Abstract
Recent simulations and experiments have shown that shearthickening of dense particle suspensions corresponds to a frictional transition. Based on this understanding, nonmonotonic rheological laws have been proposed and successfully tested in rheometers. These recent advances offer a unique opportunity for moving beyond rheometry and tackling quantitatively hydrodynamic flows of shearthickening suspensions. Here, we investigate the flow of a shearthickening suspension down an inclined plane and show that, at large volume fractions, surface kinematic waves can spontaneously emerge. Curiously, the instability develops at low Reynolds numbers, and therefore does not fit into the classical framework of Kapitza or ‘rollwaves’ instabilities based on inertia. We show that this instability, that we call ‘Oobleck waves’, arises from the sole coupling between the nonmonotonic (Sshape) rheological laws of shearthickening suspensions and the flow free surface.
Similar content being viewed by others
Introduction
How microscopic interactions affect the macroscopic flow behavior of complex fluids is at the core of soft matter physics. Recently, it has been shown that shearthickening in dense particulate suspensions corresponds to a frictional transition at the microscopic scale; when the imposed shear stress exceeds the interparticle shortrange repulsive force, the grain contact interaction transits from frictionless to frictional^{1,2,3,4,5,6,7,8}. During this transition, the proliferation of frictional contacts can be so massive that it triggers a remarkable macroscopic rheological response: the rate of shear of the suspension decreases when the imposed shear stress is increased. As a result, highly concentrated shearthickening suspensions have peculiar Sshape rheological laws^{9,10}, which have been rationalized by a frictional transition model^{3,11,12}. So far, the consequences of the frictional transition and its associated Sshape rheology have been essentially investigated in rheometers, where instabilities, shear bands and spatiotemporal patterns have been documented^{11,13,14,15}. By contrast, very little is known about the behavior of shearthickening suspensions in real hydrodynamic flow configurations beyond rheometry, in spite of the numerous applications^{16,17,18}.
An archetypical case, which is widely encountered in industrial and geophysical applications, is the incline plane flow configuration. As previously reported^{19} and illustrated in Fig. 1a (see also Supplementary movie 2), when a thin layer of shearthickening suspension flows down an inclined plane, surface waves of wavelengths much larger than the thickness can develop spontaneously and grow as they propagate downstream. This longwave freesurface instability may seem reminiscent of the Kapitza instability observed when a thin liquid film flows down a slope^{20,21}, or more generally of the so called “roll waves” instability observed from turbulent flows in open channels^{22,23,24}, to avalanches of complex fluids like mud^{25,26} or granular media^{27}. These latter two instabilities rely on the same primary mechanism: the amplification of kinematic surface waves at high velocity owing to inertial effects^{28}. For a Newtonian liquid in the laminar regime, the destabilization occurs only when the Reynolds number of the flow, Re = ρu_{0}h_{0}/η, where ρ is the fluid density, u_{0} its mean velocity, h_{0} the flow thickness and η the fluid viscosity, exceeds the Kapitza threshold, \(R{e}_{K}=5/(6\tan \theta )\), which is typically much larger than 1 for a small tilting angle θ of the incline^{29,30,31}. By contrast, the growth of surface waves observed in Fig. 1a for a dense shearthickening suspension occurs at a Reynolds number of only ≈1, i.e., far below the Kapitza threshold Re_{K} ≈ 5 predicted for θ = 10^{∘} (see also ref. ^{19}). This suggests that a different instability mechanism is at play for dense shearthickening suspensions, yet its origin remains an open question.
Here, we investigate the origin of this instability by studying the flow of a shearthickening suspension down an inclined plane over a wide range of volume fractions and flow rates. We confirm that this instability is not inertial and fundamentally different than the classical Kapitza or Roll waves instabilities. We provide experimental evidence together with a theoretical explanation, which show that this destabilization arises from the coupling between the flow free surface and the nonmonotonic (Sshape) rheological laws of shearthickening suspensions.
