We study the steady flow properties of different three-dimensional aqueous foams in a wide gap Couette geometry. From local velocity measurements through Magnetic Resonance Imaging techniques and from viscosity bifurcation experiments, we find that these foams do not exhibit any observable signature of shear banding. This contrasts with two previous results (Rodts et al., Europhys. Lett., 69 (2005) 636 and Da Cruz et al., Phys. Rev. E, 66 (2002) 051305); we discuss possible reasons for this dicrepancy. Moreover, the foams we studied undergo steady flow for shear rates well below the critical shear rate recently predicted (Denkov et al., Phys. Rev. Lett., 103 (2009) 118302). Local measurements of the constitutive law finally show that these foams behave as simple Herschel-Bulkley yield stress fluids.
We have developed a realistic simulation of 2D dry foams under quasi-static shear. After a short transient, a shear-banding instability is observed. These results are compared with measurements obtained on real 2D (confined) foams. The numerical model allows us to exhibit the mechanical response of the material to a single plastication event. From the analysis of this elastic propagator, we propose a scenario for the onset and stability of the flow localization process in foams, which should remain valid for most athermal amorphous systems under creep flow.
The occurence of shear bands in a complex fluid is generally understood as resulting from a structural evolution of the material under shear, which leads (from a theoretical perspective) to a non-monotonic stationnary flow curve related to the coexistence of different states of the material under shear. In this paper we present a scenario for shear-banding in a particular class of complex fluids, namely foams and concentrated emulsions, which differs from other scenarii in two important ways. First, the appearance of shear bands is shown to be possible both without any intrinsic physical evolu…
Shear banding is an important feature of flow in complex fluids. Essentially, shear bands refer to the coexistence of flowing and non-flowing regions in driven material. Understanding the possible sources of shear banding has important implications for a wide range of flow applications. In this regard, quasi-two dimensional flow offers a unique opportunity to study competing factors that result in shear bands. One proposal is the competition between intrinsic dissipation and an external source of dissipation. In this paper, we report on the experimental observation of the transition between di…
We investigate theoretically shear banding in large amplitude oscillatory shear (LAOS) of polymers and wormlike micelles. In LAOStrain we find banding at low frequencies and sufficiently high strain rate amplitudes in fluids for which the underlying constitutive curve of shear stress as a function of shear rate is non-monotonic. This is the direct analogue of quasi steady state banding seen in slow strain rate sweeps along the flow curve. At higher frequencies and sufficiently high strain amplitudes we report a different but related phenomenon, which we call `elastic' shear banding. This is as…
We study shear banding in soft glassy materials subject to a large amplitude oscillatory shear flow (LAOS). By numerical simulations of the widely used soft glassy rheology model, supplemented by more general physical arguments, we demonstrate strong banding over an extensive range of amplitudes and frequencies of the imposed shear rate $\dotγ(t)=\dotγ_0\cos(ωt)$, even in materials that do not permit banding as their steady state response to a steady imposed shear flow $\dotγ=\dotγ_0=$const. Highly counterintuitively, banding persists in LAOS even in the limit of zero frequency $ω\to 0$, where…
We assess the possibility of shear banding of semidilute rod-like colloidal suspensions under steady shear ow very close to the isotropic-nematic spinodal, using a combination of rheology, small angle neutron scattering, and laser Doppler velocimetry. Model systems are employed which allow for a length and stiffness variation of the particles. The rheological signature reveals that these systems are strongly shear thinning at moderate shear rates. It is shown that the longest and most flexible rods undergo the strongest shear thinning and have the greatest potential to form shear bands. Althou…
Using a micro particle imaging velocity technique, we resolve for the first time the three dimensionnal structure of wormlike shear banding flows in straight microchannels. The study revealed two effects, which should be generic for shear banding flows: the first is a strong amplification of the confinement induced by the edge of the channel, the second is an instability of the interface between the shear bands. A detailed quantitative comparison of our experimental measurements with a theoretical study of the diffusive Johnson Segalman model leads to excellent agreement. Our study clarifies t…
This precis is aimed as a practical field-guide to situations in which shear banding might be expected in complex fluids subject to an applied shear flow. Separately for several of the most common flow protocols, it summarises the characteristic signatures in the measured bulk rheological signals that suggest the presence of banding in the underlying flow field. It does so both for a steady applied shear flow, and for the time-dependent protocols of shear startup, step stress, finite strain ramp, and large amplitude oscillatory shear. An important message is that banding might arise rather wid…
Recent experimental techniques used to investigate shear banding are reviewed. After recalling the rheological signature of shear-banded flows, we summarize the various tools for measuring locally the microstructure and the velocity field under shear. Local velocity measurements using dynamic light scattering and ultrasound are emphasized. A few results are extracted from current works to illustrate open questions and directions for future research.
