), Blenkinsopp and Chaplin (2010) (field observations on breaking waves), Callaghan et al. (2014) (experiments on breaking waves), Wang et al. (2016) (DNS-VOF simulations of breaking waves, time-averaged), Deike et al. (2016) (DNS-VOF simulations of breaking waves, time-averaged), Chan et al. [264] (DNS-VOF simulations of breaking waves, time-averaged), Mostert et al. (2022) (DNS-VOF simulations of breaking-waves at higher Reynolds numbers), Di Giorgio et al. (2022) (DNS-VOF simulations of breaking waves at higher Reynolds numbers), Crialesi-Esposito et al. (2023) (drops in homogeneous isotropic turbulence, DNS-VOF, 10% volume fraction), Mukherjee et al. (2019) (drops in homogeneous isotropic turbulence, DNS-LBM, case P2), Soligo et al. (2019a) (surfactant-laden drops in turbulent channel flow, higher surfactant strength case), Chan et al. (2021b) (DNS-VOF simulations of breaking waves) and Cannon et al. (2024) (DNS-VOF of droplet morphology in homogeneous isotropic turbulence). Analytic scaling laws for coalescence- and breakage-dominated regimes, $d^{-3/2}$ and $d^{-10/3}$, are included for reference. A good agreement is observed in the breakage-dominated regime, particularly for drops larger than the Kolmogorov–Hinze scale.</figcaption></figure>

turbulent flows (Chan et al., 2021a,b). In addition, we observe the emergence of two distinct regimes influenced by the Kolmogorov–Hinze (KH) scale, $d_H$, which represents the critical diameter below which a drop or bubble is unlikely to break, based on the KH framework (Kolmogorov, 1949; Hinze, 1955). For drops and bubbles smaller than the Kolmogorov–Hinze scale, a coalescence-dominated regime emerges (left side of Fig. 6). In this regime, drop breakage is infrequent since the size is below the critical threshold; instead, size changes occur primarily through coalescence with other drops. Conversely, for drops and bubbles larger than the Kolmogorov–Hinze scale, a breakage-dominated regime is observed, where size change is governed predominantly by breakage.

In the coalescence-dominated regime, data trends are difficult to infer due to the variability across experimental, numerical, and analytical data sources and specifically, at very small diameters, a noticeable deviation appears between the phase-field simulation results and the analytical scaling law, which can be attributed to two main factors: (i) an overestimation of coalescence efficiency in interface-capturing simulations and (ii) the grid resolution limits of Eulerian methods, which restrict the smallest resolvable drop or bubble size. Additionally, it should be noted that due to the grid requirements of direct numerical simulations (DNS), the scale separation between the Kolmogorov–Hinze and dissipative Kolmogorov scales is limited to about one order of magnitude in the latest simulations. This constraint restricts the observable range for the coalescence-dominated regime.

In the breakage-dominated regime, all simulation results show strong agreement with experimental data, analytical scaling laws, and numerical results from various methods, underscoring the accuracy of the current interface-resolved simulation techniques in capturing breakage events. Notably, despite the different flow configurations (e.g., breaking waves, homogeneous isotropic turbulence, and turbulent channel flows), a good alignment across studies is observed, suggesting that the turbulent break-up cascade model is reasonably applicable across a wide range of drop- and bubble-laden turbulent flows (Chan et al., 2021a,b). An important point to underline is the capability of VOF methods to capture better, compared to phase field methods, the drop size distribution at the smallest scales.

# 5. Dynamics, heat/mass transfer and chemical reaction at the droplet scale

We report on recent results and discuss open questions related to droplet motion coupled to heat or mass transfer in an unbounded fluid flow, as depicted in Fig. 2.c.

## 5.1. Droplet motion

The successive bifurcations of the wake behind a fixed object or its path instabilities when freely moving have been widely studied for both solid spheres and bubbles (see Ern et al. (2011) for a detailed review on the subject). These observations can be applied to droplets of very high and small viscosity ratio $\mu^*$, respectively. Fig. 7 (bottom row) shows an example of a 3D flow that develops around a viscous drop (here $\mu^* = 20$) with a large Reynolds number ($Re = 300$) showing a very similar structure of the streamlines when compared to a solid sphere. The case of the intermediate viscosity ratio $\mu^* = O(1)$ has been much less considered. However, recent investigations have revealed the development of an internal 3D bifurcation for such a viscosity ratio (Edelmann et al., 2017; Rachih et al., 2020) as illustrated in Fig. 7, which surprisingly results in an increase in both the drag (Godé et al., 2025) and the lift forces (Shi et al., 2024). Both effects are the consequence of the contraction of streamlines in the near wake generated by the 3D recirculation inside the droplet.