Results and discussion
Evidence of an instability distinct from the classical Kapitza or roll waves instabilities
We perform experiments with shearthickening aqueous suspensions of commercial native cornstarch (Maisita®, http://www.agrana.com). We vary the particle volume fraction over a wide range (0.30 < ϕ < 0.48) and characterize the onset of stability (the value of ϕ refers to the dry volume of cornstarch computed from its dry weight and density, 1550 kg m^{−3}). We use a 1 m long and 10 cm wide inclined plane covered with a diamond lapping film (6633M with roughnesses of ~45 μm) to insure rough boundary conditions. The suspension is released from a reservoir at the top of the plane through a gate with an adjustable aperture (Fig. 1b). A scale, placed at the bottom end of the incline (not shown in the schematic), provides the instantaneous flow rate q of the suspension. To probe the stability of this freesurface flow for moderate volume fractions ϕ ≤ 0.4, the gate is mounted on a translating stage imposing a small sinusoidal modulation of its aperture (3 Hz, ±100 μm). At large volume fractions however (ϕ ≳ 0.4), no forcing is required because the flow is so unstable that it is dominated by noise amplification of its most unstable mode. Two lowincident laser sheets and two cameras are used to measure the mean film thickness h_{0}~2−10 mm, and the cresttocrest amplitude of the waves upstream and downstream of the incline (Fig. 1c). The calibration of the laser sheet projections on the film surface yields a local measurement of h_{0} with a precision of 10 μm.
We use a protocol designed to characterize the instability onset with a single experiment for each volume fraction ϕ and inclination θ. The flow rate is varied quasisteadily, either using the progressive drainage of the reservoir (decreasing flow rate) or by slowly increasing the gate aperture (increasing flow rate). At each instant, an effective Reynolds number of the flow is computed from the instantaneous flow rate q and from the mean film thickness h_{0} using the relation \(Re=3{q}^{2}/(g{h}_{0}^{3}\sin \theta )\), where g is the gravitational acceleration. Note that the flow rate is varied sufficiently slowly that at each instant, the flow rate is constant along the incline. With such a definition based on mean quantities only, the Reynolds number can be applied to any rheology and is directly related to the Froude number \(F={u}_{0}^{2}/(g{h}_{0}\cos \theta )\) commonly used to described roll waves^{24} by \(Re={3F}/{\tan} \theta\), where u_{0} = q/h_{0} is the mean flow velocity. Re also reduces to the standard expression Re = ρu_{0}h_{0}/η for a Newtonian fluid of viscosity η and density ρ in the laminar regime (Nusselt velocity profile). Figure 1d shows the concomitant evolution of Re and of the wave amplitude upstream (Δh_{1}) and downstream (Δh_{2}) the incline during the drainage of the reservoir, starting from an unstable situation where Δh_{2} > Δh_{1}. The stability onset is precisely reached when Δh_{1} = Δh_{2}, which provides us with the critical Reynolds number Re_{c} (dashed line in Fig. 1c), the critical flow thickness h_{c}, and the critical basal shear stress, \({\tau }_{c}=\rho g{h}_{c}\,\sin \theta\). We have verified that the same instability onset is obtained by carrying successive steadystate measurements at various constant flow rates. A new freshly prepared suspension of cornstarch is used for each measurement. Experiments are repeated four times for each volume fraction.
Figure 2a shows the critical Reynolds number Re_{c}, normalized by the Kapitza threshold for a Newtonian fluid, as a function of ϕ. For the lowest volume fraction investigated, ϕ = 0.30, the stability threshold is close to Re_{K} (it is typically 50% above, owing to the finite forcing frequency and finite width of the plane^{32,33}). As the volume fraction is increased, Re_{c} becomes increasingly larger relative to Re_{K}, reaching ≈5Re_{K} at ϕ = 0.41. This behavior is actually expected from Kapitza’s inertial mechanism for a medium that is continuously shearthickening^{34}, like our cornstarch suspension over that range of volume fractions (0.35 ≲ ϕ ≲ 0.41). More strikingly, for ϕ ≳ 0.41 the relative critical Reynolds number drops drastically, down to two orders of magnitude below the Kapitza threshold at the largest volume fraction investigated (ϕ = 0.48, for which Re ≈ 0.15 and F ≈ 10^{−2}). Clearly, in this domain, the flow destabilization can no longer be explained within the Kapitza framework since inertial effects are negligible. Note that inertia is also negligible at the particle scale, since the Stokes number, \(St \sim {(d/{h}_{0})}^{2}Re\), where d ~ 10 μm is the particle size, is ~10^{5} times smaller than Re. As shown in Fig. 2b, this qualitative change in the onset of instability around ϕ = 0.41 is also observed in the evolution of the critical shear stress τ_{c} with ϕ. Similarly, the surface wave speed at the instability threshold changes significantly and abruptly. Its value c_{c}, normalized by the mean flow velocity u_{0} drops from 3, which is Kapitza’s prediction for a Newtonian layer in the long wavelength limit, to ~2 for volume fractions exceeding ≈0.41 (see Fig. 2c). These results confirm that above ϕ ≈ 0.41, a longwave freesurface instability, which is fundamentally distinct from the Kapitza instability, emerges. In the following, we call this instability, which to our knowledge as no equivalent in classical fluids, “Oobleck waves”.
