# Quick starch guide: a perspective on shear thickening in dense non-Brownian suspensions

Cécile Clavaud, Abhinendra Singh

► To cite this version:

Cécile Clavaud, Abhinendra Singh. Quick starch guide: a perspective on shear thickening in dense non-Brownian suspensions. Rheologica Acta, 2025, 64 (9-10), pp.563-582. 10.1007/s00397-025-01519-6. hal-05371276

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Submitted on 20 Nov 2025

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# Quick starch guide: a perspective on shear thickening in dense non-Brownian suspensions

Cécile Clavaud<fnref n="1" /> · Abhinendra Singh<fnref n="2" />

Received: 7 June 2025 / Revised: 18 July 2025 / Accepted: 28 July 2025 / Published online: 1 November 2025
© The Author(s) 2025

## Abstract

In this article, we provide a brief perspective on recent developments in the study of shear thickening in dense suspensions. We give a rapid overview of the state of the art and discuss current models aiming to describe this particular rheology. Although most of the experiments and simulation studies are conducted in “ideal” flows, where the sample is confined without an open boundary condition, we have decided to highlight more realistic flow conditions. We further provide an overview on how to relate the recently proposed constitutive models to these more practical flow conditions like pipe flow or flow down an incline.

**Keywords** Shear thickening · Dense suspensions · Force network · Network analysis · Mean-field models · Friction

## Introduction

Dense suspensions are prototypical complex fluids. Their flow behavior is scientifically challenging to predict, and they are ubiquitous in various industrial, geotechnical, and biological phenomena, with examples ranging from cement transport, mudflow, and red blood cells (Denn and Morris 2014; Stickel and Powell 2005; Guazzelli and Pouliquin 2018; Morris 2020; Jerolmack and Daniels 2019; Beris et al. 2021). In this article, we focus on shear thickening, one of the many complex rheological behaviors they can exhibit. We will limit our discussion to suspensions of non-Brownian hard spheres suspended in a density-matched Newtonian liquid, and will therefore neglect the effects of thermal fluctuations, gravity, and particle deformability. Additionally, we will only consider viscous flows, and will thus not discuss

any inertial and viscoelastic effects. Finally, as stated, we focus on shear thickening in dense suspensions. There is no clear and established definition of “dense” suspensions, but we can say that a suspension is dense when its particle volume fraction is “large enough” that its behavior is dominated by short-range interactions between particles, which are separated by distances smaller than their typical size (Ness et al. 2022).

Shear thickening in dense suspensions has been the subject of many reviews over the years, either directly or as one of the many existing complex rheological behaviors (see, for example, Barnes 1989; Wagner and Brady 2009; Denn and Morris 2014 and more recently Denn et al. 2018; Guazzelli and Pouliquin 2018; Singh 2023; Ness et al. 2022; Lemaire et al. 2023; Morris 2020). This article is dedicated to surveying recent progress in the study of strong shear thickening observed in dense suspensions. It focuses on features encountered in realistic flows and on how to model or explain them. As it is a perspective article, it will not review all the existing literature but rather give our viewpoint on the current state of the art in the field and the promising directions for future research. It is necessarily incomplete, as we had to make choices among all the topics we would have liked to address, but we hope it will be helpful and spark new interests, ideas, and discussions. In particular, we hope to convey that shear thickening has opened up a playground for rheologists, physicists, engineers, chemists, tribologists, and mathematicians to play on.

Cécile Clavaud and Abhinendra Singh. These authors contributed equally to this work.

<figure><img src="image_2.png" /></figure>

<footnote>Abhinendra Singh
abhinendra.singh@case.edu</footnote>

<footnote>Cécile Clavaud
cecile.clavaud@univ-rennes.fr</footnote>

<footnote n="1">¹ Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35000 Rennes, France</footnote>

<footnote n="2">² Department of Macromolecular Science and Engineering, Case Western Reserve University, Cleveland 10040, OH, USA</footnote>
## Shear thickening: what it is, and what it is not

Shear thickening (ST) is an increase in the viscosity $\eta = \tau/\dot{\gamma}$ of a complex fluid with the applied shear rate $\dot{\gamma}$ or shear stress $\tau$. In dense shear-thickening suspensions, this increase, as a function of the applied shear rate, can be either continuous, in which case the behavior is called continuous shear thickening (CST), or discontinuous, giving discontinuous shear thickening (DST). CST is usually observed at lower volume fractions $\phi$. Shear thickening becomes stronger as $\phi$ increases, until DST is observed, where $\eta$ increases abruptly by orders of magnitude over a narrow range of $\dot{\gamma} : \partial\eta/\partial\dot{\gamma} \to \infty$. The conditions ($\phi, \dot{\gamma}$) at which DST is observed depend sensitively on the particle properties, i.e., roughness, size, shape, etc., as well as on the chemical physics of both the particle and fluid phases (Singh et al. 2019).

## The many features of shear thickening in dense suspensions

The first papers (to our knowledge) to report on this intriguing rheology are from Williamson and Hecker (1931) and Freundlich and Röder (1938), and it is worth discussing them a little. In both papers, the authors observe a saturation of their studied suspension’s flow rate with the applied stress. Williamson and Heckert conducted an extensive study of cornstarch suspensions in various suspending fluids, while Freundlich and Röder studied suspensions of quartz powder, rice starch, and potato starch in water, with or without added electrolytes. In both studies, the authors note that the behavior is strongly dependent on the fluid-particles interactions: if the particles are not well-dispersed in the fluid, they do not observe the flow rate saturation. They also both postulate that contacts between the particles play an important role. Freundlich and Röder report that the flow rate saturation comes with a periodic flow instability that causes the suspensions to fracture. Finally, Williamson and Heckert postulate the existence of a critical stress scale in the system to explain the behavior. We will come back to the observations and ideas put forward in these two papers throughout this perspective article.

Since these two articles, there have been quite a lot of experimental studies on the subject, revealing different features of shear-thickening behavior in dense suspensions. Metzner and Whitlock reported on the ability of such suspensions to fracture, stating that “*the fracturing of the fluid was not only visible but also audible*” (Metzner and Whitlock 1958). Numerous experiments pointed to DST being associated not only with an abrupt increase in viscosity, but also with giant fluctuations in stress (or viscosity) at the onset of DST (Hoffman 1972, 1974; Lootens et al. 2003, 2004; Bender and Wagner 1996; Boersma et al. 1991; D’Haene et al. 1993; Xu et al. 2020). Saint-Michel et al.

(2018), using cornstarch suspensions in a concentric cylinder and simultaneously imaging the flow with ultrasonic echography, observed and provided a statistical analysis of temporal fluctuations in the shear rate. They correlated it with spatiotemporal local particle dynamics and reported that the bulk suspension remains homogeneous during the DST transition, along with the existence of transient localized bands that travel along the vorticity direction. Rathee et al. (2017, 2020, 2022), in a series of studies using boundary stress microscopy, showed the existence of an inhomogeneous stress field as the suspension shear thickens. Along with this, the authors also revealed the existence of high local stress regions that propagate along the flow direction of the top boundary. Along these lines, Ovarlez et al. (2020) also reported density fluctuations that appear as periodic waves in the flow direction and break the azimuthal symmetry, accompanied by high normal stress fluctuations of the same periodicity. Finally, one of the most famous aspects of shear-thickening suspensions is their response to impact and their ability to transiently support loads and dissipate energy. This has also been the focus of many papers (see, for example, Waitukaitis and Jaeger 2012; Roché et al. 2013; Han et al. 2018; Brassard et al. 2021; Peters et al. 2016; James et al. 2018; Chen et al. 2023), showing the development and travel of a jamming front under impact or studying the cracks formed after impact. Much of the early experimental, theoretical, and numerical work on shear-thickening suspensions has focused on first understanding the average bulk behavior in standard rheological geometries and in steady flow, but it is interesting to note that the unstable nature of the flow in the DST regime was already discussed in Freundlich and Röder (1938). We illustrate some of the different features of shear thickening in dense suspensions in Fig. 1.

The previous paragraph concerns features related to flow behavior. In addition to this, both Refs. (Williamson and Hecker 1931) and Freundlich and Röder (1938) discussed the importance of fluid-particle interactions in the presence or absence of shear thickening. This point seems to have received less interest, although it is of crucial importance. Indeed, fluid-particle interactions, in addition to the physical characteristics of the particles, are responsible for setting the particle-particle interactions (or constraints, see Section “Constrained versus unconstrained particle motion: one point of view to unite them all”). Some recent studies on suspensions of small (~ 0.1µm to 4 µm) particles explore the effect of particle-particle interactions on CST and DST, either through directly manipulating the particles’ roughness (Hsu et al. 2018, 2021; Lootens et al. 2005; Hsiao et al. 2017; Pradeep et al. 2021), changing their physical and chemical interactions (James et al. 2018, 2019; Kim et al. 2024; Chen et al. 2023), or both (Bourrianne et al. 2022). In the concrete industry, the effect of superplasticizers (polyelectrolytes that can adsorb on the surface of the particles in the
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Some of the (many) features of shear thickening in dense shear-thickening suspensions. Suspension used: commercial potato starch suspended in tap water. Left: behavior at a first packing fraction $\phi_1$ ($\phi_1 \simeq 0.55$ wt%). (A) Scooping-up of the suspension at medium speed, versus (B) fast scooping-up, and (C) subsequent flow of the suspension out of the spoon (irrespective of the scooping-up speed). We can see signs of wrinkling, fracture that rapidly relaxes in the wake of the spoon, and fracture in the filament flowing out of the spoon. Right:</figcaption></figure>

cement slurry) on the suspension rheology has long been documented (see, for example, Neuville et al. 2012; Bossis et al. 2017), and this effect is also studied in a more formal context (see for example Richards et al. 2021). James et al. (2018) emphasized the pivotal role of physiochemical interactions in the manifestation of DST in suspensions containing both Brownian and non-Brownian (cornstarch) particles. In the case of non-Brownian suspensions, Clavaud et al. showed in (2017) that changing the suspending fluid could change fluid-mediated particle-particle interactions, thus changing the rheological behavior of the suspension. In line with the results from Williamson and Hecker (1931), Oyarte Gálvez et al. (2017) showed that the nature of the suspending fluid of cornstarch suspensions had a dramatic effect on the suspension rheology. With all this in mind, we wanted to highlight the specific case of suspensions of cornstarch in water, often taken to be the prototypical example of non-Brownian shear-thickening suspensions. Gauthier et al. showed in (2023) that they exhibit signs of adhesion, and that their behavior is not always the same as that of other shear-thickening suspensions. In light of these observations, we felt it was important to mention the critical importance of the fluid-particle interactions, and remind the reader to be careful before assuming the generality of an observed behavior.