Based on numerical modeling with the diffusive Giesekus constitutive equation, a recent Letter [Phys. Rev. Lett. 120, 138002 (2018), arXiv:1801.08798] claimed that the experimental reports of shear banding in entangled polymeric fluids were due to the disturbances from the sample edge. The conclusion that true bulk shear banding is precluded in the presence of edge effects is at odds with many past experimental, theoretical, numerical, and computer simulation studies. We briefly discuss some problems of this work.
Wormlike micelle solutions are submitted to small-amplitude oscillatory shear superimposed to steady shear in the shear banding regime. By imposing a shear oscillation, the interface between high- and low-shear regions oscillates in time. A two-fluid semi-phenomenological model is proposed for superposition rheology in the shear banding regime, which allows us to extract a characteristic velocity for the interface dynamics from experiments involving only a standard rheometer. Estimates of the stress diffusion coefficient ${\cal D}$ can also be inferred from such superposition experiments. The …
We study numerically the formation of long-lived transient shear bands during shear startup within two models of soft glasses (a simple fluidity model and an adapted `soft glassy rheology' model). The degree and duration of banding depends strongly on the applied shear rate, and on sample age before shearing. In both models the ultimate steady flow state is homogeneous at all shear rates, consistent with the underlying constitutive curve being monotonic. However, particularly in the SGR case, the transient bands can be extremely long lived. The banding instability is neither `purely viscous' n…
Homogeneous polymer solutions are well-known to exhibit viscoelastic flow instabilities: purely elastic when inertia is negligible, inertio-elastic otherwise. Recently, shear-banding wormlike micelles solutions were also discovered to follow a similar phenomenology. In the shear-banding regime, inertia is usually negligible so only purely elastic flows have been reported. Here, we investigate a non-shear-banding solution where inertia becomes significant, leading to flow patterns akin to the inertio-elastic regime of dilute polymer solutions. We show that the instability follows a supercritica…
We study theoretically shear banding in soft glassy materials subject to large amplitude time-periodic shear flows, considering separately the protocols of large amplitude oscillatory shear strain, large amplitude square or triangular or sawtooth strain rate, and large amplitude oscillatory shear stress. In each case, we find shear banding to be an important part of the material's flow response for a broad range of values of the frequency $ω$ and amplitude of the imposed oscillation. Crucially, and highly counterintuitively, in the glass phase this persists even to the lowest frequencies acces…
Edge instabilities are believed to be one of the possible causes of shear banding in entangled polymeric fluids. Here, we investigate the effect of edge disturbance on the shear-induced dynamics of well-entangled DNA solutions. Using a custom high-aspect-ratio planar-Couette cell, we systematically measure the velocity profiles of sheared DNA samples at different distances away from the edge of the shear cell. Under a weak oscillatory shear with the corresponding Weissenberg number (Wi) smaller than 1, where DNA solutions exhibit linear velocity profiles with strong wall slip, the penetration …
We review and compare the phenomenological aspects and physical origin of shear-localization and shear-banding in various material types, namely emulsions, suspensions, colloids, granular materials and micellar systems. It appears that shear-banding, which must be distinguished from the simple effect of coexisting static-flowing regions in yield stress fluids, occurs in the form of a progressive evolution of the local viscosity towards two significantly different values in two adjoining regions of the fluids in which the stress takes slightly different values. This suggests that from a global …
Using Dynamic Light Scattering in heterodyne mode, we measure velocity profiles in a much studied system of wormlike micelles (CPCl/NaSal) known to exhibit both shear-banding and stress plateau behavior. Our data provide evidence for the simplest shear-banding scenario, according to which the effective viscosity drop in the system is due to the nucleation and growth of a highly sheared band in the gap, whose thickness linearly increases with the imposed shear rate. We discuss various details of the velocity profiles in all the regions of the flow curve and emphasize on the complex, non-Newtoni…
We observe a novel type of shear banding in the rheology of thixotropic yield-stress fluids that is due to the coupling of both non-locality and thixotropy. The latter is known to lead to shear banding even in homogeneous stress fields, but the bands observed in the presence of non-local effects appear different as the shear rate varies continuously over the shear band. Here, we introduce a simple non-local model for the shear banding (NL-SB), and we implement it for the analysis of micron-scale rheo-MRI velocimetry measurements of a milk microgel suspension in a cone-and-plate geometry. The p…