Oobleck waves arise from the Sshape rheology and kinematic wave propagation
To understand the origin of Oobleck waves, we characterize the rheology of the cornstarch suspension in a cylindricalCouette rheometer (Fig. 3). We find that the volume fraction at which Oobleck waves appear (ϕ ≈ 0.41) corresponds precisely to ϕ_{DST}, the volume fraction at which the shearthickening transition becomes discontinuous. Indeed, we analyze our rheological data along Wyart & Cates’s model, which assumes that the effective viscosity of the suspension, \(\eta (\phi ,\tau )={\eta }_{s}{({\phi }_{J}(\tau )\phi )}^{2}\), diverges at a critical volume fraction, ϕ_{J}, that depends on the applied shear stress, τ, according to \({\phi }_{J}(\tau )={\phi }_{0}(1{e}^{{\tau }^{* }/\tau })+{\phi }_{1}{e}^{{\tau }^{* }/\tau }\), where (η_{s}, τ^{*}, ϕ_{0}, ϕ_{1}) are material constant. Here, η_{s} is a prefactor proportional to the solvent viscosity, τ^{*} is the shortrange repulsive stress scale above which the frictional transition occurs, which may be tuned by changing the particle roughness or surface chemistry, ϕ_{0} (resp. ϕ_{1}) is the jamming volume fraction at which the suspension viscosity diverge at low (resp. large) stress, with ϕ_{1} being dependent on the interparticle friction coefficient^{3}. By fitting our measurements with this model, we find that the rheological curve (shear stress τ versus shear rate \(\dot{\gamma }\)) becomes Sshaped when ϕ ≥ 0.41 ± 0.005 = ϕ_{DST} (see Fig. 3 and Methods for the fitting procedure). This suggests that negatively sloped portion in the rheological curve (\({\rm{d}}\dot{\gamma }/{\rm{d}}\tau \, < \, 0\)) is a key ingredient of the instability. Sshaped flow curves, and more generally rheograms with a negatively sloped region, are known to produce unstable flow conditions^{35,36}. Previous studies on shearthickening suspensions have based their analysis on this feature to explain for instance the emergence of random fluctuations^{37} reported initially by Boersma et al.^{38}, and the oscillations observed when an object moves in a shearthickening fluid^{39,40} or in rheometric configurations^{15,41}. However, all these models require inertia to predict an instability. By contrast, here, the instability seems to be of a fundamentally different nature. First, it can occur at very low Reynolds and Froude numbers, for which inertial effects are negligible. Second, at the instability onset, the unstable mode propagates at the speed of the surface kinematic waves defined by c_{kin} ≡ dq/dh_{0}^{28} (Fig. 2c inset). This indicates that the coupling between the flow and the freesurface deformation, which was not considered in previous studies, is essential to explain the emergence of Oobleck waves.
Oobleck waves instability mechanism
We now show how the negative slope in the rheology coupled with a gravitydriven freesurface flow can yield to an instability without invoking inertial effects. In the zeroReynolds number limit, the force balance on a slice of suspension, as depicted in Fig. 4a, imposes that the basal stress, τ_{b}, is equal to the sum of the projected weight of the slice, \(\rho gh\sin \theta\), where h(x, t) is the local flow thickness, and of the longitudinal pressure gradient induced by the freesurface deflection. For wavelengths much larger than the flow thickness and the capillary length, the pressure profile perpendicular to the plane can be assumed to be hydrostatic \(P(x,z,t)=\rho g\cos \theta (h(x,t)z)\), where x is the flow direction and z the height within the flowing layer in the perpendicular direction^{21}. The depthaveraged force balance is then given by
Let us now consider a perturbation of a base flow of constant thickness, as illustrated in Fig. 4b. A local increase of the flow thickness implies that ∂h/∂x becomes positive upstream of the perturbation and negative downstream. To satisfy the force balance (1), the basal shear stress τ_{b} upstream must therefore decrease, whereas it must increase downstream. However, owing to the Sshape rheology of the suspension, when \({\rm{d}}\dot{\gamma }/{\rm{d}}\tau \, < \, 0\), a decrease (resp. increase) in τ_{b} implies a local increase (resp. decrease) of the flow rate \(\dot{\gamma }\). Therefore, the shear rate increases upstream and decreases downstream, inducing a net inward mass flux underneath the bump and the amplification of the initial perturbation (red arrows in Fig. 4b).