## Constitutive laws for dense suspensions in simple shear and steady-state: why shear thickening is unexpected

In simple shear and in the steady state, and in the framework of this paper, dimensional analysis dictates the following

behavior at a packing fraction $\phi_2 > \phi_1$ ($\phi_2 \simeq 0.57$ wt%). (D) Slow scooping-up of the suspension and (E) subsequent flow out of the spoon, for a small spoon height. The suspension here behaves like a very viscous liquid. (F) Fast scooping-up of the suspension, and (G) subsequent flow out of the spoon. Again we can see signs of fracture, with (slower) relaxation where the suspension is no longer under stress. (H) Fracture of the suspension under large stress. (I) Solidification under impact

constitutive laws, which have been experimentally confirmed (see Guazzelli and Pouliquen 2018 and references therein):

$$ \begin{cases} \tau = \eta_f \dot{\gamma} \eta_s(\phi) \\ P = \eta_f \dot{\gamma} \eta_n(\phi) \end{cases} \quad \text{or} \quad \begin{cases} \tau = \mu(J)P \\ \phi = \phi(J) \end{cases} \quad (1) $$

in, respectively, a volume-imposed or a pressure-imposed geometry. Here, $\eta_f$ is the suspending fluid's viscosity and $\dot{\gamma}$ is the imposed shear rate. In a volume-imposed geometry, the particle volume fraction $\phi$ is imposed, and the shear stress $\tau$ and particle pressure $P$ depend on $\phi$ through the functions $\eta_s$ and $\eta_n$, which diverge at the jamming packing fraction $\phi_c^{\mu_p}$ above which the suspension cannot flow. This jamming packing fraction depends on the microscopic friction coefficient $\mu_p$, that is, the particle-particle friction coefficient, and we have $\phi_c^{\mu_p \neq 0} < \phi_c^{\mu_p = 0}$. In a pressure-imposed geometry, the particle pressure $P$ is imposed, and both $\phi$ and the ratio $\tau/P$ are controlled by the viscous number $J = \eta_f \dot{\gamma}/P$ (note that $J$ is the inverse of the normal viscosity $\eta_n$ as defined in an imposed volume geometry). By analogy to Coulomb's law of friction, the function $\mu(J)$ is called the macroscopic friction coefficient of the suspension. We have $\phi_c^{\mu_p} = \lim_{J \to 0} \phi(J)$, and $\mu$ has a minimum value $\mu_c^{\mu_p}$ at low $J$, which also depends on $\mu_p$. In the absence of any additional ingredient, as summarized in Fig. 2, the rheology of such a suspension is thus quasi-Newtonian: its viscosity is independent of the shear rate. The existence of shear thickening points to a missing parameter in the description of shear-thickening suspensions. We will come back to this in Section “The frictional transition scenario and the Wyart and Cates model”.
<figure><img src="image_4.png" /><figcaption>**Fig. 2** Dense suspension behavior under imposed volume and pressure. Steady-state simple shear rheology of a suspension of non-Brownian hard spheres in a density matched Newtonian liquid, undergoing a viscous flow. Left: volume-imposed case. The suspension's viscosity η is a function of only one parameter, its particle volume fraction φ: η(φ) = η<sub>f</sub>η<sub>s</sub>(φ). Right: pressure-imposed case. A certain pressure P is applied only to the particulate phase of the suspension, as represented here with a grid that allows the fluid to pass through</figcaption></figure>

Note that what we have presented here is a simplified view of the rheology of dense suspensions, for two main reasons. First, the definition of $\phi_c^{\mu_p \neq 0}$ is experimental, mostly based on a fit to experimental data, and hides a more complex phenomenology. There are, a priori, three critical packing fractions in dense suspensions: $\phi_{\min. rig.}^{\mu_p \neq 0}$, at which minimally rigid frictional clusters start to appear in the suspension (see Section “Beyond a mean-field description: fluctuations and frictional network”), $\phi_{SJ}^{\mu_p \neq 0}$ above which the suspension can only flow until it accumulates a certain amount of strain, which creates an anisotropic frictional microstructure preventing flow in the strain direction (shear-induced jamming, see Section “The frictional transition scenario and the Wyart and Cates model and Beyond a mean-field description: fluctuations and frictional network”), and $\phi_{IJ}^{\mu_p \neq 0}$, the packing fraction above which the system is isotropically jammed. All three a priori depend on the particle-particle friction coefficient, and we postulate that when $\mu_p = 0$, they are all equal. Disentangling these is a complicated task, and is the focus of some recent studies (Seto et al. 2019; Naald et al. 2024). Although we will discuss rigidity and shear-jamming, we will still restrict ourselves to a formulation in which there is only one critical packing fraction: $\phi_c^{\mu_p \neq 0}$. Second, these constitutive laws do not fully reflect the rheology of dense suspensions. They do not predict, for example, the existence of normal stress differences, hence the quasi-Newtonian appellation. Additionally, they only describe steady-state behavior and do not consider the transients leading to it. Finally, the relations are written between the averaged values of the considered physical quantities, again not allowing for the description of any transient phenomenon rising from fluctuations in the system (Guazzelli and Pouliquen 2018), nor for the consideration of any nonlocal effect, which have been shown to be crucial in explaining the quasi-static

response of other particulate systems (see Kamrin et al. 2024 for a review, and see references therein). Coming back to the definition of a dense suspension, we can rephrase it as: the suspension's particle volume fraction is “close enough” to its jamming value that it leads to significant non-Newtonian behavior, even for suspensions that have a mostly rate-independent viscosity.

Dilatancy, dilatancy, and shear thickening

From the beginning, shear thickening has had a troubled relationship with dilatancy. Williamson and Heckert argued in (1931) that it was necessarily distinct from dilatancy, in response to which Freundlich and Röder stated in (1938) that it was the same phenomenon. Let us state here clearly that shear thickening and dilatancy are, indeed, two different phenomena. Dilatancy, which we discuss in this section, is a universal feature of all frictional particulate systems (both dry granular matter and suspensions), whereas shear thickening is not.

Frictional granular piles (wether immersed or not) have stable mechanical states for a range of packing fractions [$\phi_{\min}$, $\phi_{\max}$] starting below and ending above the jamming packing fraction $\phi_c^{\mu_p}$ (see Andreotti et al. 2013 and Fig. 3, top). This has important consequences for their transient behavior under deformation, as it leads to the well-known Reynolds dilatancy effect (Reynolds 1885). When sheared at fixed pressure and in quasi-static conditions, dense granular piles with initial packing fractions $\phi > \phi_c^{\mu_p}$ cannot flow without dilating. The packing fraction thus decreases with the strain $\gamma$ until it reaches $\phi_c^{\mu_p}$. Conversely, if the medium is initially loose, the packing fraction increases with $\gamma$, until it reaches $\phi_c^{\mu_p}$. After a strain $\gamma_0 \sim O(1)$, the system has forgotten its initial state (loose or dense), and simply flows at $\phi_c^{\mu_p}$ (Andreotti et al. 2013; Singh et al. 2013). This property of being mechanically stable at different packing fractions, and of compacting or dilating under shear, is due to the existence of frictional solid contacts between the particles. In contrast, for frictionless hard spheres, the conditions of mechanical equilibrium and non-overlap between

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Granular pile under mechanical equilibrium. Range of packing fractions for which mechanical equilibrium under gravity is possible for a granular pile, as a function of the microscopic solid friction coefficient $\mu_p$</figcaption></figure>
grains impose only one possible coordination number for the pile (Andreotti et al. 2013). This suggests that perfectly frictionless beads packs at equilibrium only have access to one packing fraction, and that this packing fraction is $\phi_c^{\mu_p}=0$ (see Fig. 3, bottom), a conclusion that is supported by numerical simulations (Peyneau and Roux 2008) and experimental work (Clavaud et al. 2017).

Additionally, we have seen that under steady but non quasi-static flow, the packing fraction $\phi$ decreases with the viscous number $J$. By contrast with Reynolds dilatancy, which is a transient phenomenon, this dilatancy under flow is simply the steady-state constitutive law for the packing fraction.

Both dilatancies can be coupled when shearing a suspension with an initial packing fraction $\phi_i \neq \phi(J)$ at a given $J$. In this case, the packing fraction of the suspension will increase or decrease with strain until it reaches $\phi(J)$. This transient dilation or compaction towards the steady-state value $\phi(J)$ has important consequences on the behavior of suspensions after a change in the flow conditions. Indeed, it must be accompanied by an inward or outward liquid flow, and thus a positive or negative pore pressure between the grains. The suspension’s behavior shortly after a change in $J$ is then set by this transient flow and its interaction with the grains, rather than the steady-state suspension’s rheology (Athani et al. 2022).

## The frictional transition scenario and the Wyart and Cates model

As discussed in Subsection “Constitutive laws for dense suspensions in simple shear and steady-state: why shear thickening is unexpected”, dimensional analysis requires that an additional ingredient, namely, a stress scale, is added to the parameters describing the suspension so that its viscosity can depend on the shear stress (or rate). A key idea put forward by Seto et al. (2013) to explain shear thickening in dense suspensions was the occurrence of frictional contacts between particles above a certain critical force. This frictionless-to-frictional transition led to a paradigm shift in understanding the physics behind DST, and is the core of this perspective.

The main idea on which this series of works has been focused is the stress-driven activation of interparticle friction. The authors suggest that the minimal model for shear thickening (especially DST) should consider only three forces: a short-range repulsive force $F_R$, the lubrication force $F_H$, and frictional contact force $F_C$. They assume the breakdown of the lubrication film, as identified in a previous study by Ball and Melrose (1995), which is of fundamental importance for the formation of contacts between particles. Since lubrication breakdown occurs in Stokes flow, it is assumed to

be rate-independent (Ball and Melrose 1995; Morris 2020). This, along with experimental evidence that roughness drastically affects shear-thickening behavior, motivated simulation studies to include frictional contacts (Seto et al. 2013; Mari et al. 2014). The resulting frictional transition scenario is as follows. The short-range repulsive force $F_R$ leads to a critical stress $\tau_R \sim F_R/\pi a^2$, which must be overcome for the particles to form frictional contacts (where $a$ is the particle size). Below this critical stress $\tau_R$, external shear is balanced by the repulsive force, which keeps particle surfaces apart. As shear stress (or rate) increases, that is, for $\tau > \tau_R$, the particles start forming more and more frictional contacts. In the stress states $\tau \ll \tau_R$, the particles are in an effective frictionless state, and the suspension’s rheology evolves on the frictionless branch (see Fig. 4 A), with a viscosity diverging at the frictionless jamming point $\phi_c^{\mu_p}=0$ (corresponding to random close

<figure><img src="image_6.png" /></figure>

**Fig. 4** Shear-thickening behavior in a dense suspension, in the framework of the Wyart and Cates model. (A) The two accessible rheological branches. In red, the frictional branch, diverging at $\phi_c^{\mu_p}=0$, and in blue, the frictionless branch, diverging at $\phi_c^{\mu_p}=0$. (B), (C), (D) and (E): the different flow curves resulting from the transition from the frictionless to the frictional branch, at different packing fractions. (B): CST at low values of the packing fraction (monotonic $\tau(\dot{\gamma})$ relation). (C): onset of DST at a packing fraction $\phi^*$, with the apparition of a vertical tangent in the flow curve. (D): for $\phi^* < \phi < \phi_c^{\mu_p}=0$, DST between the frictionless and the frictional branches (non-monotonic $\tau(\dot{\gamma})$ relation). Shear stresses $\tau^{*-}$ and $\tau^{*+}$: locations of the vertical tangents for $\tau(\dot{\gamma})$ (equivalently, locations of $\frac{\partial \dot{\gamma}}{\partial \tau} = 0$). (E): for $\phi > \phi_c^{\mu_p}=0$, DST going from the frictionless branch to the shear-jammed frictional state. (F): resulting phase diagram. For $\tau \gg \tau_R$, the suspension is in a frictional state, and for $\tau \ll \tau_R$, it is in an effective frictionless state, thus undergoing a frictionless flow. Above $\phi_c (\tau/\tau_R)$ (which depends on the shear stress $\tau$), no flow is possible
packing in the monodisperse case). For $\tau \gg \tau_R$, almost all contacts are frictional, and the suspension evolves on the frictional branch, which has a higher viscosity (see Fig. 4 A) (Seto et al. 2013; Mari et al. 2014). As stated in Subsection “Constitutive laws for dense suspensions in simple shear and steady-state: why shear thickening is unexpected”, the viscosity of the frictional branch diverges at a volume fraction $\phi_c^{\mu_p \neq 0} < \phi_c^{\mu_p = 0}$. The ability of shear-thickening suspensions to switch between a frictionless and a frictional rheological branch leads to the shear-thickening transition. We find it interesting to come back to Williamson and Hecker (1931) and Freundlich and Röder (1938), and point out that both suggested particle-particle contacts may play a role in shear thickening. From their observations, Williamson and Heckert also postulated the existence of a stress scale in the system that would explain the observed unusual behavior. One last thing that is critical to mention here is that friction between the particles and hydrodynamics alone will yield the rate-independent quasi-Newtonian rheology described in Subsection “Constitutive laws for dense suspensions in simple shear and steady-state: whyshear thickeningis unexpected”, Eq. 1. The repulsive force is necessary to obtain a stress-driven transition from the frictionless to the frictional state.