The motion of a droplet is generally predicted using the second Newton's law as

$$ (\rho_d + C_M \rho_f) \theta_d \frac{d\mathbf{u}_d}{dt} = (\rho_d - \rho_f) \theta_d g \quad (4) $$

$$ + 2\pi\mu d D(\mathrm{Re}, \mu^*, \rho^*) (\mathbf{U} - \mathbf{u}_d) \quad (5) $$

$$ + 3\pi\mu d \int_{t_0}^{t} \mathcal{K}(t - \tau) \left( \frac{\partial \mathbf{U}}{\partial \tau} - \frac{d\mathbf{u}_d}{d\tau} \right) d\tau \quad (6) $$

$$ + \rho_f \theta_d (1 + C_M) \left( \frac{\partial \mathbf{U}}{dt} + \mathbf{U} \cdot \nabla \mathbf{U} \right)_{\mathbf{x}_d} \quad (7) $$

$$ + C_L (\mathrm{Re}, \mu^*, \rho^*) \rho \theta_d (\mathbf{U} - \mathbf{u}_d) \times \mathbf{\Omega}. \quad (8) $$

On the left-hand side, the mass involved in the droplet inertial term is composed of its own mass and the contribution of the added-mass effect, $C_M$ being the so-called added-mass coefficient. On the right-hand
<figure><img src="image_10.png" /><figcaption>Fig. 7. Typical flow visualizations using streamlines plotted inside (red) and outside (magenta) a viscous droplet set fixed in a uniform flow from left to right (Shi et al., 2024). Top row: axisymmetric flow for $Re = 200$, $\mu^* = 0.5$ and internal Reynolds-number $Re' = 200$. Middle row: 3D flow resulting from an internal bifurcation for $Re = 200$, $\mu^* = 0.5$ ad $Re' = 1000$. Bottom row: 3D flow resulting from an external bifurcation for $Re = 300$, $\mu^* = 20$ and $Re' = 1000$.</figcaption></figure>

side, the force contributions are from top to bottom: the buoyancy, the drag force, the history force (the viscous drag correction during fast unsteady motion), the inertia force originating from the fluid acceleration (both time and space contributions) evaluated at the droplet location $x_d$ and the lift force where $\Omega = \nabla \times U$ is the vorticity of the carrying flow field U.

Appropriate expressions are required for the added mass coefficient $C_M$, the drag force parameter $D(Re, \mu^*, \rho^*)$, the history force kernel $K(Re, \mu^*, \rho^*, t)$ and the lift coefficient $C_L(Re, \mu^*, \rho^*)$. For a spherical droplet $C_M = 1/2$ whatever the value of the Reynolds number of the droplet $Re$, the density ratio $\rho^*$ and the viscosity ratio $\mu^*$. The evolution of the drag coefficient has been reported in several works (Clift et al., 1978) and correlations have been provided thanks to direct numerical simulations (Michaelides, 2003; Rachih et al., 2020). It has been shown that for spherical droplet, $D$ is mainly dependent on the Reynolds number $Re$ and the viscosity ratio $\mu^*$, and can be determined with the relation

$$D(Re, \mu^*, \rho^*) = (D^{\text{Spherical Bubble}} + \mu^* D^{\text{Solid Sphere}}) / (1 + \mu^*) \quad (9)$$

where $D^{\text{Spherical Bubble}} = (16 + 3.315Re^{1/2} + 3Re)/(16 + 3.315Re^{1/2} + Re)$ and $D^{\text{Solid Sphere}} = 3(1 + 0.15Re^{0.687})/2$ are the drag parameters for a spherical clean bubble ($\mu^* = 0$, Mei et al. (1994)) and a solid sphere ($\mu^* = \infty$, Schiller and Naumann (1933)), respectively. This expression is only correct while the wake remains 2D axisymmetric.

The history force kernel $K$ was for a long time only known in the spectral domain, but recently it has been characterized as a function of the droplet viscosity ratio using the relation between the effective slip length at the droplet interface $\lambda$ and the viscosity ratio $\lambda\mu^* = d/6$ (Godé et al., 2023a; Legendre et al., 2019).

The lift coefficient $C_L$ varies with $Re$, $\mu^*$ and the shear rate $Sr = \|\Omega\|d/\|U\|$. For a low Reynolds number, $C_L$ follows, from $\mu^* = 0$ (bubble) to $\mu^* = \infty$ (solid sphere), a monotonous evolution of the form $C_L(Re \le 1, \mu^*) \propto (2/3 + \mu^*)^2/(1 + \mu^*)^2$ (Legendre and Magnaudet, 1997), while for a larger Reynolds number, a non-monotonic evolution with $\mu^*$ is observed with a maximum at a viscosity ratio of order unity (Shi et al., 2024). This behavior is related to an internal 3D flow bifurcation (middle row in Fig. 7) that also affects the drag force, such that the expression (9) based on axi-symmetric numerical simulations is restricted to $Re \le 200$ (Godé et al., 2025) and needs to be revisited for larger Reynolds numbers.