Quantitative depthaveraged model without inertia
To go beyond this qualitative picture, we perform a linear stability analysis of a steady uniform flow of thickness h_{0} and depthaveraged velocity u_{0} using the SaintVenant approximations (long wavelength limit^{27,42}) and neglecting inertia (lubrication approximation^{21}). The equations are written using the dimensionless variables \(\tilde{h}=h/{h}_{0}\), \(\tilde{x}=x/{h}_{0}\), \(\tilde{u}=u/{u}_{0}\), \(\tilde{t}=t{u}_{0}/{h}_{0},\) \({\tilde{\tau }}_{b}={\tau }_{b}/\rho g{h}_{0}\sin \theta\) and linearized by writing \(\tilde{h}=1+{h}_{1}\), \(\tilde{u}=1+{u}_{1}\) and \({\tilde{\tau }}_{b}=1+{\tau }_{1}\) with (h_{1}, u_{1}, τ_{1} ≪ 1). Under these conditions, the mass conservation, \({\partial}_{\tilde{t}} h+{\partial}_{\tilde{x}}(hu)=0\), and the force balance (1) become \({\partial}_{\tilde{t}} {h_{1}}+{\partial}_{\tilde{x}}{h_{1}}+{\partial}_{\tilde{x}}{u_{1}}=0\) and \({\tau }_{1}={h}_{1}\tan {\theta }^{1}{\partial }_{\tilde{x}}{h}_{1}\), respectively. The linearization of the normalized shearrate \(\tilde{\dot{\gamma }}(\tilde{{\tau }_{b}})\equiv \tilde{u}/\tilde{h}\), gives Aτ_{1} = u_{1} − h_{1}, where \(A={\rm{d}}\tilde{\dot{\gamma }}/{\rm{d}}{\tilde{\tau }}_{b}{ }_{{\tilde{\tau }}_{b} = 1}\) is the slope of the rheological curve for the base state basal stress \({\tau }_{b}=\rho g{h}_{0}\sin \theta\), obtained from the integration of the flow velocity profile (see Methods). Taking the spatial derivative of the force balance and substituting τ_{1} and \({\partial}_{\tilde{x}}{u_{1}}\) using the rheology and mass balance, respectively, lead to a single partial differential equation for the freesurface perturbation h_{1}:
where \(\tilde{c}={c}_{{\rm{kin}}}/{u}_{0}=2+A\) is the dimensionless speed of the kinematic waves^{28}. Interestingly, the perturbation amplitude is found to follow a diffusion equation in the reference frame of the kinematic waves, with an effective “diffusion coefficient” \(A/\tan \theta\). When A < 0, i.e., when the slope of the rheological law \({u}_{0}/{h}_{0}=\dot{\gamma }({\tau }_{b})\) is negative, antidiffusion occurs, which leads to an amplification of all perturbations, whereas for A > 0 the flow is stable. The onset of instability is thus given by A = 0. This criteria can be expressed in terms of a critical Reynolds number Re_{c} and a critical basal shear stress τ_{c} using the Wyart & Cates rheological laws (see Methods). Figures 2a, b show that, for ϕ ≥ ϕ_{DST}, these predictions (redsolid lines) capture very well the value of the critical Reynolds Re_{c} and its dramatic drop over two decades when increasing ϕ, as well as the order of magnitude and the drop of the critical shear stress τ_{c} with ϕ. The decrease of the instability onset with increasing ϕ is a direct consequence of Wyart and Cates’ rheological laws^{3}, where the DST onset stress also decreases with ϕ. Physically, it comes from the fact that, when approaching the maximal possible packing fraction ϕ_{0}, less and less frictional contacts are required to reach the DST region. The model also predicts a weak dependence of Re_{c} on the plane inclination angle, as observed experimentally (Fig. 2a inset). This overall agreement is all the more conclusive that it is rooted on physically based constitutive laws, of which the rheological parameters are measured independently, without further fitting. Another strong prediction of the model is that, at the instability onset (A = 0), the speed of the unstable mode is equal to the speed of the kinematic waves c_{c}/u_{0} = 2. This prediction is fully consistent with the drop and value of the normalized wave speed observed experimentally for ϕ > ϕ_{DST} (see Fig. 2c). These results conclusively show that surface waves can emerge from the coupling between a negatively sloped rheology and the flow free surface, without the need for inertial effects.