This idea led to the development of a constitutive model by Wyart and Cates (2014), which captures both CST and DST. The authors integrated the activation of frictional contacts into the constitutive laws given in Eq. 1 in the volume-imposed context. To do that, they introduce $f$, the fraction of frictional contacts between the particles, and write it as a function of $\tau/\tau_R$, such that $f(\tau/\tau_R) = 0$ if $\tau/\tau_R \ll 1$ and increases smoothly with $\tau$ to $f(\tau/\tau_R) = 1$ for $\tau/\tau_R \gg 1$. They then write the jamming packing fraction of the suspension as an interpolation between the frictionless and the frictional ones:

$$ \phi_c \left( \frac{\tau}{\tau_R} \right) = f \left( \frac{\tau}{\tau_R} \right) \phi_c^{\mu_p \neq 0} + \left( 1 - f \left( \frac{\tau}{\tau_R} \right) \right) \phi_c^{\mu_p = 0}. \quad (2) $$

Writing that $\eta_s$ and $\eta_n$ scale as $(\phi - \phi_c)^{-2}$ (see for example Guazzelli and Pouliquen 2018), they finally get the rheological curves sketched in Fig. 4 B, C, D and E. Starting from the end, let us first note that $\phi > \phi_c^{\mu_p = 0}$ is not physically accessible, as stated in Subsection “Dilatancy, dilatancy, and shear thickening”. For $\phi < \phi_c^{\mu_p = 0}$, the behavior is quasi-Newtonian at stress levels below $\tau^-$, and for $\phi < \phi_c^{\mu_p \neq 0}$, it is also quasi Newtonian above $\tau^+$. With increasing $\phi$, the viscosity of the frictional branch increases faster than that of the frictionless one. As a consequence, there is a critical volume fraction $\phi^* < \phi_c^{\mu_p \neq 0}$ at which the curve joining the frictionless and frictional branches has a vertical tangent. The exact value of $\phi^*$, and of the corresponding critical shear stress $\tau^*$,

depends on the functional form chosen for $f$. The same goes for the values of $\tau^+$ and $\tau^-$. Note that $\tau^* \neq \tau_R$ (we have $\tau^* \simeq 1.5 \tau_R$, see Wyart and Cates 2014; Singh et al. 2018). Below this critical volume fraction, that is, at $\phi < \phi^*$, the curve connecting the frictionless and the frictional states is monotonic, and the suspension undergoes CST between $\tau^-$ and $\tau^+$. Above $\phi^*$, the flow curve becomes S-shaped. Above $\phi^*$ but below $\phi_c^{\mu_p \neq 0}$, the suspension undergoes CST above $\tau^-$, then DST between $\tau^{*-}$ and $\tau^{*+}$, where $\frac{\partial \tau}{\partial \dot{\gamma}} < 0$. The shear stresses $\tau^{*-}$ and $\tau^{*+}$ correspond to the two points at which $\tau(\dot{\gamma})$ has a vertical tangent. They are associated with shear rates $\dot{\gamma}^{*-}$ and $\dot{\gamma}^{*+}$. Both $\dot{\gamma}^{*-}$ and $\dot{\gamma}^{*+}$ decrease with the packing fraction, with $\dot{\gamma}^{*+} \to 0$ when $\phi \to \phi_c^{\mu_p \neq 0}$ and $\dot{\gamma}^{*-} \to 0$ when $\phi \to \phi_c^{\mu_p = 0}$. Again, note that the DST onset stress $\tau^{*-}$ is different from $\tau_R$ (for one thing, $\tau^{*-}$ depends on $\phi$). Finally, when the frictional state is jammed, that is, for $\phi > \phi_c^{\mu_p \neq 0}$, the system can only flow at low and intermediate stresses, while frictional contacts cause the suspension to shear-jam at higher stresses. In this case, the flow curve tends towards zero shear rate at high stresses.

Note that by construction, the Wyart and Cates model is a mean-field and steady-state model: it is not designed to capture any transient behavior or describe local fluctuations or heterogeneities, similarly to Eq. 1. The original construction by Wyart and Cates defined $f$ in terms of particle pressure (Wyart and Cates 2014), but these forms are interchangeable, since shear and normal stresses are related by an $O(1)$ ratio. Following this, work by Singh et al. (2018) showed that the Wyart and Cates model is able to predict the full rheological response of concentrated shear-thickening suspensions, that is: viscosity, normal stress differences, and particle pressure. The authors demonstrated that viscometric functions can be cast into functional forms that can be parameterized by the volume fraction, the applied stress relative to the repulsive force scale, and the interparticle friction coefficient.

Let us now come back to what happens in shear-thickening suspensions when $\phi > \phi_c(\tau/\tau_R)$. In accordance with the literature, we term the crossing of this $\phi_c(\tau/\tau_R)$ line with increasing stress as shear-jamming (SJ). However, this application deserves a comment. Here, as long as the applied stress $\tau$ is such that $\phi > \phi_c(\tau/\tau_R)$, the shear-thickening suspension stays jammed, but it unjams if this stress is decreased or removed. If the stress is lowered to values such that $\tau < \tau^-$, the suspension goes back to its stable frictionless, flowable state. This is different from shear-jamming in non shear-thickening suspensions, which is caused by the creation, under shear, of a microstructure that endures even at vanishing (but finite) stresses. In this case, the shear-jammed suspension stays jammed in the direction of the previously applied stress, even at very small stresses. As we have seen in Subsection “Constitutive laws for dense suspensions in
simple shear and steady-state: why shear thickening is unexpected”, non shear-thickening suspensions do not have an internal stress scale. In order to undergo shear-jamming, they need to be dense enough, and to accumulate a certain amount of strain, no matter the value of the applied stress leading to this strain. SJ for shear-thickening suspensions is shear-jamming *under stress* (stress is necessary to achieve and to maintain the jammed state), while SJ for non shear-thickening suspensions is *strain-induced shear-jamming* (the shear-jammed state is created by the accumulation of strain, and provided it stays finite, the stress can become very low without the shear-jammed state failing). For the sake of readability, we will call all these behaviors shear-jamming, but we invite the reader to keep in mind the differences between them. We felt it was important to clarify that SJ for shear-thickening and for non shear-thickening dense suspensions are conceptually different, despite being called the same. This similarity in naming is due, in part, to the fact that for a while, suspensions of cornstarch in water were seen as prototypical examples of dense suspensions. As we have seen, not all dense suspensions shear thicken, and cornstarch particles suspended in water are adhesive to a certain degree. Cornstarch suspensions in water are therefore neither prototypical dense suspensions, nor prototypical shear-thickening dense suspensions.

Finally, note that the previous discussion does not consider cases where the suspension has an open boundary. In this case, dilatancy under shear may cause the particles to deform the air-suspension interface. This induces a capillary stress on the particles protruding from the interface (Cates and Wyart 2014; Etcheverry et al. 2023). The physical aspect of the suspension’s surface is an experimental signature of this, turning from smooth and shiny to matte and corrugated (Cates and Wyart 2014; Han et al. 2018). This capillary stress can be enough to freeze the suspension in a shear-jammed state, causing granulation (Cates and Wyart 2014; Hodgson et al. 2022).

**Other models: what if lubrication didn't break down?**

In the previous section, we assumed that particles suspended in a Newtonian fluid can come close enough to each other to form frictional contacts. As discussed, this idea is based on the assumption that the lubrication films between particles cannot thin indefinitely, and have to break up at some point. However, lubrication breakdown is not always assumed. Earlier work by Jeffrey and co-workers (Jeffrey and Onishi 1984; Jeffrey 1992) led to the development of hydrodynamic interactions between a pair of particles in terms of resistance matrix and relative particle velocity. Based on this, Brady and coworkers developed a discrete particle simulation

tool, called the Stokesian Dynamics approach (Bossis and Brady 1987; Brady and Bossis 1988). This approach splits the hydrodynamic interactions into close-ranged or near-field interactions (lubrication) and far-field (full hydrodynamics). With this approximation, the leading terms in the resistance matrix associated with the normal and tangential motions of the particles diverge as 1/h and 1/ log(h) respectively, *h* being the surface separation. This method is able to well reproduce the rheology of Brownian (Morris and Katyal 2002) and non-Brownian (Foss and Brady 2000) suspensions until strong shear thickening is observed experimentally. As the shear rate (or stress) is increased, the particles come closer and closer. Since the hydrodynamic stress scales as 1/h, the dissipative forces become significantly larger than external deformation. These hydrodynamically nearly touching particles align along the dominant principal axis, where a strong lubrication force between small gaps causes strong correlated motion. This led to the proposition of the hydrocluster mechanism by Wagner and Brady (2009). The hydrocluster model successfully reproduces the onset of shear-thickening and weak continuous shear thickening; but it cannot reproduce strong CST or DST.

The main challenge with the hydrodynamics approach is the basic assumption of smooth hard-sphere particles, whereas in real life smooth hard spheres do not exist. Based on this, Jamali and Brady (2019) proposed a modified lubrication approach. Using a discrete particle simulation approach called dissipative particle dynamics, the authors simulated explicit asperities on top of smooth particles. In this case, when the particles surfaces are separated by a distance *h*, the particles asperities can be closer than *h*. This leads to asperity-asperity lubrication forces being stronger than base particle-particle lubrication forces. As the asperity size increases, stronger hydrodynamic dissipation leads to larger viscosity, eventually leading to strong CST and, more importantly, DST. Wang et al. (2020) further showed that strengthening the tangential logarithmic divergence with a polynomial one, the terms of polynomial functions being dependent on the size of the asperity, can essentially reproduce the DST. In a related but different approach, Rosales-Romero and coworkers (2024) consider the effect of confinement between asperities on the suspending fluid’s viscosity, and show that this can also lead to shear thickening.

To conclude this section, what is clear is that idealized smooth hard spheres do not exist in real life. Simulations considering static friction as well as modified hydrodynamics (considering the effect of asperities) can reproduce the DST behavior. Simulations considering static friction also reproduce the SJ behavior observed experimentally (Peters et al. 2016; James et al. 2018), since friction keeps particles in enduring contacts and will be present even in the limit of zero shear rate. However, SJ is not expected in the absence
of static friction, as the fluid-mediated forces vanish in the limit of vanishing shear rate, thus restoring a finite viscosity. The origin of DST is still debated, but one thing is clear: Near-contact interactions between particles with lubrication, surface roughness, and repulsion are the minimal ingredients. Surface roughness can either lead to friction between particles (Seto et al. 2013; Mari et al. 2014; Clavaud et al. 2017; Comtet et al. 2017; Singh et al. 2018, 2020; More and Ardekani 2020a) or lead to enhanced lubrication between asperities (Jamali and Brady 2019; Wang et al. 2020; Rosales-Romero et al. 2024).

## Constrained versus unconstrained particle motion: one point of view to unite them all

Both static friction and modified hydrodynamics ideas point in the same direction: (i) ideal smooth particles do not exist (at least in the dense limit) and (ii) an additional force, other than hydrodynamics and contact, is needed to observe shear thickening. These two aspects show that shear thickening can be seen as a *stress-activated* crossover from unconstrained to constrained pairwise tangential particle motions, as postulated by Singh et al. (2020, 2022) and Guy et al. (2018). These constraints may arise from many factors: asperity/roughness at the particle level (Hsu et al. 2018, 2021), faceted particle shapes (Hsiao et al. 2017; Pradeep et al. 2021; d'Ambrosio et al. 2023), interfacial chemistry (James et al. 2018, 2019), among many others (Fig. 5). In the same spirit as what we discussed in Section “The frictional transition scenario and the Wyart and Cates model”, this crossover happens at an onset stress $\tau_R$. At stresses $\tau \ll \tau_R$, the particles are unconstrained,

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Relating origin of constraints and rheology. (A) Relative viscosity $\eta_r$ ($\eta_r = \eta_s$ the shear viscosity defined in Subsection “Constitutive laws for dense suspensions in simple shear and steady-state: why shear thickening is unexpected”) and coordination number $Z$ vs volume fraction $\phi$. (B) Different types of constraints: hard sphere, $\{\mu_s, \mu_r\} = \{0, 0\}$; infinite sliding, $\{\mu_s, \mu_r\} = \{\infty, 0\}$; infinite sliding and rolling, $\{\mu_s, \mu_r\} = \{\infty, \infty\}$. (C) Origin of constraints from the particle perspective: constraints can originate from particle surface roughness or surface chemistry, among others</figcaption></figure>

whereas at stress levels $\tau \gg \tau_R$, the suspension transitions to a fully constrained state, with a higher viscosity.