We study the dynamics of the Taylor-Couette flow of shear banding wormlike micelles. We focus on the high shear rate branch of the flow curve and show that for sufficiently high Weissenberg numbers, this branch becomes unstable. This instability is strongly sub-critical and is associated with a shear stress jump. We find that this increase of the flow resistance is related to the nucleation of turbulence. The flow pattern shows similarities with the elastic turbulence, so far only observed for polymer solutions. The unstable character of this branch led us to propose a scenario that could acco…
Using velocity profile measurements based on dynamic light scattering and coupled to structural and rheological measurements in a Couette cell, we present evidences for a shear-banding scenario in the shear flow of the onion texture of a lyotropic lamellar phase. Time-averaged measurements clearly show the presence of structural shear-banding in the vicinity of a shear-induced transition, associated to the nucleation and growth of a highly sheared band in the flow. Our experiments also reveal the presence of slip at the walls of the Couette cell. Using a simple mechanical approach, we demonstr…
Within a highly generalised theoretical framework for the flow properties of complex fluids, we study the onset of shear banding in the three most common time-dependent experimental protocols: step stress, step strain and shear startup. By means of a linear stability analysis we derive a fluid-universal criterion for the onset of banding, separately for each protocol, that depends only on the shape of the experimentally measured time-dependent rheological response function, independent of the constitutive law and internal state variables of the particular fluid in question. Our predictions thu…
Using high-resolution confocal rheometry, we study the shear profiles of well-entangled DNA solutions under large amplitude oscillatory shear (LAOS) in a rectilinear planar shear cell. With increasing Weissenberg number (Wi), we observe successive transitions from normal Newtonian linear shear profiles to wall-slip dominant shear profiles and finally to shear-banding profiles at high Wi. To investigate the microscopic origin of the observed shear banding, we study the dynamics of micron-sized tracers embedded in DNA solutions. Surprisingly, tracer particles in the shear frame exhibit transient…
We report on the spatio-temporal dynamics of the interface in shear-banding flow of a wormlike micellar system (cetyltrimethylammonium bromide and sodium nitrate in water) during a start-up experiment. Using the scattering properties of the induced structures, we demonstrate the existence of an instability of the interface between bands along the vorticity direction. Different regimes of spatio-temporal dynamics of the interface are indentified along the stress plateau. We build a model based on the flow symetry which qualitatively describes the observed patterns.
The linear stability analysis of an uniform shear flow of granular materials is revisited using several cases of a Navier-Stokes'-level constitutive model in which we incorporate the global equation of states for pressure and thermal conductivity (which are accurate up-to the maximum packing density $ν_{m}$) and the shear viscosity is allowed to diverge at a density $ν_μ$ ($< ν_{m}$), with all other transport coefficients diverging at $ν_{m}$. It is shown that the emergence of shear-banding instabilities (for perturbations having no variation along the streamwise direction), that lead to shear…
We investigate the occurrence of shear banding in nematogenic fluids under planar Couette flow, based on mesoscopic dynamical equations for the orientational order parameter and the shear stress. We focus on parameter values where the sheared homogeneous system exhibits regular oscillatory orientational dynamics, whereas the equilibrium system is either isotropic (albeit close to the isotropic--nematic transition) or deep in its nematic phase. The numerical calculations are restricted to spatial variations in shear gradient direction. We find several new types of shear banded states characteri…
We study a simple model of shear banding in which the flow-induced phase is destabilised by coupling between flow and microstructure (wormlike micellar length). By varying the strength of the instability and the applied shear rate, we find a rich variety of oscillatory and rheochaotic shear banded flows. At low shear and weak instability, the induced phase pulsates in width next to one wall of the flow cell. For stronger instability, single or multiple high shear pulses ricochet across the cell. At high shear rates we observe oscillating bands on either side of a defect. In some cases, multipl…
We study the dynamics of shear-band formation and evolution using a simple rheological model. The description couples the local structure and viscosity to the applied shear stress. We consider in detail the Couette geometry, where the model is solved iteratively with the Navier-Stokes equation to obtain the time-evolution of the local velocity and viscosity fields. It is found that the underlying reason for dynamic effects is the non-homogeneous shear distribution, which is amplified due to a positive feedback between the flow field and the viscosity response of the shear thinning fluid. This …