The Oobleck waves instability mechanism highlighted in this study is not limited to shearthickening suspensions and could be extended to any other complex fluids having a rheology with a negatively sloped region (e.g., granular materials and geomaterials exhibiting velocityweakening rheology^{43,44}, concentrated polymers or surfactant solutions^{35,36}, liquid crystals^{45}, active selfpropelled suspensions^{46}). More generally, our analysis shows that gravity forces, which are usually stabilizing for gravitydriven freesurface flows, can become destabilizing in the presence of a nonmonotonic rheology. Our result could thus be extended to other stabilizing forces such as capillary forces arising from the freesurface deformation. We thus anticipate that other interesting instabilities may be explained directly, or in the light of our study.
Finally, in a broader context, our study reveals that kinematic waves can be unstable in an overdamped medium, where inertia is negligible. These waves, which are observed in a wide range of situations (e.g., traffic and pedestrian flows^{47}, sediment transport^{48}, fluidized bed^{49}, surge, and floods^{23}), result from mass conservation and a general relationship between a local flow rate and a local concentration^{50,51} (e.g., number of vehicles or pedestrians on a road, solid packing fraction in a suspension, depth of the flow). However, in all these systems, their emergence from a uniform state requires an inertial lag between the flow rate and the concentration^{28}. In our system, the situation is very different as these waves are unstable not from inertia, but from the intrinsic constitutive flow rule of the material. Whether this description of kinematic waves could be extended to more complex systems, such as crowds^{47} or active selfpropelled particles^{46}, are interesting topics to address in future studies.
Methods
Rheological data fitting procedure
Rheograms of the aqueous suspension of cornstarch are measured for various volume fractions in a narrowgap cylindricalCouette cell (Fig. 5a) using a rheometer (Anton Paar MCR 501). The height of the shearcell (40 mm) is sufficiently large to neglect sedimentation effects of the particles during the measurement. Similarly to the procedure followed by Guy et al.^{4}, the viscosity below (τ ≪ τ^{*}) and above (τ ≫ τ^{*}) the shearthickening transition are extracted and plotted versus ϕ (Fig. 5b). The low viscosity branch (frictionless branch) is first fitted with \(\eta (\phi )={\eta }_{s}{({\phi }_{0}\phi )}^{2}\), with η_{s} and ϕ_{0} as fitting parameters. This yields η_{s} = 0.91 ± 0.01 mPa.s and ϕ_{0} = 0.52 ± 0.005. The large viscosity branch (frictional branch) is then fitted with \(\eta (\phi )={\eta }_{s}{({\phi }_{1}\phi )}^{2}\), using the previous estimation of η_{s}, and letting ϕ_{1} as the only fitting parameter. Note that the rheograms are obtained using both very rough (square symbols) and rough walls (circle symbols), by covering the cellwalls with sand papers of different grades (roughnesses of ≈80 μm and ≈15 μm, respectively). The two measurements overlap, except in the frictional branch at high volume fraction (shaded symbols in Fig. 5b). For instance, the data from ϕ = 0.4 and 0.41 are included in the fitting procedure, while 0.42 and 0.43 are not, because the first two points overlap, independently of the roughness of the boundaries, whereas for 0.42 and 0.43 systematic deviations are observed indicating slippage or other artefacts. All data points which are interpreted as biased measurements (transparent symbols) are discarded from the fitting procedure; this yields ϕ_{1} = 0.43 ± 0.005. Once the values of η_{s}, ϕ_{0} and ϕ_{1} are set, we determine the value of τ^{*} by fitting the full rheograms \(\tau (\dot{\gamma })\) with Wyart & Cates laws: \(\tau ={\eta }_{s}{({\phi }_{J}(\tau )\phi )}^{2}\dot{\gamma }\), with \({\phi }_{J}(\tau )={\phi }_{0}(1{e}^{{\tau }^{* }/\tau })+{\phi }_{1}{e}^{{\tau }^{* }/\tau }\). The best fit, shown in Fig. 5c, is obtained for τ^{*} = 12 ± 2 Pa. The value of τ^{*} represents the critical shear stress required to overcome the interparticle repulsive force and activate frictional contacts between particles. For an interparticulate force f and a particle size d, the critical shear stress is expected to be of order f/d^{2}. The value we obtain (≈12 Pa) is consistent with the values already reported in the literature for cornstarch in water.