The original Wyart and Cates model, along with many simulations, are based on static sliding friction with hydrodynamics (Seto et al. 2013; Mari et al. 2014, 2015a, b; Ness and Sun 2016; Boromand et al. 2018; Singh et al. 2018; More and Ardekani 2020a, b). Singh et al. introduced rolling friction (2020; 2022), thus presenting the following picture. The suspended particles impose (i) hard sphere, (ii) sliding, (iii) rolling, and (iv) twisting constraints, as sketched in Fig. 5. The hard sphere constraint simply means that no two particles can occupy the same volume. If it is the only one taken into account, any occurring contact is considered to be frictionless. The other terms refer to the constraints on the relative motion between particles, pointing to the theoretical limits of no sliding, no rolling (together with no sliding) and no twisting (with no sliding and rolling). Note that in two-dimensions, only sliding and rolling need to be considered, since the twist mode can only exist in three dimensions. The friction coefficients (or modes) that are introduced for the sliding and rolling constraint terms restrict the relative motion of the particles and stabilize the contact network. Instead of having a single particle-particle friction coefficient, we thus now have both a sliding particle friction coefficient $\mu_s$ and a rolling particle friction coefficient $\mu_r$. From this, Maxwell counting arguments (van Hecke 2009; Behringer and Chakraborty 2018; Song et al. 2008; Santos et al. 2020) give a lower and an upper bound to the isostatic number of contacts per particle $Z_{iso}$. For finite values of friction, mechanical stability in dimension $d$ is achieved for $d(d + 1)/(2d - 1) \le Z_{iso}^{\{\mu_s, \mu_r\}} \le 2d$. This isostatic condition then modulates the jamming volume fraction $\phi_c^{\{\mu_s, \mu_r\}}$, which decreases as the relative motion is constrained. In three dimensions, the jamming volume fraction ranges from the lower bound $0.365 \le \phi_c^{\{\mu_s, \mu_r\}} \le 0.65$ (see Fig. 5). The upper limit is roughly close to the so-called random close packing (RCP) limit in dry granular physics (van Hecke 2009; Berryman 1983; Visscher and Bolsterli 1972). Simulations conducted in this framework are capable of reproducing CST, DST, and SJ, and shed light on the specific effects of the different types of constraints (Singh et al. 2020, 2022).

The premise of this approach is that the physical or chemical origins of the particle-particle interactions are less important than their overall impact in constraining the ability of neighboring particles to move relative to each other. As already mentioned, asperities, faceted particle shapes, adhesive forces etc. all lead to constraints on relative particle motion in sliding, rolling, and twist modes, which in turn lower the jamming volume fraction and enhance shear-thickening behavior. In this spirit, more recently the suspension rheology is beginning to be viewed as a tool to estimate the physics at the particle level (Singh et al.
2022). As a consequence, manipulation of, for example, the shape (d'Ambrosio et al. 2023; Hsiao et al. 2017; Pradeep et al. 2021), deformability (Chen et al. 2023), or interfacial chemistry (James et al. 2018, 2019) of the particles changes the constraints and affects the bulk rheology, allowing one to tune it. First-order deviations from a spherical shape (without manipulating the chemistry) due to asperities (Hsu et al. 2018, 2021), faceted (Hsiao et al. 2017; Pradeep et al. 2021) and / or flattened globular faces (d'Ambrosio et al. 2023; Estrada et al. 2008, 2011); adhesion due to hydrogen bonding effects (James et al. 2018, 2019) or a change in pH (Laun 1984), can all be understood in terms of constraints. By design, these concepts cannot however be extended to more complex particle shapes, such as cuboids or super balls (Cwalina et al. 2017; Cwalina and Wagner 2016; Royer et al. 2015).

## Beyond a mean-field description: fluctuations and frictional network

The mean-field models discussed so far (Wyart and Cates model and constraints-based models) are quite promising, as they reproduce strain-averaged steady-state experimental (Guy et al. 2015, 2018; Lin et al. 2015; Royer et al. 2016) and simulation (Singh et al. 2018, 2019, 2020, 2022; Ness and Sun 2016; More and Ardekani 2020b, a) data sets for dense shear-thickening suspension flows. However, by construction, they do not consider large-scale or long-range spatial correlations that are often associated with jamming or rigidity transitions in particulate systems (DeGiuli et al. 2015; Andreotti et al. 2012, 2013; Olsson and Teitel 2007).

They are also agnostic to the geometrical and topological structure of frictional force chains and do not consider the role and emergence of inherent large and intermittent fluctuations in the sample, which we discuss below.

As discussed in previous sections, shear thickening is driven by stress-activated constraints, which we chose to interpret and model as solid friction. In particulate systems, it is established that friction leads to jamming and shear-induced jamming (Cates et al. 1998; Bi et al. 2011; Behringer and Chakraborty 2018). Let us therefore start by providing a granular perspective. In granular physics, the emergence of jamming has been shown to be intimately related to the emergence of enduring contacts that lead to robust mesoscale frictional contact and force networks (Jaeger et al. 1996; Farr et al. 1997; Cates et al. 1998; Behringer and Chakraborty 2018; Bi et al. 2011; Majmudar and Behringer 2005). In Cates et al. (1998), the authors define a force chain as a linear string of rigid particles in point contact. They further note that under shear or any external deformation, once the force chains emerge along the principal compressive direction, this structure, together with an orthogonal support from other force-bearing particles, leads the system to jam (or behave like a solid). This is illustrated in Fig. 6 (A). Contemporaneous experiments from Behringer and coworkers experimentally validated these ideas, using photoelastic discs that allowed visualization of the local stress (force) on the particles and popularized the force chain concept. An example of such an experimental system is presented in Fig. 6 (B). Series of studies on photoelastic disks have proven to be useful in obtaining local forces/stresses (Majmudar and Behringer 2005), establishing heterogeneity in particle forces (Howell et al. 1999), and experimental validation of

<figure><img src="image_8.png" /><figcaption>**Fig. 6** Jamming, force chains, and discontinuous shear thickening. (A) Two-dimensional schematic as proposed by Cates et al. (1998) showing force chains, other load-bearing and rattler particles, that is, particles that don't participate in frictional contacts. (B) A shear-jammed state in a simple shear experiment using photoelastic</figcaption></figure>

discs (Ren 2013), where the bright colors show force chains percolating in compressive and tensile directions. (C) Frictional contact network of low-viscosity and high-viscosity states of a shear-thickening suspension; based on simulations from Seto et al. (2013) © Ren J (2013). All rights reserved
shear-jamming (Bi et al. 2011; Ren et al. 2013; Zhao et al. 2019).

Previous experimental studies had shown that shear thickening is accompanied by large-scale temporal fluctuations: In the classical rheological perspective, in a rate-controlled set up, shear stress (or viscosity) often fluctuates by orders of magnitude at the critical rate (Hoffman 1972, 1974; Lootens et al. 2003, 2004; Bender and Wagner 1996; Boersma et al. 1991; D’Haene et al. 1993). This observation, coupled with the knowledge that frictional contacts are implicated in the emergence of shear thickening, suggests that these large-scale temporal fluctuations should arise from major changes or rearrangements in the force and contact networks (Lootens et al. 2003; Cates et al. 1998). This was demonstrated by simulations by Seto et al. (2013); Mari et al. (2014), who established that a dynamic network of system-spanning frictional contacts is responsible for the high viscosity of the shear-thickened (constrained) state. Furthermore, Mari et al. (2014) also showed that traditional microstructural responses are subtle across the pure DST regime, that is, from low-viscosity (low stress) to high-viscosity (large stress) flowing states, which does not explain the orders of magnitude change in viscosity. This work paints the following picture for the frictional network (Fig. 6 (C)): At low stress conditions $τ ≪ τ_R$, all contacts are unconstrained or frictionless. As stress increases, $τ ≈ τ_R$, frictional contacts form and force chains emerge as roughly linear structures along the compression axis only; this is referred to as low-viscosity condition in Fig. 6 (C). Eventually, at $τ ≫ τ_R$, the suspension reaches a fully shear-thickened (high-viscosity) state and is found to have force chains along the compressive and tensile directions, which stabilize the force chains in the compressive direction and prevent their buckling under shear. These observations directly point to similarity with the propositions of Cates et al. (1998); Radjai et al. (1998), and Bi et al. (2011) and reaffirm the intimate connection between DST and SJ from a frictional network perspective.

Thomas et al. (2018, 2020) showed that shear thickening is accompanied by the emergence of correlations in the force space. As previously stated, Mari et al. (2014) demonstrated that traditional correlations show only subtle differences across pure DST. This means that the network space of frictional contacts and forces, rather than the physical space of particles, holds the key to providing a mechanistic understanding of shear thickening. This led to studies using network science and graph theory tools to analyze the topology and geometry of frictional contact and force networks (Sedes et al. 2020, 2022; Gameiro et al. 2020; Boromand et al. 2018; Naald et al. 2024; D’Amico et al. 2025; Sharma et al. 2025; Edens et al. 2019, 2021; Boromand et al. 2018; Nabizadeh et al. 2022). In the context of network science, the physical space of particles is converted into a graph, where each particle is represented as a node, and the frictional

contact between particles is represented by an edge. Gameiro et al. (2020) demonstrated that pure DST is accompanied by the emergence of loop-like structures in the frictional network. The authors used persistent homology to describe the topology of the frictional force network and used total persistence as a measure of the importance of loops (see below) to link the force network with the rheology. The correlation was found to be independent of volume fraction $φ$ and applied stress $τ$, suggesting that viscosity is solely controlled by the persistence of the frictional network. In the same spirit, D’Amico et al. (2025) decomposed the frictional contact network into loops. In network science, a loop (or cycle) is defined as a closed path in a graph, which starts and ends at the same node, traversing a sequence of edges and nodes without repeating any edge or node (except the starting/ending node) (Papadopoulos et al. 2018). Thus, a loop of order $l$ can be thought of as a polygon with $l$ sides. They showed that the suspension’s relative viscosity $η_r$ can be expressed in terms of the number of third-order loops $n_3$. Moreover, this correlation is insensitive to the full phase space ($τ, φ, μ_s$) that controls the rheology of the dense suspension. This has deep connections with the rigidity concepts in graph theory. According to Laman’s theorem (Laman 1970), triangles are found to be the smallest rigid structures, that is, they do not deform under an externally applied load. More recent work has attempted to probe the concepts of rigidity of the local frictional contact network and relate it to shear thickening. Using different methods, Goyal et al. (2024) and Naald et al. (2024) probed the local rigidity of the frictional network and found the onset of system-spanning minimally rigid clusters in the vicinity of $φ^*$, the volume fraction that marks the onset of DST. The frictional contact network and the associated minimally rigid clusters for different stress levels $τ/τ_R$, at a packing fraction $φ > φ^*$, that is, in the pure DST regime, are illustrated in Fig. 7 (B). In particular, Naald et al. (2024) suggested that the onset of rigidity drives the enhanced dissipation allowing for high viscosities $η_r > 1000$, thus showing that rigidity (rather than DST) is a precursor to SJ.

Sedes et al. (2022) used the K-core network to analyze the contact network clusters in a percolated network, as the suspension shear thickens. The $k$-cores are subgraphs (or “clusters”) of the frictional subnetwork, where all nodes have $k$ or more contacts with other nodes in the subgraph. Using the Molloy-Reed criterion (Molloy and Reed 1995), the authors showed that percolation occurs in both CST and DST, which means that percolation is not the discriminating measure for strong CST and DST. They defined $χ_σ = dσ/dγ$, a rheological susceptibility, and showed that the appearance of 3-core structures coincides with the divergence of $χ_σ$. This hints at the onset of DST (that is, the $(τ^*, φ^*)$ point in the state diagram) being a non-equilibrium critical point, an aspect which needs to be further pursued. In this direction, more recently (Ramaswamy et al. 2023) have noted that
shear thickening behavior is instead governed by two separate critical points, frictional and frictionless jamming points; demonstrating a crossover scaling framework to characterize shear thickening in cornstarch and silica suspensions. Finally, Nabizadeh et al. (2022) used community detection to analyze the frictional contact network, and found that the force clusters formed in the pure DST regime are more tightly constrained by their surrounding clusters than the ones in the CST regime.