Recent numerical simulations have revealed that the drag force is affected by the development of 3D flow destabilization both in the wake of the droplet as observed for solid spheres and deformed bubble, but also inside the droplet, thus changing the drag force behavior (Rachih et al., 2020), the lift force (Shi et al., 2024) and the heat and mass transfer (Godé et al., 2023b).

## 5.2. Heat/mass transfer and chemical reaction

The transport of passive or reactive scalars in multiphase turbulent flows plays a pivotal role in numerous industrial processes and natural phenomena. Examples range from vaporization of atomized fuel jets (Gorokhovski and Herrmann, 2008; Ashgriz, 2011; Gao et al., 2022; Boyd and Ling, 2023) to rain formation and atmosphere-ocean heat/mass exchanges (Duguid and Stampfer, 1971; Deike, 2022), and even to uptake of nutrients and other biochemicals by cells in complex fluid flows (Aksnes and Egge, 1991; Magar and Pedley, 2005).

On a very local scale, small droplets in turbulent flow can, to a first approximation, be considered spherical or ellipsoidal, depending on the intensity of the flow local deformation rate. When convection takes over from diffusion of the transferred species or heat, the basic flow is considered either uniform (due to the slip velocity between phases) or linear (pure deformation, shear, rotation). For a drop in an ambient planar linear flow, A boundary layer analysis allows for a largely analytical estimate of the transport rate from/to the drop in the limit of large Peclet numbers (Krishnamurthy and Subramanian, 2018a,b). Addressing this phenomenon in turbulent multiphase flows remains a challenge (Gauding et al., 2022; Ni, 2024), while most existing studies focus on heat and mass transfer involving isolated drops and bubbles (Boussinesq, 1905; Levich, 1962; Bothe et al., 2004; Figueroa-Espinoza and Legendre, 2010). However, recent significant contributions were extended to swarms of drops and bubbles (Herlina and Wissink, 2016; Albernaz et al., 2017; Dodd et al., 2021; Farsoiya et al., 2021; Shao et al., 2022; Scapin et al., 2022; Farsoiya et al., 2023; Mangani et al., 2024).

A fundamental challenge in studying turbulent multiphase flows lies in the intricate interplay between interfaces and the fluxes across them. These interfaces, which dynamically move and deform in response to evolving flow conditions, are essential in controlling heat and mass transport, driving mixing, and facilitating phase change
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Effect of topology changes on heat transfer. Results from a Direct Numerical Simulation of turbulence with a phase-field method for the interface description. Heat transfer properties of the carrying fluid and droplets are the same. Top row illustrates a breakage event, while bottom row depicts a coalescence event. During breakage, heat transfer from the drops to the carrier fluid is enhanced due to the high surface-to-volume ratio in the pinch-off region (Mangani et al., 2024).</figcaption></figure>

processes (Deckwer, 1980; Gvozdić et al., 2018). The occurrence of the COVID-19 pandemic was a recent example of paramount importance that was attracting worldwide interest in heat and mass transfer in aerosol flows. Understanding the dynamics of droplet evaporation and dispersion became critical in predicting airborne transmission of pathogens and mitigating public health risks (Duguid, 1946; Wells, 1934; Bourouiba, 2021; Balachandar et al., 2020; Wang et al., 2021).

The interfacial topology of the droplets can influence the overall heat transfer distribution. When turbulence is capable of strongly deforming interfaces, droplet emulsions are subject to frequent break-up and coalescence events. This modifies the topology of the interface and, consequently, the spatial distribution of transferred species or heat. In Fig. 8 the dynamics of the initially hot droplets is illustrated as they cool in a turbulent fluid. The Prandtl number is the same for droplets and carrier fluid and equal to one in the examined case. The topology dynamics of the drops has a clear influence on the temperature distribution. Inertial or viscous break-up is much easier to simulate numerically than coalescence. Indeed, coalescence involves first a long-range hydrodynamic interaction leading the interfaces to move closer together and then a short-range hydrodynamic interaction owing to the presence of fluid film drainage. Finally, the effect of physical-chemical interactions will dominate when the gap is very fine. Sub-grid modeling is an option to take into account drainage and the last moments of merging (like the disjunction pressure model).