Computation of τ _{c} and R e _{c}
To compute the critical shear stress τ_{c} and the critical Reynold number Re_{c} from the instability criteria resulting from the linear stability analysis \(A\equiv {\rm{d}}\tilde{\dot{\gamma }}/{\rm{d}}\tilde{{\tau }_{b}}{ }_{\tilde{{\tau }_{b}} = 1}=0\), we need to relate the shear rate \(\dot{\gamma }\equiv {u}_{0}/{h}_{0}\), defined as the ratio of the depthaveraged flow velocity to the flow thickness, to the basal shear stress τ_{b} and the basal suspension viscosity η(τ_{b}).
For a steady uniform flow down an inclined plane of slope θ, the momentum equation applied to a surface layer of thickness h_{0} − z gives
where the second equality uses the definition of viscosity, \(\eta =\tau /({\rm{d}}\hat{u}/{\rm{d}}z)\), and \(\hat{u}(z)\) is the local velocity parallel to x. From the proportionality between τ and h_{0} − z, the local velocity can be expressed as
Using the definition of the depthaveraged flow velocity, \({u}_{0}=\mathop{\int}\nolimits_{0}^{{\tau }_{b}}\hat{u}(\tau ^{\prime} )\ {\rm{d}}\tau ^{\prime} /{\tau }_{b}\), we obtain the expression of the depthaveraged shear rate
where
embeds the shearthickening of the suspension. By definition, \({\mathcal{G}}=1\) for a Newtonian fluid.
where we have used \(\frac{{\rm{d}}}{{\rm{d}}\tau }(\mathop{\int}\nolimits_{0}^{\tau }\mathop{\int}\nolimits_{{\tau }^{\prime}}^{\tau }\frac{{\tau }^{^{\prime\prime} }}{\eta ({\tau }^{^{\prime\prime} })}{\rm{d}}{\tau }^{^{\prime\prime} }{\rm{d}}{\tau }^{\prime})={\tau }^{2}/\eta (\tau )\). Note that in (7), the basal stress τ_{b}, the shear rate \(\dot{\gamma }\) and the derivative are taken at the base state, i.e., for \(\dot{\gamma }={u}_{0}/{h}_{0}\) and \({\tau }_{b}=\rho g\sin \theta {h}_{0}\).
Finally, the critical shear stress τ_{c}, at which the flow destabilizes (blacksolidline plotted in Fig. 2b of the main text), is obtained numerically by finding the value of τ_{b} for which A(τ_{c}) = 0, i.e., \({\mathcal{G}}({\tau }_{b}={\tau }_{c})=3/2\). From the value of τ_{c}, we obtain the critical Reynolds number (blacksolidline plotted in Fig. 2(a) of the main text)
Data availability
The data that support the findings of this study are available from https://zenodo.org/record/4247592 (DOI 10.5281/zenodo.4247592).
References
Seto, R., Mari, R., Morris, J. F. & Denn, M. M. Discontinuous shear thickening of frictional hardsphere suspensions. Phys. Rev. Lett. 111, 218301 (2013).
Mari, R., Seto, R., Morris, J. F. & Denn, M. M. Shear thickening, frictionless and frictional rheologies in nonbrownian suspensions. J. Rheol. 58, 1693–1724 (2014).
Wyart, M. & Cates, M. E. Discontinuous shear thickening without inertia in dense nonbrownian suspensions. Phys. Rev. Lett. 112, 098302 (2014).
Guy, B. M., Hermes, M. & Poon, W. C. K. Towards a unified description of the rheology of hardparticle suspensions. Phys. Rev. Lett. 115, 088304 (2015).