Results from network science allow to populate the state diagram from Fig. 4 (as originally proposed by Wyart and Cates), presented here in Fig. 7 (C). In this form of diagram, a new rigidity line emerges where minimally rigid clusters begin to span the system. This state in shear-thickening suspensions lies below the onset of SJ and we believe is the true precursor to SJ, rather than DST, as previously hypothesized (Peters et al. 2016; Morris 2020).

To conclude this section, let us mention that the effect of constraints (sliding and rolling modes) on the rheology is also beginning to be established. How it affects the frictional network has, however, not yet been explored. Preliminary work suggests that systems with similar viscosity originating from various combinations of {$\mu_s, \mu_r$} can have very differ-

ent frictional networks (Sharma et al. 2025). However, the proposition of Cates et al. (1998) (Fig. 6 (A)) only considers sliding constraints and thus an orthogonal support is needed for the force chains to not buckle under external deformation. More work is needed to consider how constraints affect the minimal requirement for the force chains or the particulate system to be mechanically stable.

## Beyond simple shear: realistic flows and fluid dynamics

In the previous sections, we discussed mean-field models such as the Wyart and Cates model, as well as some of their limitations, and pointed to network science tools and how they allow us to build our understanding of dense suspensions rheology when mean-field models are no longer enough. In the context of steady-state flows and either in simple shear or in rheometer experiments designed to be as close as possible to a simple shear flow, mean-field models have been shown to capture both experimental (Guy et al. 2015, 2018) and simulation data sets (Mari et al. 2014; Ness and Sun 2016; Singh et al. 2018, 2020). Additionally, even in an “ideal” flow in a rheometer, the presence of S-shaped flow curves or the onset of DST leads to instabilities, shear banding, and spatiotemporal fluctuations (Hermes et al. 2016; Saint-Michel et al. 2018). In this section, we discuss recent experimental studies that explore the behavior of shear-thickening suspensions in even more complex types of flow. In particular, we will discuss the extent to which the Wyart and Cates model can still provide some basis to the analysis of these flows.

### Flow down an incline

Flow down an incline is among the archetypical classic problems in fluid mechanics, where the Reynolds number framework based on pure fluid predicts (Kapitza and Kapitza 1948) or roll-waves instabilities due to turbulence (inertia) in open channels. Such instabilities are also observed in other types of complex fluids, such as power-law fluids (Hwang et al. 1994; Allouche et al. 2017), mud flow, and granular materials (Forterre and Pouliquen 2003; Forterre 2006). In all of these cases, the instability is driven by inertia.

In the case of a Newtonian liquid, the force of gravity down the inclined plane can be written as $g \sin(\theta)$, where $\theta$ and $g$ are the angle of plane and acceleration due to gravity, respectively. This would be balanced by the viscous force $\eta u_{zz}$, where $\eta$ is the fluid’s viscosity. Here, we focus on a two-dimesional Cartesian coordinate system with $z = 0$ denoting the inclined plane and $x$ pointing downslope, with $(u, w)$ the velocity field (see Fig. 8 B.). This implies $\eta U / \rho H^2 \sim g \sin(\theta)$, that is, $Re = \rho U H / \eta \sim U^2 / (g H \sin(\theta))$, with $\rho$ the fluid’s density. In this case, the Kapitza threshold is

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Frictional contact network, rigidity and SJ. (A) $\tau(\dot{\gamma})$ flow-curve showing DST, based on simulations of Naald et al. (2024); (B) Frictional network (left) and associated rigid minimally rigid cluster (right) for stress levels that correspond to onset of CST, onset of thickened-state, and fully thickened-state at a packing fraction $\phi > \phi^*$, that is, in pure DST regime. (C) Flow state diagram delineating the flow behaviors of shear-thickening dense suspensions in the $\tau-\phi$ space. The lines separate different flow regimes based on the mean-field Wyart-Cates model: shear-jammed (grey), pure DST (pink), DST-SJ (purple), and isotropic frictionless jamming (black). DST and SJ lines are based on the Wyart-Cates model giving the locus of, respectively, states with $\frac{\partial \dot{\gamma}}{\partial \tau} = 0$ and $\phi_c(\tau/\tau_R)$ (Eq. 2). Blue and red colors show frictionless and fully frictional states. The dashed region with black lines indicate the system-spanning minimally rigid states. Adapted from Naald et al. (2024)</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 8** “Oobleck wave” down an incline plane. (A) Photo of the developing waves. (B) Sketch of the problem. (C) Measured rheology, and fits to the Wyart and Cates model. (D) Oobleck waves mechanism. Adapted</figcaption></figure>

expected to be $Re_K = 5/(6 \tan(\theta))$, which is found to be such that $Re_K \gg 1$ for the typical tilt angles studied.

Earlier experimental observations of the flow of a shear-thickening suspension down an incline by Balmforth et al. (2005) showed the occurrence of instability waves at Reynolds number $Re \lesssim 1$, much smaller than expected and therefore unexplained. The authors concluded their article by “What-ever the physical origin of the observed phenomenon, we hope that our study will motivate further investigations of this curious material.”. This problem has recently been revisited by Darbois Texier et al. (2020, 2023) in the context of material instability that originates from the rheology of cornstarch and its coupling with the free surface. They reported the emergence of an unstable kinematic wave termed the “Oobleck wave” that originated at the onset of DST ($\tau^*, \phi^*$), as predicted by the Wyart and Cates model. The S-shaped flow curves with negative slope (see Fig. 8 C), coupled with the free surface, trigger an instability in the medium that does not require inertia. A mismatch between hydrostatic pressure and basal stress $\tau_b$ leads to the amplification of the kinematic surface wave, and the unstable mode propagates at the speed of the surface kinematic waves. The mechanism proposed in Darbois Texier et al. (2020) is illustrated in Fig. 8.

## Flow through a constriction and liquid migration

Flow of dense particulate materials through an extruder or through a constriction often leads to liquid migration (LM) (O’Neill 2019) (also known as self-filtration Haw 2004). Solids that build up near the constriction can cause jamming, thus impeding the flow, and the material becomes more dilute. O’Neill (2019) performed extrusion experiments on a mixture of cornstarch and water, and coupled their findings with stress-controlled steady-shear rheology. They

from Darbois Texier et al. (2020) (article and figures licensed under a Creative Commons Attribution 4.0 International License, copy of the license: https://creativecommons.org/licenses/by/4.0/)

showed that the volume fraction $\phi_{out}$ of the extruded material is simply controlled by the scaling variable $Q/R_d^3$, with $Q$ and $R_d$ being the volumetric flow rate and the radius of the die, respectively. The term $Q/R_d^3$ sets the shear rate scale in the die. The LM state diagram shows the master curve as a “phase boundary” for different initial volume fraction $\phi_{in}$, barrel radius $R_b$ and die radius $R_d$ (see Fig. 9). For low volume fractions $\phi_{in}$ of input material, $\phi_{out} \simeq \phi_{in}$. For an initial volume fraction $\phi_{in}$ larger than a certain critical value $\phi_{in}^c$, liquid migration occurs: $\phi_{out} < \phi_{in}$. Using steady-shear rheology, $\phi_{in}^c$ was found to coincide with the onset of DST $\phi^*$ as predicted by the Wyart and Cates model.

## Flow through a pipe

Pipe flow is one of the most usual chemical engineering problems, often encountered in industrial and natural settings. In the case of a Newtonian liquid of viscosity $\eta$ flowing through a pipe of radius $R$ with a pressure gradient $\Delta P$, the flow rate in the laminar regime is $Q \propto R^4 \Delta P / \eta$. Bougouin et al. (2024) studied the case of shear-thickening suspensions,

<figure><img src="image_11.png" /><figcaption>**Fig. 9** Liquid migration in the flow of a shear-thickening suspension through a constriction. Left: experimental data. Right: one-dimensional model, using interpolated jamming packing fraction from the Wyart and Cates model. The quantity $\Delta\langle u \rangle/u$ quantifies the particle phase motion relative to the suspending fluid. Adapted from O’Neill (2019)</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 10 Flow of shear-thickening suspensions through a pipe:** flow-rate saturation due to the nucleation and propagation of a frictional soliton. Top left: Sketch of the problem. Bottom left: flow rate saturation, comparison between the experimental data (symbols) and the predictions of the Wyart and Cates model (lines). Right: Proposed mechanism for the nucleation and upstream propagation of the frictional soliton. Adapted from Bougouin et al. (2024) (article and figures distributed under Creative Commons Attribution License 4.0 (CC BY))</figcaption></figure>

where they observed some intriguing physics. They showed that for volume fractions $\phi < \phi^*$, the flow rate increases linearly with the mean shear stress at the wall, as expected from a laminar flow. Beyond the onset of DST, at $\phi > \phi^*$, and at wall stress $\tau_w$ levels below $\tau^{-}(\phi)$, the flow rate still increases linearly, like that of a Newtonian liquid. The near-wall flow is uniform and steady and almost an order of magnitude smaller than the mean-flow velocity, in line with what is observed for a Poiseuille-like laminar flow. At stress levels $\tau_w > \tau^{-}(\phi)$, the flow rate becomes independent of forcing (even for an increase of an order of magnitude). This is illustrated in Fig. 10, bottom left. A spatiotemporal inspection reveals the propagation of a frictional soliton that separates two laminar flow regions (illustrated in Fig. 10, right). In these regions, the velocities are orders of magnitude below the mean velocity, as in the low-forcing regime. The frictional

soliton however is a plug-like flow. The authors show that the frictional soliton is longitudinally localized, propagates upstream, and preserves its shape spanning the cross sections of the pipe, with the dissipation concentrating over a flow length of the order of pipe radius. They propose a minimal mechanism for the upward propagation of the soliton that is based on a transient Reynolds-like dilation and a subsequent Darcy backflow. This mechanism is illustrated in Fig. 10, right. The coupling between, on one hand, transient dilation in a shear-thickening suspension forced into its frictional state, and on the other hand, the subsequent Darcy backflow is further discussed in Athani et al. (2025), showing that while frictional contacts dominate the steady-state behavior of shear-thickening suspensions, the fluid phase and hydrodynamics are very important to describe transients.

Once again, the Wyart and Cates model predicts the onset of the deviation from the typical Newtonian behaviour. However, it incorrectly describes what happens once the flow rate saturates. This is to be expected, since in this regime the suspension is separated into two distinct phases, while the Wyart and Cates model considers the suspension as one continuous phase.

Note that although other solitary waves or intermittent flow structures had already been observed in suspensions pipe flows, the frictional soliton is a new kind of structure. Its specificity is detailed in the extended Discussion section in Bougouin et al. (2024).

## Air injection in a shear-thickening suspension

Two different papers, by Ozturk et al. (2020) and Lilin et al. (2023), explored the effect of air injection into a shear thickening suspension. In Ozturk et al. (2020), the authors study the problem of air invasion in a Hele-Shaw cell (see Fig. 11).

<figure><img src="image_13.png" /><figcaption>**Fig. 11 Air invasion in a Hele-Shaw cell filled with suspensions of cornstarch in water.** Top: measured rheological curves. Bottom: phase diagram of the different response regimes, in the injected pressure $P$-packing fraction $\phi$ plane. The pure DST, DST-SJ, and rigidity regimes cannot be placed here, and the positions of $P^*$ (equivalent of $\tau^*$) and</figcaption></figure>

$\phi^*$ are only estimated. The purple line is the equivalent of that of Fig. 4, with $P_R$ being the equivalent of $\tau_R$. Adapted from Ozturk et al. (2020) (article and figures licensed under a Creative Commons Attribution 4.0 International License, copy of the license: https://creativecommons.org/licenses/by/4.0/)
They first measure the steady-state rheological flow-curve of cornstarch suspensions in water, and use this to estimate the values of $\phi^*$ and $\phi_c^{\mu_p \neq 0}$, the packing fractions marking the onset of the DST and SJ regimes respectively (see Fig. 11, top). They show that below a critical air injection pressure $P^*$ (equivalent of $\tau^*$ as defined in Section “The frictional transition scenario and the Wyart and Cates model”), the suspension exhibits viscous fingering, as would be the case for a Newtonian fluid. Above $P^*$, the behavior depends on the suspension’s packing fraction. Below $\phi^*$, it still exhibits viscous fingering, behaving like a Newtonian fluid. Between $\phi^*$ and $\phi_c^{\mu_p \neq 0}$, the authors observe what they call dendritic fracturing. Upon air invasion, the system fractures locally into a dendritic network, which then relaxes partially. The pattern is conserved, but the edges of the initial fractures are smoothed out. This can be rationalized using the Wyart and Cates model. Upon air invasion, if the air pressure is larger than $P^*$, the system is locally forced into its frictional state, and transiently jams, as would a suspension of frictional hard spheres subjected to a sudden high stress. Once the high stress front has passed, the suspension can relax to a liquid state, because its frictional, high viscosity branch still has flowable states. As for the frictional soliton discussed in Subsection “Flow through a pipe”, the transient behavior here is quite important to understand the entire response of the suspension, and ideas described and developed in Athani et al. (2025), particularly the effect of particle confinement, allow to rationalize the observations. Then, above $\phi_c^{\mu_p \neq 0}$, the authors observe a large-scale fracturing regime, in which the system also fractures upon air invasion, but forms a fracturing pattern with a much lower number of branches, that doesn’t relax. At these packing fractions, the air pressure prevents the system from evolving on its frictionless branch, so it is forced to stay in a shear-jammed frictional state.