In turbulent flows, a droplet is subject to numerous gradients (velocity and pressure gradients, which can induce major deformations of the interface or changes in topology, rupture, coalescence), gradients of the species transferred (which can generate physical-chemical or thermal Marangoni effects) and gradients of volume fraction. This takes the real-life situation further away from the validity of theoretical predictions—correlations for isolated drops in a uniform medium.

Direct numerical simulation is therefore the reference methodology for studying, understanding and modeling these complex systems, as experimentation is ill-equipped to characterize local effects on small scales. However, a further major limitation is the separation of scales between the size of the droplet and the smallest scales characterizing gradients in the boundary layer, or the Batchelor scale representative of mixing lengths in turbulent flows. This is even more true as the Schmidt number increases. The presence of chemical reactions generally makes this constraint even more drastic for rapid kinetics, since the consumption of species steepens concentration gradients near interfaces (increasing interfacial mass transfer).

These problems are schematically depicted in Fig. 9, where a large deformable drop is cooling down in a colder carrier fluid. The situation is complex, with turbulence playing an important role in transporting the fluid and the heat inside and outside of the drop. The interface allows for the transfer of heat (or chemical species) but not of fluid. The concentration distribution is strongly heterogeneous, and in addition, more different length scales enter the problem in turbulence, such as the Batchelor length for mixing, the passive tracer wake, and thin boundary layer at high Sc number. The fine resolution of very thin boundary layers can be envisaged with several modeling and simulation strategies. It is becoming increasingly common to use Adaptive Mesh Refinement (AMR) techniques in areas of high gradient (multilevel octree representation allows computational meshes to be concentrated in regions of the flow where the physics is most complex to capture). Highly optimized methods have been implemented within open source code (Popinet, 2022). This methodology reaches its limits when there are many regions where such refinement is required (high volume fraction, uniformly distributed interface in the computational domain), as the cost of refinement/de-refinement becomes greater than the time saved by having a non uniform mesh. Vreman (2016) proposed a
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Challenges in resolving transfer processes at the interface of a deformable drop. Panel (a) shows the temperature field (upper half) and velocity field (lower half) within a large, deforming drop with a Prandtl number of 8. The differences between the temperature and velocity structures are evident and governed by the smallest turbulence scales. When turbulence has reached an equilibrium, the local Batchelor scale $\eta_{\theta}$ is related to the local Kolmogorov scale $\eta_{k}$ following $\eta_{\theta} = \eta_{k}Pr^{-1/2}$. Panel (b) presents a schematic of thickness and growth of momentum boundary layer, $\delta$, and of thermal boundary layer, $\delta_{\theta}$, on the outer side of the drop for $Pr > 1$. As the Prandtl number increases, progressively thinner boundary layers emerge, posing significant computational challenges for current numerical methods.</figcaption></figure>

numerical method that uses a highly refined mesh attached to the particle interface moving across a fixed Eulerian grid. Another strategy is to consider thin boundary layers as poorly resolved, and thus to supplement the flow explicitly resolved by simulation with sub-grid models for transfer at interfaces (in the spirit of what is done for large-scale simulation). To solve accurately heat and mass transfers using interface tracking and capturing methods, sub-grid modeling takes advantage of analytic solutions from the boundary layer theory. A quasi-static approach based on local fitting of instantaneous heat or concentration profiles was proposed by Weiner and Bothe (2017) or Cai et al. (2021). Another option is to use a transient method to track the unsteady energy balance in the boundary layer (Claassen et al., 2019; Weiner et al., 2022). Similarly, a number of studies are proposing closure models for sub-grid deformation associated with the small scales of turbulent flow, since correction of the transfer coefficient and interfacial area must be approached in a coherent manner yielding a general framework for multiphase Large Eddy Simulations coupled to heat and mass transfer.

Heat transfer with an interface is a key element in many industrial or natural processes involving boiling or condensation. These often rapid kinetic phenomena require a thorough understanding of interface-wall interaction issues such as wetting, rebound, or droplet splashing. The contact time may be short, but the amount of heat transferred is very high. In particular, the transition to the Leidenfrost regime (Mialhe et al., 2023) is crucial to the design of processes such as spray drying or spray cooling. There are therefore difficulties in terms of small-scale spatial resolution and a strong constraint on the time step needed to capture these brief but intense events.