Lin, N. Y. C. et al. Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions. Phys. Rev. Lett. 115, 228304 (2015).
Clavaud, C., Bérut, A., Metzger, B. & Forterre, Y. Revealing the frictional transition in shearthickening suspensions. Proc. Natl. Acad. Sci. 114, 5147–5152 (2017).
Comtet, J. et al. Pairwise frictional profile between particles determines discontinuous shear thickening transition in noncolloidal suspensions. Nat. Commun. 8, 15633 (2017).
Clavaud, C., Metzger, B. & Forterre, Y. The darcytron: a pressureimposed device to probe the frictional transition in shearthickening suspensions. J. Rheol. 64, 395 (2020).
Pan, Z., de Cagny, H., Weber, B. & Bonn, D. Sshaped flow curves of shear thickening suspensions: direct observation of frictional rheology. Phys. Rev. E 92, 032202 (2015).
Mari, R., Seto, R., Morris, J. F. & Denn, M. M. Nonmonotonic flow curves of shear thickening suspensions. Phys. Rev. E 91, 052302 (2015).
Chacko, R. N., Mari, R., Cates, M. E. & Fielding, S. M. Dynamic vorticity banding in discontinuously shear thickening suspensions. Phys. Rev. Lett. 121, 108003 (2018).
Singh, A., Mari, R., Denn, M. M. & Morris, J. F. A constitutive model for simple shear of dense frictional suspensions. J. Rheol. 62, 457–468 (2018).
Hermes, M. et al. Unsteady flow and particle migration in dense, nonbrownian suspensions. J. Rheol. 60, 905–916 (2016).
SaintMichel, B., Gibaud, T. & Manneville, S. Uncovering instabilities in the spatiotemporal dynamics of a shearthickening cornstarch suspension. Phys. Rev. X 8, 031006 (2018).
Richards, J. A., Royer, J. R., Liebchen, B., Guy, B. M. & Poon, W. C. K. Competing timescales lead to oscillations in shearthickening suspensions. Phys. Rev. Lett. 123, 038004 (2019).
Abdesselam, Y. et al. Rheology of plastisol formulations for coating applications. Polym. Eng. Sci. 57, 982–988 (2017).
Blanco, E. et al. Conching chocolate is a prototypical transition from frictionally jammed solid to flowable suspension with maximal solid content. Proc. Natl. Acad. Sci. 116, 10303–10308 (2019).
LaFarge. Superplasticizers: the wonder of fluid concrete. https://www.youtube.com/watch?v=CSZxjQwDKF038 (2013).
Balmforth, N. J., Bush, J. W. M. & Craster, R. V. Roll waves on flowing cornstarch suspensions. Phys. Lett. A 338, 479–484 (2005).
Kapitza, P. L. & Kapitza, S. P. Wave flow of thin viscous fluid layers. Zh. Eksp. Teor. Fiz. 18, 3–28 (1948).
Craster, R. & Matar, O. Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131 (2009).
Jeffreys, H. The flow of water in an inclined channel of rectangular section. Philos. Mag. 49, 793–807 (1925).
Dressler, R. F. Mathematical solution of the problem of rollwaves in inclined opel channels. Commun. Pure Appl. Math. 2, 149–194 (1949).
Needham, D. J. & Merkin, J. H. On roll waves down an open inclined channel. Proc. R. Soc. Lond. A. Math. Phys. Sci. 394, 259–278 (1984).
Trowbridge, J. Instability of concentrated free surface flows. J. Geophys. Res. Oceans 92, 9523–9530 (1987).
Liu, K. & Mei, C. C. Roll waves on a layer of a muddy fluid flowing down a gentle slopea bingham model. Phys. Fluids 6, 2577–2590 (1994).
Forterre, Y. & Pouliquen, O. Longsurfacewave instability in dense granular flows. J. Fluid Mech. 486, 21–50 (2003).
Whitham, G. B. Linear and nonlinear waves, vol. 42 (John Wiley & Sons, 2011).
Benjamin, T. B. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–573 (1957).
Yih, C.S. Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963).
Liu, J., Paul, J. D. & Gollub, J. P. Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101 (1993).
Georgantaki, A., Vatteville, J., Vlachogiannis, M. & Bontozoglou, V. Measurements of liquid film flow as a function of fluid properties and channel width: evidence for surfacetensioninduced longrange transverse coherence. Phys. Rev. E 84, 026325 (2011).