In Lilin et al. (2023), the authors study the formation, growth, and rise of bubbles in a dense cornstarch in water suspension (see Fig. 12). They show that at low enough mass fractions, the bubble behavior is the same as in a Newtonian liquid, while at very large mass fractions the suspension always locally jams and fractures upon air injection. In this case, the “bubble” is really a growing fracture. At intermediate volume fractions, they observe an interesting behavior: at short times, the local shear rate at the bubble border is larger than the critical shear rate $\dot{\gamma}$ above which the suspension shear-thickens, which leads to the suspension initially fracturing. At larger times, the local shear rate $\dot{\gamma}_{\text{loc}}$ at the bubble border decreases. Indeed, $\dot{\gamma}_{\text{loc}} \propto \dot{\Omega}/\Omega$, with $\Omega$ being the bubble volume. As time passes, the bubble volume increases, and the authors show that $\dot{\Omega}$ is time-independent: $\dot{\gamma}_{\text{loc}}$ thus necessarily decreases with time. If this decrease is large enough, the suspension at the border of the bubble relaxes, leading to the rise of a bubble which was initially a fracture. Again, this

<figure><img src="image_14.png" /><figcaption>**Fig. 12** Bubble formation, growth, and in certain cases rise, in suspensions of cornstarch in water, for different mass packing fractions $\phi_m$. Top: Sketch of the experimental system. Middle: two different types of bubbles, depending on the mass packing fraction, and their evolution in time. Bottom: comparison of different bubbles and their surface roughness (quantified by $\nabla h^2$), depending on the mass packing fraction, at a fixed time. Adapted from Lilin et al. (2023) (article and figures distributed under the terms of the Creative Commons CC BY)</figcaption></figure>

can be rationalized within the Wyart and Cates model, in the same way as for the previous paper.

## Conclusion and perspectives

In this perspective article, we have discussed some of the recent advances and current understanding of shear thickening (and particularly DST) observed in dense suspensions. The first studies date back to the 1930s, and it is surprising that the mechanistic basis still remains debated. Still, a minimal model picture has emerged to explain this behavior: a two-state model involving a stress-activated transition, at a certain stress-scale $\tau_R$ resulting from the presence of a short-range repulsion between the particles, from unconstrained to constrained pairwise relative tangential motion. This model can quantitatively predict shear thickening. This general picture points to distinct physics driving the rheology at low ($\tau \ll \tau_R$) and high ($\tau \gg \tau_R$) stresses, i.e., unthickened and thickened states. The low-stress “unconstrained” state is that of classical hydrodynamics with the lubrication film between the particles being intact, while in the high-stress state the lubrication breaks down and the particles come close enough to constrain the pairwise relative motion.

For dense suspensions in general, this paints a picture that we illustrate in Fig. 13. At the smallest scale of the problem
<figure><img src="image_15.png" /><figcaption>**Fig. 13** A hierarchical multiscale view of dense suspensions. Dense suspensions rheology is measured at the macroscopic scale (steady-state strain averaged observation), and can be understood in terms of a hierarchical interrelationship across length-and time-scales. At the smallest length scale, we have a single particle and surrounding fluid, where asperities or roughness (orders of magnitude smaller than the</figcaption></figure>

particles) and interfacial chemistry (with the fluid) lead to microscopic interactions. These microscopic interactions are present at the level of two or more particles and lead to the collective motion of particles, which often originate from constraints between particles. Finally, the correlated motion or the collective behavior (order of 10 or 20 particles) dictates the behavior of dense suspensions

is a single particle immersed in a fluid, where roughness, faceted shapes, surface irregularities, adhesion ... in summary: surface properties, either intrinsic to the particle or rising from particle-fluid interactions, constrain the relative motion between two particles. These microscopic constraints lead, at high enough volume fraction, to correlated non-affine motion at the mesoscale, which can be quantified in terms of frictional force and contact networks. The behavior of these networks, which are as stated mesoscale objects, in turn determines the macroscopic rheology of the suspension. In the case of shear-thickening suspensions, the constraints only play a role at high enough stresses, that is, for $τ ≫ τ_R$.

The state-of-the-art constitutive models for shear-thickening suspensions are mean-field in nature, which means they relate the microscopic constraints directly to the macroscopic bulk response of the material. This inherently assumes that the mean simple shear flow exists; thus ignoring many effects: fluctuations, instability, wall slip, free surface, and presence of boundary. The modeling approach, even though mean-field in nature and devoid of any boundary effects, still interestingly predicts the onset of material instability in various non-viscometric flows, as briefed in Section 7. Of course, it cannot (and should not be expected to) predict the local material state, which would be highly heterogeneous in such cases.

The ideas developed so far only hold for spherical quasi-monodisperse particles (single peaked size), which do not exist in reality. The Wyart and Cates formalism in its original form breaks down with a modest polydispersity in size, although some progress has been made to expand it Singh et al. (2024); Guy et al. (2020); Pednekar et al. (2018); Malbranche et al. (2023). Interestingly, suspensions of quasi-monodisperse particle sizes considering spherical (Guy et al. 2015, 2018; Royer et al. 2016) or irregularly shaped (corn-starch or mineral crystals Richards et al. 2021) particles all show qualitatively similar shear-thickening behavior. However, the theoretical and simulation understanding in terms of jamming and isostaticity is limited to spherical shape. Simulation and theoretical studies that go beyond spherical shapes will be particularly useful, but challenges associated with both implementation of non-spherical granular particles and the correct lubrication forces associated with face-face, edge-edge, and face-edge interactions impede the development.

Real-world application to multiphase flows will also involve understanding the effect of boundaries or walls, which are present in extrusion of dense paste or redox flow batteries, as well as open-surface (e.g. flow down incline) boundary conditions. In the dense limit, wall slip,
localization near walls, particle migration, etc. will be impor-
tant – none of which is currently considered theoretically.
Additionally, current modeling approaches do not consider
transients. However, in real-world flows these need to be con-
sidered (Bougouin et al. 2024; Balmforth et al. 2005; Athani
et al. 2022, 2025). Another point to consider is that in order to
apply these concepts to multiphase flows, additional effects
like surface tension, capillary effects, etc. need to be consid-
ered as well.

Finally, dense suspensions also present a model system to
probe the statistical framework for out-of-equilibrium driven
systems, which remains an open question. Some recent stud-
ies suggest that DST (in a rate-controlled framework) is a
critical point. Extending ideas related to criticality and floppy
modes from glassy and dry granular physics to dense sus-
pensions can open exciting avenues, as the last decade has
witnessed. In this spirit, generalizing the isostatic conditions
to a broad polydisperse setting will be extremely useful.

Take-home messages

In this final section, we list some of the important take-home
messages we wish to put forward in light of everything dis-
cussed so far in this perspective.

1. For our first point, let us go back to Section “The many features of shear thickening in dense suspensions”, where we first discussed the importance of fluid-particles interactions. As we have seen, the nature of these interactions can have a dramatic effect on the overall suspension’s rheology. Yet we don’t always know, let alone control, what these interactions are. In light of this, we believe it is essential to caution the reader against generalizing behaviors observed in a specific experimental system to all dense shear-thickening suspensions. In the same spirit, choices made on how to model different interactions in a simulation code may lead to behaviors that do not necessarily reflect how the modeled interaction influences the “real-world” suspension, or may add another interaction. Modeling friction, for example, can be achieved by introducing some elasticity to the particles, making them slightly elastic (Chen et al. 2023). Therefore, we would like to ask the reader to also be wary of generalizing simulation results to all dense shear-thickening suspensions.

2. Shear thickening in dense, non-Brownian suspensions seems to be viewed as a phenomenon in which the unexpected part is the thickening, and the thickened and shear-jammed states. We consider that it is the opposite, and consequently propose to shift perspective. The Wyart and Cates and constraints-based model both postulate

that shear thickening stems from stress-activated con-
straints or frictional contacts. The short-range repulsive
force is critical to achieve this unconstrained–constrained
transition. It is the existence of this repulsive force
between the particles that allows the suspension to be
able to evolve onto two different rheological branches,
and this ability is what is specific to shear-thickening sus-
pensions. Both limit cases, the fully unconstrained state
and the fully constrained state, have their own particu-
lar rheological behavior. In particular, the rheology of
the fully constrained state is “simply” that of frictional
granular suspensions (Lemaire et al. 2023) (keeping in
mind that it needs high enough stresses to exist). It is
thus not surprising that the fully constrained state behaves
as a dense frictional suspension: that is exactly what
it is. Which means it is also not surprising that above
the stress-dependent frictional jamming packing fraction
($\phi_c$ ($\tau/\tau_R$) as defined in Section “The frictional transition
scenario and the Wyart and Cates model”), the suspen-
sion is shear-jammed in its fully constrained state. What
is specific to shear-thickening suspensions is (i) that they
can, at low stresses, relax back to an unconstrained state
which can flow, and (ii) that they can exist in transitory
states that are neither fully unconstrained nor fully con-
strained: the CST, pure-DST, and DST-SJ states.

3. The previous point warrants a word of caution on using experimental results obtained on the thickened state of a shear-thickening suspension as results that would be general to dense frictional suspensions. It is not always easy to determine whether the suspension is indeed in its fully constrained state, and there might be some regions of the $\tau-\phi$ state diagram which we do not yet understand well enough. In line with the first point of this section, we urge the reader to be careful about hasty generalizations.

4. Although a unified picture is emerging for dense sus-
pensions and dry granular materials, a note of caution is
warranted. In both systems, flow behavior is governed by
the frictional contact network. In dry granular materials, a
percolating frictional network is directly associated with
the onset of shear jamming (Bi et al. 2011; Ren et al. 2013;
Otsuki and Hayakawa 2011; Zhao et al. 2019). In con-
trast, in dense shear-thickening suspensions, percolation
occurs near the pure-DST regime (Boromand et al. 2018;
Sedes et al. 2020; Naald et al. 2024), which lies close to,
but still below, the SJ threshold. Likewise, while rigidity
in dry granular materials aligns with the shear jamming
point (Henkes et al. 2016), the emergence of a rigid fric-
tional network in shear-thickening suspensions appears
at or just before the packing fraction $\phi^*$ where pure DST
is first observed (Naald et al. 2024).

5. Most of the topological and geometrical approaches are
limited to sliding constraints and two-dimensional set-
tings only. As an example, the pebble game method to

<figure><img src="image_16.png" /></figure>
study the minimal rigidity is only applicable to sliding
constraints (available for frictionless and frictional parti-
cle contacts) and does not have a three-dimensional coun-
terpart. Real-world suspensions are three-dimensional.
Thus, to make meaningful predictions in the real world,
efforts are needed to be extended to 3D. Additionally,
given that traditional discrete particle simulations are
computationally expensive, large system sizes close to
jamming are computationally intractable. In this spirit,
machine learning is just starting to be a promising avenue,
which needs more attention (Mandal et al. 2022; Amini-
majd et al. 2025a, b).