## 6. Role of surfactants

Surfactant behavior can be classified into two main categories: insoluble and soluble surfactants (Manikantan and Squires, 2020). Surfactants do not dissolve in gases and their solubility in liquids varies depending on the surfactant and liquid properties (Levich and Seeger, 1963; Manikantan and Squires, 2020). Surfactants are molecules that accumulate at the interface between two fluids, altering the intermolecular forces and changing interfacial tension (Levich and Seeger, 1963). Their presence induces complex interfacial phenomena: not only do

they reduce capillary forces, but surfactant gradients along the interface also generate tangential Marangoni forces (Scriven and Sternling, 1960), having a strong impact on the drag force experienced by droplets and thus on their relative velocity. Due to their significance, surfactant modeling has gained increasing scientific attention over the past twenty years. It remains challenging to conduct experiments in very well controlled conditions, so efficient numerical methods need to be developed considering that the choice of numerical approach depends on the surfactant type, system configuration, and fluid properties. Research on surfactants in bubbles (gas-liquid systems) is extensive due to their critical role in foam systems. However, understanding surfactant effects in liquid-liquid systems (drops) has great relevance for emulsions, where surfactants influence drop stability and coalescence

### 6.1. Modeling insoluble and soluble surfactants

The distinction between insoluble and soluble surfactants affects interface contamination and, consequently, the modeling approach. Insoluble surfactants are treated as a species confined to the interface, while soluble surfactants are considered present both at the interface and within the bulk phases. For insoluble surfactants, a single governing equation is solved at the interface, within a narrow band, or throughout the domain (Lai et al., 2008; Lu et al., 2017; Xu and Ren, 2014; Jain, 2024b). For soluble surfactants, either a single equation (Laradji et al., 1992; Soligo et al., 2019b) or a two-equation model is used (Muradoglu and Tryggvason, 2014; Jain, 2024b). Fig. 10 illustrates the main categories of numerical approaches for tracking surfactant concentration.

### 6.2. Surfactant with asymmetric solubility

Surfactants can exhibit varying solubility not only in gas/liquid flows (e.g., air and water) but also in liquid/liquid flows, making their study relevant for diverse applications. Within the phase-field method, two approaches address the different solubility or insolubility of surfactants in one or both phases. The first, typically used with the Cahn-Hilliard equation, introduces a skewed term in the free energy functional to penalize the surfactant's presence in one phase (Liu and Zhang, 2010). The second, used with the conservative Allen-Cahn
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Approaches commonly employed to describe the interface concentration of a surfactant. A first classification can be obtained considering the type of surfactant: insoluble or soluble. For insoluble surfactants, the methods commonly used rely on the solution of a single equation at the interface, in a narrow band about the interface or in the entire domain. For soluble surfactants, there are two possibilities: single equation models, where a single equation account for the bulk-interface exchanges and vice-versa or two equation models, where coupled equations are used to describe these exchanges.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Panel a shows a sketch of the migration of a bubble due to the Marangoni stresses (black arrows, from point A towards point B). Surfactant molecules (blue hydrophilic head and red hydrophobic tails) are present in the carrier phase and at the interface while they cannot be absorbed in the bubble. Panel b shows the corresponding phase-field method simulation. Streamlines (left) are used to visualize the induced flow circulation while the contour map (right) shows the surfactant concentration: surfactant is absent in the bubble; a non-zero concentration is found in the carrier phase while a higher concentration is found in the upper part of the bubble (point A).</figcaption></figure>

equation, employs modified transport equations for surfactant concentration (Jain, 2024b,a). One or two equations are chosen depending on solubility requirements (Jain, 2024b).

Uneven surface tension generates Marangoni stresses proportional to surface tension gradients. This phenomenon is illustrated in Fig. 11 in the instance of a migrating bubble. Panel (a) shows a sketch: surfactant molecules, with hydrophobic tails and hydrophilic heads, are present in the carrier phase and at the interface, where they align with heads facing the water phase. Uneven surfactant distribution causes non-uniform surface tension, generating Marangoni stresses that drive fluid from low surface tension regions (high surfactant concentration, point A) to high surface tension regions (low surfactant concentration, point B). This propels the bubble upwards. Panel (b) displays a phase-field simulation (Soligo et al., 2019b), where the left side visualizes circulation via streamlines, and the right side shows surfactant concentration: absent inside the bubble, non-zero in the carrier phase, and highest at the upper bubble interface. A similar effect occurs with temperature gradients, producing thermocapillary effects (Levich and Krylov, 1969; Scriven and Sternling, 1960; Young et al., 1959).

## 7. Challenges and future directions in drop-laden flows

Drop-laden flows are critical in a wide range of natural phenomena and industrial applications, but understanding their intricate physics remains challenging. Despite advancements in numerical simulations and experimental techniques, the multiscale nature of these flows presents significant hurdles. Numerical models currently resolve only a fraction of the scales, while experimental methods face limitations in capturing the smallest turbulence structures and validating complex interactions. To bridge these gaps, novel approaches combining computational and experimental efforts are needed.