In 1975 Doi and Edwards predicted that entangled polymer melts and solutions can have a constitutive instability, signified by a decreasing stress for shear rates greater than the inverse of the reptation time. Experiments did not support this, and more sophisticated theories incorporated Marrucci's idea (1996) of removing constraints by advection; this produced a monotonically increasing stress and thus stable constitutive behavior. Recent experiments have suggested that entangled polymer solutions may possess a constitutive instability after all, and have led some workers to question the val…
Soft Glassy Materials, Non Linear Rheology, Lattice Kinetic models, frustrated phase separation} We present numerical simulations based on a Boltzmann kinetic model with competing interactions, aimed at characterizating the rheological properties of soft-glassy materials. The lattice kinetic model is shown to reproduce typical signatures of driven soft-glassy flows in confined geometries, such as Herschel-Bulkley rheology, shear-banding and histeresys. This lends further credit to the present lattice kinetic model as a valuable tool for the theoretical/computational investigation of the rheolo…
We study the emergence of a yield stress in dense suspensions of non-Brownian particles, by combining local velocity and concentration measurements using Magnetic Resonance Imaging with macroscopic rheometric experiments. We show that the competition between gravity and viscous stresses is at the origin of the development of a yield stress in these systems at relatively low volume fractions. Moreover, it is accompanied by a shear banding phenomenon that is the signature of this competition. However, if the system is carefully density matched, no yield stress is encountered until a volume fract…
We demonstrate that application of an increasing shear field on a glass leads to an intriguing dynamic first order transition in analogy to equilibrium transitions. By following the particle dynamics as a function of the driving field in a colloidal glass, we identify a critical shear rate upon which the diffusion time scale of the glass exhibits a sudden discontinuity. Using a new dynamic order parameter, we show that this discontinuity is analogous to a first order transition, in which the applied stress acts as the conjugate field on the system's dynamic evolution. These results offer new p…
We review the experimental and theoretical results obtained during the past decade on the structure and rheology of wormlike micellar solutions. We focus on the linear and nonlinear viscoelasticity and emphasize the analogies with polymers. Based on a comprehensive survey of surfactant systems, the present study shows the existence of standard rheological behaviors for semidilute and concentrated solutions. One feature of this behavior is a shear banding transition associated with a stress plateau in the nonlinear mechanical response. For concentrated solutions, we show that in the plateau reg…
We report a large set of experimental data which demonstrates that a simple yield stress fluid, i.e. which does not present aging or thixotropy, exhibits transient shear banding before reaching a steady state characterized by a homogeneous, linear velocity profile. The duration of the transient regime decreases as a power law with the applied shear rate $\dotγ$. This power law behavior, observed here in carbopol dispersions, does not depend on the gap width and on the boundary conditions for a given sample preparation. For $\dotγ\lesssim 0.1$ s$^{-1}$, heterogeneous flows could be observed for…
We study numerically the nonlinear dynamics of a shear banding interface in two dimensional planar shear flow, within the non-local Johnson Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for sufficiently small ratio of the interfacial width $\ell$ to cell length $L_x$. The instability saturates in finite amplitude interfacial fluctuations. For decreasing $\ell/L_x$ these undergo a non equilibrium transition from simple travelling interfacial waves with constant average wall stress, to pe…
Bacterial suspensions--a premier example of active fluids--show an unusual response to shear stresses. Instead of increasing the viscosity of the suspending fluid, the emergent collective motions of swimming bacteria can turn a suspension into a "superfluid" with zero apparent viscosity. Although the existence of active "superfluids" has been demonstrated in bulk rheological measurements, the microscopic origin and dynamics of such an exotic phase have not been experimentally probed. Here, using high-speed confocal rheometry, we study the dynamics of concentrated bacterial suspensions under si…
We study the dynamics of shear startup of the Johnson-Segalman and non-stretching Rolie-Poly models using nonlinear simulations. We consider cases where the startup is from zero shear rate to shear rates in both the monotonic and nonmonotonic regions of the constitutive curve. For the Johnson-Segalman model, which exhibits a shear stress overshoot during startup, our nonlinear simulations show that transient shear banding is absent regardless of whether the start-up shear rate is in the monotonic or nonmonotonic regions of the constitutive curve. In the latter case, while there is clearly an i…
Jammed systems all have a yield stress. Among these materials some have been shown to shear-band but it is as yet unclear why some materials develop shear-band and some others do not. In order to rationalize existing data concerning the flow characteristics of jammed systems and in particular understand the physical origin of such a difference we propose a simple approach for describing the steady flow behaviour of yield stress fluids, which retains only basic physical ingredients. Within this frame we show that in the liquid regime the behaviour of jammed systems turns from that of a simple y…