**Acknowledgements** The authors thank everyone who participated in the “The Plot Thickens” seminar series, as a speaker or as an attendee. This work is in large part the result of the many interesting discussions held during this monthly gathering. The authors also thank the masked geometer for their punny contribution. A. S. acknowledges the Case Western Reserve University for start-up funding.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References

Allouche MH, Botton V, Millet S, Henry D, Dagois-Bohy S, Güzel B, Ben Hadid H (2017) Primary instability of a shear-thinning film flow down an incline: experimental study. J Fluid Mech 821:1

Aminimajd A, Maia J, Singh A (2025) Scalability of a graph neural network in accurate prediction of frictional contact networks in suspensions. Soft Matter 21(15):2826–2835

Aminimajd A, Maia J, Singh A (2025) Robust prediction of frictional contact network in near-jamming suspensions employing deep graph neural networks. Phys Fluids 37:073306

Andreotti B, Barrat J-L, Heussinger C (2012) Shear flow of non-Brownian suspensions close to jamming. Phys Rev Lett 109:105901

Andreotti B, Forterre Y, Pouliquen O (2013) Granular media: between fluid and solid. Cambridge University Press

Athani S, Metzger B, Forterre Y, Mari R (2022) Transient flows and migration in granular suspensions: key role of reynolds-like dilatancy. J Fluid Mech 949:9

Athani S, Metzger B, Forterre Y, Mari R (2025) Transients in shear thickening suspensions: When hydrodynamics matters. Phys Rev Fluids 10:043301

Ball RC, Melrose JR (1995) Lubrication breakdown in hydrodynamic simulations of concentrated colloids. Adv Colloid Interf Sci 59:19–30

Balmforth N, Bush J, Craster R (2005) Roll waves on flowing cornstarch suspensions. Phys Lett A 338(6):479–484

Barnes HA (1989) Shear-thickening (“dilatancy”) in suspensions of nonaggregating solid particles dispersed in Newtonian liquids. J Rheol 33(2):329–366

Behringer RP, Chakraborty B (2018) The physics of jamming for granular materials: a review. Rep Progress Phys 82(1):012601

Bender J, Wagner NJ (1996) Reversible shear thickening in monodisperse and bidisperse colloidal dispersions. J Rheol 40:899–916

Beris AN, Horner JS, Jariwala S, Armstrong MJ, Wagner NJ (2021) Recent advances in blood rheology: a review. Soft Matter 17(47):10591–10613

Berryman JG (1983) Random close packing of hard spheres and disks. Phys Rev A 27(2):1053

Bi D, Zhang J, Chakraborty B, Behringer RP (2011) Jamming by shear. Nature 480:355–358

Boersma WH, Baets PJM, Laven J, Stein HN (1991) Time-dependent behavior and wall slip in concentrated shear thickening dispersions. J Rheol 35(6):1093–1120

Boromand A, Jamali S, Grove B, Maia JM (2018) A generalized frictional and hydrodynamic model of the dynamics and structure of dense colloidal suspensions. J Rheol 62(4):905–918

Bossis G, Brady JF (1987) Self-diffusion of Brownian particles in concentrated suspensions under shear. J Chem Phys 87(9):5437–5448

Bossis G, Boustingorry P, Grasselli Y, Meunier A, Morini R, Zubarev A, Volkova O (2017) Discontinuous shear thickening in the presence of polymers adsorbed on the surface of calcium carbonate particles. Rheologica Acta

Bougouin A, Metzger B, Forterre Y, Boustingorry P, Lhuissier H (2024) A frictional soliton controls the resistance law of shear-thickening suspensions in pipes. Proc Natl Acad Sci 121(17):2321581121

Bourrianne P, Niggel V, Polly G, Divoux T, McKinley GH (2022) Tuning the shear thickening of suspensions through surface roughness and physico-chemical interactions. Phys Rev Res 4:033062

Brady JF, Bossis G (1988) Stokesian dynamics. Ann Rev Fluid Mech 20:111–157

Brassard M-A, Causley N, Krizou N, Dijksman JA, Clark AH (2021) Viscous-like forces control the impact response of shear-thickening dense suspensions. J Fluid Mech 923:38

Cates ME, Wyart M (2014) Granulation and bistability in non-brownian suspensions. Rheol Acta 53

Cates ME, Wittmer JP, Bouchaud J-P, Claudin P (1998) Jamming, force chains, and fragile matter. Phys Rev Lett 81:1841–1844

Chen C, Naald M, Singh A, Dolinski ND, Jackson GL, Jaeger HM, Rowan SJ, Pablo JJ (2023) Leveraging the polymer glass transition to access thermally switchable shear jamming suspensions. ACS Central Sci 9(4):639–647

Clavaud C, Bérut A, Metzger B, Forterre Y (2017) Revealing the frictional transition in shear-thickening suspensions. Proc Natl Acad Sci USA 1517–1515

Comtet J, Chatté G, Niguès A, Bocquet L, Siria A, Colin A (2017) Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions. Nat Comm 8:15633

Cwalina CD, Wagner NJ (2016) Rheology of non-brownian particles suspended in concentrated colloidal dispersions at low particle reynolds number. J Rheol 60(1):47–59

Cwalina CD, Harrison KJ, Wagner NJ (2017) Rheology of cubic particles in a concentrated colloidal dispersion suspending medium. AIChE J 63(3):1091–1101

d'Ambrosio E, Koch DL, Hormozi S (2023) The role of rolling resistance in the rheology of wizarding quidditch ball suspensions. J Fluid Mech 974:36

D'Amico A, Tu S, Singh A (2025) Topological insights into dense frictional suspension rheology: Third order loops drive discontinuous shear thickening. J Chem Phys 162:214906
Darbois Texier B, Lhuissier H, Forterre Y, Metzger B (2020) Surface-wave instability without inertia in shear-thickening suspensions. Commun Phys 3(1):232

Darbois Texier B, Lhuissier H, Metzger B, Forterre Y (2023) Shear-thickening suspensions down inclines: from kapitza to oobleck waves. J Fluid Mech 959:27

DeGiuli E, Düring G, Lerner E, Wyart M (2015) Unified theory of inertial granular flows and non-Brownian suspensions. Phys Rev E 91:062206

Denn MM, Morris JF (2014) Rheology of non-Brownian suspensions. Annu Rev Chem Biomol Eng 5(1)

Denn MM, Morris JF, Bonn D (2018) Shear thickening in concentrated suspensions of smooth spheres in Newtonian suspending fluids. Soft Matter 14(2):170–184

D'Haene P, Mewis J, Fuller GG (1993) Scattering dichroism measurements of flow-induced structure of a shear thickening suspension. J Colloid Interface Sci 156(2):350–358

Edens LE, Pednekar S, Morris JF, Schenter GK, Clark AE, Chun J (2019) Global topology of contact force networks: insight into shear thickening suspensions. Phys Rev E 99(1):012607

Edens LE, Alvarado EG, Singh A, Morris JF, Schenter GK, Chun J, Clark AE (2021) Shear stress dependence of force networks in 3d dense suspensions. Soft Matter 17(32):7476–7486

Estrada N, Taboada A, Radjai F (2008) Shear strength and force transmission in granular media with rolling resistance. Phys Rev E 78(2):021301

Estrada N, Azéma E, Radjai F, Taboada A (2011) Identification of rolling resistance as a shape parameter in sheared granular media. Phys Rev E 84(1):011306

Etcheverry B, Forterre Y, Metzger B (2023) Capillary-stress controlled rheometer reveals the dual rheology of shear-thickening suspensions. Phys Rev X 13:011024

Farr RS, Melrose JR, Ball RC (1997) Kinetic theory of jamming in hard-sphere startup flows. Phys Rev E 55:7203–7211

Forterre Y (2006) Kapiza waves as a test for three-dimensional granular flow rheology. J Fluid Mech 563:123–132

Forterre Y, Pouliquen O (2003) Long-surface-wave instability in dense granular flows. J Fluid Mech 486:21–50

Foss DR, Brady JF (2000) Structure, diffusion and rheology of Brownian suspensions by Stokesian Dynamics simulation. J Fluid Mech 407:167–200

Freundlich H, Röder H (1938) Dilatancy and its relation to thixotropy. Trans Faraday Soc 34:308–316

Gameiro M, Singh A, Kondic L, Mischaikow K, Morris JF (2020) Interaction network analysis in shear thickening suspensions. Phys Rev Fluids 5(3):034307

Gauthier A, Ovarlez G, Colin A (2023) Shear thickening in presence of adhesive contact forces: the singularity of cornstarch. J Colloid Interf Sci 650:1105–1112

Goyal A, Martys NS, Del Gado E (2024) Flow induced rigidity percolation in shear thickening suspensions. J Rheol 68(2):219–228

Guazzelli É, Pouliquen O (2018) Rheology of dense granular suspensions. J Fluid Mech 852

Guy BM, Hermes M, Poon WCK (2015) Towards a unified description of the rheology of hard-particle suspensions. Phys Rev Lett 115:088304

Guy BM, Richards J, Hodgson D, Blanco E, Poon WCK (2018) Constraint-based approach to granular dispersion rheology. Phys Rev Lett 121(12):128001

Guy BM, Ness C, Hermes M, Sawiak LJ, Sun J, Poon WC (2020) Testing the wyart-cates model for non-brownian shear thickening using bidisperse suspensions. Soft Matter 16(1):229–237

Han E, Wyart M, Peters IR, Jaeger HM (2018) Shear fronts in shear-thickening suspensions. Phys Rev Fluids 3:073301

Haw M (2004) Jamming, two-fluid behavior, and “self-filtration” in concentrated particulate suspensions. Phys Rev Lett 92(18):185506

Hecke M (2009) Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J Phys Condens Matter 22(3):033101

Henkes S, Quint DA, Fily Y, Schwarz JM (2016) Rigid cluster decomposition reveals criticality in frictional jamming. Phys Rev Lett 116:028301

Hermes M, Guy BM, Poon WCK, Poy G, Cates ME, Wyart M (2016) Unsteady flow and particle migration in dense, non-Brownian suspensions. J Rheol 60(5):905–916

Hodgson DJM, Hermes M, Blanco E, Poon WCK (2022) Granulation and suspension rheology: a unified treatment. J Rheol 66(5):853–858

Hoffman RL (1972) Discontinuous and dilatant viscosity behavior in concentrated suspensions. I. Observation of a flow instability. Trans Soc Rheol 16:155–173

Hoffman RL (1974) Discontinuous and dilatant viscosity behavior in concentrated suspensions. II. Theory and experimental tests. J Colloid Interface Sci 46:491–506

Howell D, Behringer RP, Veje C (1999) Stress fluctuations in a 2d granular couette experiment: a continuous transition. Phys Rev Lett 82(26):5241

Hsiao LC, Jamali S, Glynos E, Green PF, Larson RG, Solomon MJ (2017) Rheological state diagrams for rough colloids in shear flow. Phys Rev Lett 119(15):158001

Hsu C-P, Ramakrishna SN, Zanini M, Spencer ND, Isa L (2018) Roughness-dependent tribology effects on discontinuous shear thickening. Proc. Nat. Acad, Sci

Hsu C-P, Mandal J, Ramakrishna SN, Spencer ND, Isa L (2021) Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles. Nat, Com

Hwang C-C, Chen J-L, Wang J-S, Lin J-S (1994) Linear stability of power law liquid film flows down an inclined plane. J Phys D: Appl Phys 27(11):2297

Jaeger HM, Nagel SR, Behringer RP (1996) The physics of granular materials. Phys Today 49(4):32–38

Jamali S, Brady JF (2019) Alternative frictional model for discontinuous shear thickening of dense suspensions: Hydrodynamics. Phys Rev Lett 123(13):138002

James NM, Han E, Cruz RAL, Jureller J, Jaeger HM (2018) Interparticle hydrogen bonding can elicit shear jamming in dense suspensions. Nat Mater 17(11):965

James NM, Hsu C-P, Spencer ND, Jaeger HM, Isa L (2019) Tuning interparticle hydrogen bonding in shear-jamming suspensions: Kinetic effects and consequences for tribology and rheology. J Phys Chem Lett 10(8):1663–1668

Jeffrey DJ (1992) The calculation of the low Reynolds number resistance functions for two unequal spheres. Phys Fluids A 4:16–29