Key challenges to address include improving resolution of droplet dynamics and interface phenomena, advancing coalescence models, and incorporating surfactant dynamics and molecular-scale interactions. Developing multiscale models, data-driven approaches, and enhanced experimental diagnostics will be essential for unlocking a deeper understanding of turbulent drop-laden flows and their impact across scientific and industrial fields.

The following graphical summary outlines the current state of the art and the key future challenges to tackle in this field.
## State of the Art

1. **Multiscale Nature of Drop-Laden Turbulent Flows**
Drop-laden turbulent flows encompass a vast range of scales, from macroscopic problem domains to Kolmogorov scales and down to molecular-level interface dynamics, illustrating their inherently multiscale nature.

2. **Limitations of Experimentally-Based Models** Existing experimentally-based models for droplet fragmentation, developed from limited datasets, show significant variability in predictions. This underscores the need for further validation and refinement to improve their reliability.

3. **Advances in Numerical Methods** Numerical simulations have significantly enhanced understanding of droplet dynamics and topological changes. These methods now predict droplet size distributions at sub-Kolmogorov-Hinze scales, offering valuable insights into fragmentation processes.

4. **Current Computational Limitations** Resolving the full spectrum of multiphase turbulence—from macroscopic to molecular scales—remains beyond current computational capabilities, which are limited to resolving three to four orders of magnitude out of the required eight to nine.

5. **Adaptive Mesh Refinement (AMR)** AMR techniques improve resolution in capturing interface phenomena such as breakup and coalescence. However, these techniques achieve only moderate success in resolving the smallest interfacial topology features, leaving gaps in accuracy.

6. **Modeling Unresolved Physics** Models for surfactant effects, evaporation, and contact angle dynamics approximate unresolved physics but do not directly simulate the underlying molecular mechanisms, limiting their predictive accuracy.

7. **Limitations of Coalescence Models** Current coalescence models approximate interface merging at mesoscopic scales but fail to capture critical molecular-scale phenomena, such as interface rupture and the formation of coalescence bridges.

## Future Issues to Address

1. **Advanced Experimental Techniques** Recent advancements in experimental techniques have demonstrated the ability to simultaneously track the shape evolution of fragmenting droplets and accurately measure the local flow field. Further efforts are needed to analyze these results and develop universally applicable models.

2. **Resolved Interface Simulation Methods** Given current and projected limitations in computational resources, resolved interface simulations will require the incorporation of closure models to address heat and mass transfer in highly convective regimes and evaporation processes.

3. **Coalescence Models and Physico-Chemical Interactions** Future developments in coalescence models should account for physico-chemical interactions by tracking interface deformations in the extremely thin gaps between interfaces, often falling below grid resolution, to improve accuracy and applicability.

4. **Surfactant Dynamics, Interface Impurities, and Marangoni Effects** Advancing models of additives, such as surfactants and particle-based impurities, will require the development of closure models that link molecular-scale interactions to interface-scale phenomena. These models must capture wettability, interfacial elasticity, and the Marangoni effect while also accounting for their influence on heat and mass transfer processes.

5. **Interfacial Rheology** Experimental and simulation efforts should continue to explore the complex rheological behavior of interfaces, including viscoelastic properties and the impact of surfactants and particles on interfacial dynamics, to enhance predictive capabilities.

6. **Data-Driven Modeling Approaches** Data-driven techniques have the potential to support the development of physically informed models and extend their application across scales. These methods can integrate hierarchical data from experimentally and computationally resolvable scales to finer, unresolved scales, improving model accuracy and universality.

## CRediT authorship contribution statement

O.M.H. Rodriguez: Writing – review & editing, Writing – original draft, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. P. Angeli: Writing – review & editing, Writing – original draft, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. D. Legendre: Writing – review & editing, Writing – original draft, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. E. Climent: Writing – review & editing, Writing – original draft, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. A. Soldati: Writing – review & editing, Writing – original draft, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors acknowledge the TU Wien University Library for financial support through its Open Access Funding Program. O. R. would like to thank Johann E. Castro-Bolivar and Pedro J. Miranda-Lugo for their help with some graphics, the National Council for Scientific and

Technological Development — CNPq for the productivity grant Proc. 311057/2020-9, and The São Paulo Research Foundation — FAPESP Proc. 2024/16839-4. P.A. is grateful to the UK Research and Innovation (UKRI) for supporting projects, Programme Grant PREMIERE (PREdictive Modeling with Quantification of Uncertainty for Multiphasic Systems, EP/T000414/1) and SusAgriChem - Sustainable Formulation of Agri-Chemicals via Dynamic Molecular Interfaces (EP/V032909/1). A.S. is grateful to the Austrian Science Fund (FWF) [Modeling And Simulation of emulsiOns (MAIO), Grant DOI: 10.55776/PAT9292123], and to the Italian Ministry for Research [The fluid dynamics of interfaces: mesoscale models for bubbles, droplets, and membranes and their coupling to large scale flows, PRIN Project No. 2022R9B2MW_PE8].