Pollak, T., Haas, A. & Aksel, N. Side wall effects on the instability of thin gravitydriven films from longwave to shortwave instability. Phys. Fluids 23, 094110 (2011).
Ng, C.O. & Mei, C. C. Roll waves on a shallow layer of mud modelled as a powerlaw fluid. J. Fluid Mech. 263, 151–184 (1994).
Goddard, J. Material instability in complex fluids. Annu. Rev. Fluid Mech. 35, 113–133 (2003).
Divoux, T., Fardin, M. A., Manneville, S. & Lerouge, S. Shear banding of complex fluids. Annu. Rev. Fluid Mech. 48, 81–103 (2016).
Sedes, O., Singh, A. & Morris, J. F. Fluctuations at the onset of discontinuous shear thickening in a suspension. J. Rheol. 64, 309–319 (2020).
Boersma, W., Baets, P., Lavn, J. & Stein, H. Timedependent behavior and wall slip in concentrated shear thickening dispersions. J. Rheol. 35, 1093–1120 (1991).
von Kann, S., Snoeijer, J. H., Lohse, D. & van der Meer, D. Nonmonotonic settling of a sphere in a cornstarch suspension. Phys. Rev. E 84, 060401 (2011).
Von Kann, S., Snoeijer, J. H. & Van Der Meer, D. Velocity oscillations and stopgo cycles: The trajectory of an object settling in a cornstarch suspension. Phys. Rev. E 87, 042301 (2013).
Nakanishi, H. & Mitarai, N. Shear thickening oscillation in a dilatant fluid. J. Phys. Soc. Jpn. 80, 033801 (2011).
SaintVenant, A. J. C. Théorie du mouvement nonpermanent des eaux, avec application aux crues des rivières et è l’introduction des marées dans leur lit. C. R. Acad. Sc. Paris 73, 147–154 (1871).
Perrin, H., Clavaud, C., Wyart, M., Metzger, B. & Forterre, Y. Interparticle friction leads to nonmonotonic flow curves and hysteresis in viscous suspensions. Phys. Rev. X 9, 031027 (2019).
Lucas, A., Mangeney, A. & Ampuero, J. P. Frictional velocityweakening in landslides on earth and on other planetary bodies. Nat. Commun. 5, 1–9 (2014).
Orihara, H. et al. Negative viscosity of a liquid crystal in the presence of turbulence. Phys. Rev. E 99, 012701 (2019).
Loisy, A., Eggers, J. & Liverpool, T. B. Active suspensions have nonmonotonic flow curves and multiple mechanical equilibria. Phys. Rev. Lett. 121, 018001 (2018).
Bain, N. & Bartolo, D. Dynamic response and hydrodynamics of polarized crowds. Science 363, 46–49 (2019).
Charru, F., Andreotti, B. & Claudin, P. Sand ripples and dunes. Annu. Rev. Fluid Mech. 45, 469–493 (2013).
Anderson, T. & Jackson, R. Fluid mechanical description of fluidized beds. stability of state of uniform fluidization. Ind. Eng. Chem. Fundamentals 7, 12–21 (1968).
Lighthill, M. J. & Whitham, G. B. On kinematic waves i. flood movement in long rivers. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 229, 281–316 (1955).
Lighthill, M. J. & Whitham, G. B. On kinematic waves ii. a theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 229, 317–345 (1955).
Acknowledgements
This work was supported by the European Research Council under the European Union Horizon 2020 Research and Innovation program (ERC grant agreement no. 647384), by the Labex MEC (ANR10LABX0092) under the A*MIDEX project (ANR11IDEX000102) funded by the French government program Investissements d’Avenir, and by ANR ScienceFriction (ANR18CE300024).
Author information
Authors and Affiliations
Contributions
B.D.T., H.L., Y.F., and B.M. designed and performed experiments and analyzed the data. Y.F. developed the model. All authors discussed the results and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
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 license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license 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 license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Darbois Texier, B., Lhuissier, H., Forterre, Y. et al. Surfacewave instability without inertia in shearthickening suspensions. Commun Phys 3, 232 (2020). https://doi.org/10.1038/s42005020005004
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005020005004
This article is cited by

Making waves without inertia
Nature Reviews Physics (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.