Jeffrey DJ, Onishi Y (1984) Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J Fluid Mech 139:261–290

Jerolmack DJ, Daniels KE (2019) Viewing earth's surface as a soft-matter landscape. Nat Rev Phys 1(12):716–730

Kamrin K, Hill KM, Goldman DI, Andrade JE (2024) Advances in modeling dense granular media. Annu Rev Fluid Mech 56(1):215–240

Kapitza PL, Kapitza SP (1948) Wave flow of thin viscous fluid layers. Zh Eksp Teor Fiz 18:3–28

Kim H, Naald M, Braaten FA, Witten TA, Rowan SJ, Jaeger HM (2024) Shear thickening in suspensions of particles with dynamic brush layers. Soft Matter 20:6384–6389

Laman G (1970) On graphs and rigidity of plane skeletal structures. J Eng Math 4(4):331–340

Laun HM (1984) Rheological properties of aqueous polymer dispersions. Angew Makromol Chem 123(1):335–359
Lemaire E, Blanc F, Claudet C, Gallier S, Lobry L, Peters F (2023) Rheology of non-Brownian suspensions: a rough contact story. Rheologica Acta

Lilin P, Elkhoury JE, Peters IR, Bischofberger I (2023) Fracture and relaxation in dense cornstarch suspensions. PNAS Nexus 3(1):451

Lin NYC, Guy BM, Hermes M, Ness C, Sun J, Poon WCK, Cohen I (2015) Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions. Phys Rev Lett 115(22):228304

Lootens D, Van Damme H, Hébraud P (2003) Giant stress fluctuations at the jamming transition. Phys Rev Lett 90:178301

Lootens D, Hébraud P, Lécolier E, Van Damme H (2004) Gelation, shear-thinning and shear-thickening in cement slurries. Oil Gas Sci Technol- Rev IFP 59(1):31–40

Lootens D, Damme H, Hémar Y, Hébraud P (2005) Dilatant flow of concentrated suspensions of rough particles. Phys Rev Lett 95:268302

Majmudar TS, Behringer RP (2005) Contact force measurements and stress-induced anisotropy in granular materials. Nature 435(7045):1079–1082

Malbranche N, Chakraborty B, Morris JF (2023) Shear thickening in dense bidisperse suspensions. J Rheol 67(1):91–104

Mandal R, Casert C, Sollich P (2022) Robust prediction of force chains in jammed solids using graph neural networks. Nat Commun 13(1):1–7

Mari R, Seto R, Morris JF, Denn MM (2014) Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions. J Rheol 58(6):1693–1724

Mari R, Seto R, Morris JF, Denn MM (2015) Nonmonotonic flow curves of shear thickening suspensions. Phys Rev E 91(5):052302

Mari R, Seto R, Morris JF, Denn MM (2015) Discontinuous shear thickening in Brownian suspensions by dynamic simulation. Proc Natl Acad Sci USA 112(50):15326–15330

Metzner AB, Whitlock M (1958) Flow behavior of concentrated (dilatant) suspensions. Trans Soc Rheol 2:239–253

Molloy M, Reed B (1995) A critical point for random graphs with a given degree sequence. Random Struct Algorithms 6(2–3):161–180

More R, Ardekani A (2020) Roughness induced shear thickening in frictional non-brownian suspensions: a numerical study. J Rheol 64(2):283–297

More R, Ardekani A (2020) A constitutive model for sheared dense suspensions of rough particles. J Rheol 64(5):1107–1120

Morris JF (2020) Shear thickening of concentrated suspensions: recent developments and relation to other phenomena. Annu Rev Fluid Mech 52:121–144

Morris JF, Katyal B (2002) Microstructure from simulated Brownian suspension flows at large rate. Phys Fluids 14:1920–37

Naald M, Singh A, Eid TT, Tang K, Pablo JJ, Jaeger HM (2024) Minimally rigid clusters in dense suspension flow. Nat Phys 1–7

Nabizadeh M, Singh A, Jamali S (2022) Structure and dynamics of force clusters and networks in shear thickening suspensions. Phys Rev Lett 129(6):068001

Ness C, Sun J (2016) Shear thickening regimes of dense non-Brownian suspensions. Soft Matter 12(3):914–924

Ness C, Seto R, Mari R (2022) The physics of dense suspensions. Annu Rev Cond Matter Phys 13:97–117

Neuville M, Bossis G, Persello J, Volkova O, Boustingorry P, Mosquet M (2012) Rheology of a gypsum suspension in the presence of different superplasticizers. J Rheol 56(2):435–451

Olsson P, Teitel S (2007) Critical scaling of shear viscosity at the jamming transition. Phys Rev Lett 99:178001

O'Neill RE, Royer JR, Poon WC (2019) Liquid migration in shear thickening suspensions flowing through constrictions. Phys Rev Lett 123(12):128002

Otsuki M, Hayakawa H (2011) Critical scaling near jamming transition for frictional granular particles. Phys Rev E 83:051301

Ovarlez G, Vu Nguyen Le A, Smit WJ, Fall A, Mari R, Chatté G, Colin A (2020) Density waves in shear-thickening suspensions. Sci Adv 6(16):5589

Oyarte Gálvez L, Beer S, Meer D, Pons A (2017) Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology. Phys Rev E 95:030602

Ozturk D, Morgan ML, Sandnes B (2020) Flow-to-fracture transition and pattern formation in a discontinuous shear thickening fluid. Commun Phys

Papadopoulos L, Porter MA, Daniels KE, Bassett DS (2018) Network analysis of particles and grains. J Complex Netw 6(4):485–565

Pednekar S, Chun J, Morris JF (2018) Bidisperse and polydisperse suspension rheology at large solid fraction. J Rheol 62(2):513–526

Peters IR, Majumdar S, Jaeger HM (2016) Direct observation of dynamic shear jamming in dense suspensions. Nature 532(7598):214–217

Peyneau P-E, Roux J-N (2008) Frictionless bead packs have macroscopic friction, but no dilatancy. Phys Rev E 78(1):011307

Pradeep S, Nabizadeh M, Jacob AR, Jamali S, Hsiao LC (2021) Jamming distance dictates colloidal shear thickening. Phys Rev Lett 127(15):158002

Radjai F, Wolf DE, Jean M, Moreau J-J (1998) Bimodal character of stress transmission in granular packings. Phys Rev Lett 80(1):61

Ramaswamy M, Griniasty I, Liarte DB, Shetty A, Katifori E, Del Gado E, Sethna JP, Chakraborty B, Cohen I (2023) Universal scaling of shear thickening transitions. J Rheol 67(6):1189–1197

Rathee V, Blair DL, Urbach JS (2017) Localized stress fluctuations drive shear thickening in dense suspensions. Proc Natl Acad Sci 114(33):8740–8745

Rathee V, Blair DL, Urbach JS (2020) Localized transient jamming in discontinuous shear thickening. J Rheol 64(2):299–308

Rathee V, Miller J, Blair DL, Urbach JS (2022) Structure of propagating high-stress fronts in a shear-thickening suspension. Proc Natl Acad Sci 119(32):2203795119

Ren J (2013) Nonlinear dynamics and network properties in granular materials under shear. PhD thesis, Duke University

Ren J, Dijksman JA, Behringer RP (2013) Reynolds pressure and relaxation in a sheared granular system. Phys Rev Lett 110(1):018302

Reynolds, O.: LVII. On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 20(127), 469–481 (1885)

Richards JA, O'Neill RE, Poon WCK (2021) Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction. Rheologica Acta

Roché M, Myftiu E, Johnston MC, Kim P, Stone HA (2013) Dynamic fracture of nonglassy suspensions. Phys Rev Lett 110:148304

Rosales-Romero A, Vázquez-Quesada A, Prasanna Kumar SS, López-Aguilar JE, Ellero M (2024) Effects of confinement-induced non-newtonian lubrication forces on the rheology of a dense suspension. J Non-Newt Fluid Mech 329:105248

Royer JR, Burton GL, Blair DL, Hudson SD (2015) Rheology and dynamics of colloidal superballs. Soft Matter 11(28):5656–5665

Royer JR, Blair DL, Hudson SD (2016) Rheological signature of frictional interactions in shear thickening suspensions. Phys Rev Lett 116(18):188301

Saint-Michel B, Gibaud T, Manneville S (2018) Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension. Phys Rev X 8(3):031006

Santos AP, Bolintineanu DS, Grest GS, Lechman JB, Plimpton SJ, Sri-vastava I, Silbert LE (2020) Granular packings with sliding, rolling, and twisting friction. Phys Rev E 102(3):032903

Sedes O, Singh A, Morris JF (2020) Fluctuations at the onset of discontinuous shear thickening in a suspension. J Rheol 64(2):309–319

Sedes O, Makse HA, Chakraborty B, Morris JF (2022) K-core analysis of shear-thickening suspensions. Phys Rev Fluids 7(2):024304
Seto R, Mari R, Morris JF, Denn MM (2013) Discontinuous shear thickening of frictional hard-sphere suspensions. Phys Rev Lett 111:218301

Seto R, Singh A, Chakraborty B, Denn MM, Morris JF (2019) Shear jamming and fragility in dense suspensions. Gran Matt 21(3):82

Sharma S, Sharma A, Singh A (2025) Frictional contact network in dense suspension flow. arXiv:2505.22747

Singh A (2023) Hidden hierarchy in the rheology of dense suspensions. MRS Commun 13(6):971–979

Singh A, Mari R, Denn MM, Morris JF (2018) A constitutive model for simple shear of dense frictional suspensions. J Rheol 62(2):457–468

Singh A, Pednekar S, Chun J, Denn MM, Morris JF (2019) From yielding to shear jamming in a cohesive frictional suspension. Phys Rev Lett 122(9):098004

Singh A, Ness C, Seto R, Pablo JJ, Jaeger HM (2020) Shear thickening and jamming of dense suspensions: the “roll” of friction. Phys Rev Lett 124:248005

Singh A, Jackson GL, Naald M, Pablo JJ, Jaeger HM (2022) Stress-activated constraints in dense suspension rheology. Phys Rev Fluids 7(5):054302

Singh A, Ness C, Sharma AK, Pablo JJ, Jaeger HM (2024) Rheology of bidisperse non-brownian suspensions. Phys Rev E 110(3):034901

Singh A, Magnanimo V, Luding S (2013) Effect of friction and cohesion on anisotropy in quasi-static granular materials under shear. In: AIP Conference Proceedings. American Institute of Physics, vol 1542, pp 682–685

Song C, Wang P, Makse HA (2008) A phase diagram for jammed matter. Nature 453(7195):629–632

Stickel JJ, Powell RL (2005) Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 37:129–149

Thomas JE, Ramola K, Singh A, Mari R, Morris JF, Chakraborty B (2018) Microscopic origin of frictional rheology in dense suspensions: correlations in force space. Phys Rev Lett 121(12):128002

Thomas JE, Goyal A, Singh Bedi D, Singh A, Del Gado E, Chakraborty B (2020) Investigating the nature of discontinuous shear thickening: beyond a mean-field description. J Rheol 64(2):329–341

Visscher WM, Bolsterli M (1972) Random packing of equal and unequal spheres in two and three dimensions. Nature 239(5374):504–507

Wagner NJ, Brady JF (2009) Shear thickening in colloidal dispersions. Phys Today 62:27–32

Waitukaitis SR, Jaeger HM (2012) Impact-activated solidification of dense suspensions via dynamic jamming fronts. Nature

Wang M, Jamali S, Brady JF (2020) A hydrodynamic model for discontinuous shear-thickening in dense suspensions. J Rheol 64(2):379–394

Williamson RV, Hecker WW (1931) Some properties of dispersions of the quicksand type. Ind Eng Chem 23(6):667–670

Wyart M, Cates ME (2014) Discontinuous shear thickening without inertia in dense non-Brownian suspensions. Phys Rev Lett 112:098302

Xu Q, Singh A, Jaeger HM (2020) Stress fluctuations and shear thickening in dense granular suspensions. J Rheol 64(2):321–328

Zhao Y, Barés J, Zheng H, Socolar JE, Behringer RP (2019) Shear-jammed, fragile, and steady states in homogeneously strained granular materials. Phys Rev Lett 123(15):158001

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