## Appendix. Compilation of correlations for drop size in pipe flow

A comprehensive compilation of phenomenological models and empirical correlations used for drop size calculations in pipe flow is presented in Table A.1. These models provide useful tools for analysis, which are organized following various approaches, ranging from simplified empirical relationships to more detailed phenomenological descriptions, offering a broad perspective on drop size prediction across different flow regimes. It can serve as a valuable reference for researchers and engineers seeking to evaluate or compare methodologies for predicting droplet diameter in pipe flow. Each model or correlation is typically characterized by its assumptions, applicable conditions, and key parameters, allowing users to identify the most suitable approach for their specific system.
<table><caption>**Table A.1**
Available correlations for drops size in pipe flow.</caption><thead><tr><th>Author</th><th>Phenomenological model or empirical correlation</th><th>Application</th><th>Range</th></tr></thead><tbody><tr><td>Hinze (1955)</td><td>d<sub>max</sub> (ρ<sub>o</sub>/σ)<sup>3/5</sup> e<sup>-2/5</sup> = 0.725<br>d<sub>max</sub>/D = 0.871 W e<sup>-0.6</sup> Re<sup>0.1</sup> w [ρ<sub>o</sub>(1-ε<sub>o</sub>)]<sup>-0.4</sup></td><td>Dilute dispersions (original)<br>Oil-in-water dispersions in pipes, With low viscosity</td><td>l<sub>k</sub> &lt;&lt; d<sub>max</sub> &lt; l<sub>e</sub><br>d<sub>max</sub> &gt; 1.8 Re<sup>-0.7</sup> w and d<sub>max</sub>/D &lt; 0.1</td></tr><tr><td>Sleicher (1962)<sup>a</sup></td><td>d<sub>max</sub>/D = 38 W e<sup>-1</sup> w Re<sup>1/2</sup> w [ρ<sub>o</sub><sup>2</sup>/σρ<sub>o</sub>D]<sup>1/2</sup> [1 + 0.7 Ca<sup>0.7</sup>]</td><td>Oil-in-water dispersions in pipes. Turbulent flow</td><td>d<sub>max</sub> &gt; l<sub>e</sub></td></tr><tr><td>Paul and Sleicher (1965)<sup>a</sup></td><td>d<sub>max</sub>/σρ<sub>o</sub><sup>2</sup> = √(ρ<sub>o</sub><sup>2</sup>/σρ<sub>o</sub>D) = C<sub>1</sub> (1 + 0.7 ρ<sub>o</sub><sup>2</sup>/σ)<sup>0.7</sup></td><td>Oil-liquid viscous dispersions (octane or benzene)</td><td>d<sub>max</sub> &lt; l<sub>k</sub></td></tr><tr><td>Ward and Knudsen (1967)<sup>a</sup></td><td>Re<sub>w</sub> (D<sub>o</sub>/D)<sup>2</sup> ρ<sub>o</sub>/ρ<sub>o</sub> ≤ 2</td><td>Oil-in-water dispersions</td><td>10<sup>4</sup> ≤ Re<sub>w</sub> ≤ 10<sup>5</sup></td></tr><tr><td>Kubie and Gardner (1977)</td><td>d<sub>max</sub>/D = 5.53 ρ<sub>o</sub>D/(fρ<sub>o</sub><sup>2</sup>) where f = 0.076 Re<sup>-0.25</sup></td><td>Oil-in-water dispersions in pipes. Turbulent Flow</td><td>d<sub>max</sub> &gt; l<sub>e</sub></td></tr><tr><td>Karabelas (1978)<sup>a</sup></td><td>d<sub>max</sub>/D = 2.645 (σ/Dρ<sub>o</sub>U<sup>2</sup>)<sup>0.6</sup> Re<sup>0.15</sup> c<sup>0.4</sup> (R/R)<sup>0.4</sup></td><td>Dilute dispersions</td><td>R ≥ 35 and ε<sub>j</sub> = ε̃</td></tr><tr><td>Acrivos and Lo (1978)<sup>b</sup></td><td>d<sub>max</sub>/D = 0.074 (σ/μ<sub>o</sub>U<sup>2</sup>)<sup>1/6</sup></td><td>Water-in-oil