We analyze the temporal fluctuations of the flow field associated to a shear-induced transition in a lyotropic lamellar phase: the layering transition of the onion texture. In the first part of this work [Salmon et al., submitted to Phys. Rev. E], we have evidenced banded flows at the onset of this shear-induced transition which are well accounted for by the classical picture of shear-banding. In the present paper, we focus on the temporal fluctuations of the flow field recorded in the coexistence domain. These striking dynamics are very slow (100--1000s) and cannot be due to external mechanic…
Motivated by the need for a theoretical study in a planar geometry that can easily be implemented experimentally, we study the pressure driven Poiseuille flow of a shear banding fluid. After discussing the "basic states" predicted by a one dimensional calculation that assumes a flat interface between the bands, we proceed to demonstrate such an interface to be unstable with respect to the growth of undulations along it. We give results for the growth rate and wavevector of the most unstable mode that grows initially, as well as for the ultimate flow patterns to which the instability leads. We …
We have generated a novel form of shear banding in a 2D foam and measured the relative magnitude of drag forces on soap films at different lubrication layers. We injected air part way along a flowing bubble field in a narrow Hele-Shaw cell. The injected air inflates bubbles as they flow by, and these bubbles form a shear band down the middle of the Hele-Shaw cell. This channel appears to select a height that minimizes the total dissipation. Fitting data to a simple theoretical model, we show that the drag force on a soap film in contact with the wetting layer on a plate of glass is two orders …
We have designed 3D numerical simulations of a soft spheres model, with size polidispersity and in athermal conditions, to study the transient shear banding that occurs during yielding of jammed soft solids. We analyze the effects of different types of drag coefficients used in the simulations and compare the results obtained using Lees-Edwards periodic boundary conditions with the case in which the same model solid is confined between two walls. The specific damping mechanism and the different boundary conditions indeed modify the load curves and the velocity profiles in the transient regime.…
Main characteristics of colloidal systems that develop fluid phases with different mechanical properties, namely shear-banding fluids, are briefly reviewed both from experimental and theoretical (modelling) point of view. A non-monotonic shear stress vs. shear rate constitutive relation is presented. This relation derives from a phenomenological model of a shear ratedependent viscosity describing structural changes and involves the possibility of multivalued shear rates under a given shear stress. In the case of a stress-dependent viscosity, the same model allows one to predict vorticity bandi…
Amorphous solids are critical in the design and production of nanoscale devices, but under strong confinement these materials exhibit changes in their mechanical properties which are not well understood. Phenomenological models explain these properties by postulating an underlying defect structure in these materials but do not detail the microscopic properties of these defects. Using machine learning methods, we identify mesoscale defects that lead to shear banding in polymer nanopillars well below the glass transition temperature as a function of pillar diameter. Our results show that the pri…
In this work, we use flow visualization and rheometry techniques to study the dynamics and evolution of secondary flows in a model wormlike micellar solution sheared between concentric cylinders, i.e., in a Taylor-Couette (TC) cell. The wormlike micellar solution studied in this work contains cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal). This system can be shear banding and highly elastic, non-shear banding and moderately elastic, or nearly Newtonian as the temperature is varied over a narrow range. The effect of elasticity on transitions and instabilities is probed by c…
We report on a non trivial dynamics of the interface between shear bands following a start-up of flow in a semi-dilute wormlike micellar system investigated using a combination of mechanical and optical measurements. During the building of the banding structure, we observed the stages of formation, migration of the interface between bands and finally the destabilization of this interface along the vorticity axis. The mechanical signature of these processes has been indentified in the time series of the shear stress. The interface instability occurs all along the stress plateau, the asymptotic …
After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant solutions, we present flow phase diagrams spanned by shear stress (or strain-rate) and concentration in the two-fluid, non-local Johnson-Segalman (d-JS-phi) model. We also present macroscopic flow curves for a range of (average) concentrations. For any concentration high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence ``plateau'' that slopes upwards. As the concentration is reduced, the width …