dispersions. Laminar flow</td><td>Re<sub>o</sub> &lt; 2100 and C<sup>crit</sup><sub>w,l</sub> ≤ 0.25</td></tr><tr><td>Davies (1987)<sup>a</sup></td><td>d<sub>max</sub> = C<sub>1</sub> (1 + 0.7 ρ<sub>o</sub>μ<sub>o</sub>/σ)<sup>0.6</sup><br>d<sub>max</sub>/D = C<sub>1</sub> [1 + 0.14 Ca<sub>o</sub> Re<sup>-0.125</sup>]</sup><sup>0.6</sup></td><td>Dilute Oil-in-water dispersions. Agitated tanks, stirred vessels<br>Oil-in-water dispersions in pipes. Turbulent flow</td><td>l<sub>k</sub> &lt;&lt; d<sub>max</sub> &lt; l<sub>e</sub><br>l<sub>k</sub> &lt;&lt; d<sub>max</sub> &lt; l<sub>e</sub></td></tr><tr><td>Angeli and Hewitt (2000)<sup>a</sup></td><td>d<sub>max</sub>U<sub>c</sub><sup>1.8</sup> = 4.2 * 10<sup>-2</sup> f<sup>-3.13</sup><br>d<sub>32</sub>U<sub>c</sub><sup>1.8</sup> = 2 * 10<sup>-2</sup> f<sup>-3.12</sup></td><td>Oil-in-water dispersions in horizontal pipes</td><td>0.05 &lt; U<sub>d</sub> &lt; 0.11<br>1.1 &lt; U<sub>c</sub> &lt; 1.7</td></tr><tr><td>Brauner (2001)</td><td>d<sub>max</sub>/D = 7.61 C<sub>H</sub> W e<sup>-0.6</sup> w Re<sup>0.08</sup> w (ρ<sub>o</sub>/ρ<sub>l</sub>)<sup>0.08</sup> (ρ<sub>o</sub>/ρ<sub>l</sub>)<sup>0.4</sup> (ε<sub>o</sub><sup>0.6</sup>(1-ε<sub>o</sub>))<sup>0.3</sup></td><td>Oil-in-water dispersions in pipes. Turbulent flow.</td><td>1.82 Re<sup>-0.7</sup> w &lt; d<sub>max</sub>/D &lt; 0.1</td></tr><tr><td>Kouba (2003)</td><td>d<sub>min</sub> = σg/ρ<sub>o</sub>U<sup>2</sup>; d<sub>max</sub> = 20σg/ρ<sub>o</sub>U<sup>2</sup></td><td>Oil-in-water horizontal flows</td><td></td></tr><tr><td>Razzaghi and Shahraki (2010)<sup>a</sup></td><td>d<sub>32</sub>/D = 0.28 (1 + 0.92ε<sub>o</sub>) W e<sup>-0.6</sup> w<br>d<sub>32</sub>/D = 0.14 (1 + 0.48ε<sub>o</sub>) W e<sup>-0.6</sup> w</td><td>Water-in-Oil dispersions<br>Oil-in-Water dispersions</td><td>ε<sub>d</sub> &lt; 0.6</td></tr><tr><td>Torres et al. (2016)</td><td>d<sub>max</sub>/D = 0.549<br>d<sub>max</sub>/D = (ε<sub>d</sub>/1-ε<sub>d</sub>)<sup>0.6</sup> W e<sup>0.2</sup> f<sup>0.4</sup></td><td>Oil-in-water dilute dispersions<br>Oil-in-water dense dispersions</td><td></td></tr><tr><td>Castro-Bolivar and Rodriguez (2021)</td><td>d<sub>max</sub>/D = 303.79 W e<sup>-0.1</sup> w Re<sup>0.25</sup> w (ρ<sub>o</sub>/ρ<sub>l</sub>)<sup>0.6</sup> f (μ<sub>o</sub>)<sup>0.6</sup><br>f (μ<sub>o</sub>) = 1 + 0.819 (ρ<sub>o</sub>/ρ<sub>l</sub>)<sup>1/2</sup> [ρ<sub>o</sub>/ρ<sub>o</sub>(1-ε<sub>o</sub>)]<sup>0.2</sup> W e<sup>-0.2</sup> Re<sup>-0.05</sup> w Ca<sub>d</sub><br>W e<sub>w</sub> = ρ<sub>o</sub>μ<sub>o</sub><sup>2</sup>D/σ; Re<sub>w</sub> = ρ<sub>o</sub>μ<sub>o</sub>D<sub>w</sub>/ρ<sub>o</sub><br>D<sub>w</sub> = 46/D (D - δ<sub>l</sub>)</td><td>Oil-in-water dispersions in oil-water-gas flows</td><td>d<sub>max</sub>/D &gt; l<sub>k</sub></td></tr></tbody></table>

## Data availability

Data will be made available on request.

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<footnote>b Only for laminar flow, i.e., Water-in-oil dispersions.</footnote>

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