We report on the rheology of a lamellar lyotropic surfactant solution (SDS/dodecane/pentanol/water), and identify a discontinuous transition between two shear thinning regimes which correspond to the low stress lamellar phase and the more viscous shear induced multi-lamellar vesicle, or ``onion'' phase. We study in detail the flow curve, stress as a function of shear rate, during the transition region, and present evidence that the region consists of a shear banded phase where the material has macroscopically separated into bands of lamellae and onions stacked in the vorticity direction. We in…
The interplay between shear band (SB) formation and boundary conditions (BC) is investigated in wormlike micellar systems (CPyCl--NaSal) using ultrasonic velocimetry coupled to standard rheology in Couette geometry. Time-resolved velocity profiles are recorded during transient strain-controlled experiments in smooth and sand-blasted geometries. For stick BC standard SB is observed, although depending on the degree of micellar entanglement temporal fluctuations are reported in the highly sheared band. For slip BC wall slip occurs only for shear rates larger than the start of the stress plateau.…
One prototypical instability in granular flows is the shear-banding instability, in which a uniform granular shear flow breaks into alternating bands of dense and dilute clusters of particles having low and high shear (shear stress or shear rate), respectively. In this work, the shear-banding instability in an arbitrarily inelastic granular shear flow is analyzed through the linear stability analysis of granular hydrodynamic equations closed with Navier--Stokes-level constitutive relations. It is shown that the choice of appropriate constitutive relations plays an important role in predicting …
Many soft materials, including foams, dense emulsions, micro gel bead suspensions, star polymers, dense packing of surfactant onion micelles, and textured morphologies of liquid crystals, share the basic "glassy" features of structural disorder and metastability. These in turn give rise to several notable features in the low frequency shear rheology (deformation and flow properties) of these materials: in particular, the existence of a yield stress below which the material behaves like a solid, and above which it flows like a liquid. In the last decade, intense experimental activity has also r…
A general phenomenological reaction-diffusion model for flow-induced phase transitions in complex fluids is presented. The model consists of an equation of motion for a nonconserved composition variable, coupled to a Newtonian stress relations for the reactant and product species. Multivalued reaction terms allow for different homogeneous phases to coexist with each other, resulting in banded composition and shear rate profiles. The one-dimensional equation of motion is evolved from a random initial state to its final steady-state. We find that the system chooses banded states over homogeneous…
Cooperativity effects have been proposed to explain the non-local rheology in the dynamics of soft jammed systems. Based on the analysis of the free-energy model proposed by L. Bocquet, A. Colin \& A. Ajdari ({\em Phys. Rev. Lett.} {\bf 103}, 036001 (2009)), we show that cooperativity effects resulting from the non-local nature of the fluidity (inverse viscosity), are intimately related to the emergence of shear-banding configurations. This connection materializes through the onset of inhomogeneous compact solutions (compactons), wherein the fluidity is confined to finite-support subregions of…
We study theoretically the formation of shear bands in time-dependent flows of polymeric and wormlike micellar surfactant fluids, focussing on the protocols of step shear stress, step shear strain (or in practice a rapid strain ramp), and shear startup, which are commonly studied experimentally. For each protocol we perform a linear stability analysis to provide a fluid-universal criterion for the onset of shear banding, following our recent Letter [Phys. Rev. Lett. 110 (2013) 086001]. In each case this criterion depends only on the shape of the experimentally measured rheological response fun…
Wang [Phys. Rev. Lett. 103, 219801 (2009)] makes the following points about our Letter [Phys. Rev. Lett. 102, 067801 (2009), arXiv:0805.0679]: (1) He infers that, "contrary to its title, shear banding emerged from monotonic curves only if there was a stress gradient", and he points out that nonquiescent relaxation was found (experimentally) after step strain in geometries without a stress gradient. (2) He disagrees with the values of the parameters we used. (3) In some recent experiments the flow was homogeneous after cessation of step strain, and only subsequently developed nonquiescent macro…
We study the early stages of the shear banding instability in semidilute wormlike micelles using the non-local Johnson-Segalman model with a two-fluid coupling of the concentration (phi) to the shear rate (gamma_dot) and micellar strain (tensor{W}). We calculate the ``spinodal'' limit of stability for sweeps along the homogeneous intrinsic flow curve. For startup ``quenches'' into the unstable region, the instability in general occurs before the homogeneous startup flow can attain the intrinsic flow curve. We predict the selected time and length scales at which inhomogeneity first emerges. In …
