Linear Instability of Planar Shear Banded Flow

Fielding, Suzanne M.

doi:10.1103/physrevlett.95.134501

Cited by 77 publications

(129 citation statements)

References 34 publications

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“…(For 3D-related effects in situations of shear-banding, see [38].) This choice for the heterogeneity direction is motivated by the classical geometry of steady shear bands in thickening materials [5,6] where, as shown in Fig.…”

Section: A Model Equationsmentioning

confidence: 99%

Minimal model for chaotic shear banding in shear thickening fluids

Aradian

Cates

2006

Phys. Rev. E

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We present a minimal model for spatiotemporal oscillation and rheochaos in shear-thickening complex fluids at zero Reynolds number. In the model, a tendency towards inhomogeneous flows in the form of shear bands combines with a slow structural dynamics, modelled by delayed stress relaxation. Using Fourier-space numerics, we study the nonequilibrium 'phase diagram' of the fluid as a function of a steady mean (spatially averaged) stress, and of the relaxation time for structural relaxation. We find several distinct regions of periodic behavior (oscillating bands, travelling bands, and more complex oscillations) and also regions of spatiotemporal rheochaos. A low-dimensional truncation of the model retains the important physical features of the full model (including rheochaos) despite the suppression of sharply defined interfaces between shear bands. Our model maps onto the FitzHugh-Nagumo model for neural network dynamics, with an unusual form of long-range coupling.

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Section: A Model Equationsmentioning

confidence: 99%

Minimal model for chaotic shear banding in shear thickening fluids

Aradian

Cates

2006

Phys. Rev. E

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“…In other words, the amplitude of the interface instability oscillates with time. This could be due, among others, to tridimensional flow or to destabilization of the interface in the velocity direction [12].…”

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“…This model equation can also be derived using perturbation theory. To this end, it is necessary to adopt a model for the rheology of the fluid [12,20]. It is a computation that we did using the model [20] and we will report it elsewhere [21] with the complete exploration of the stress plateau.…”

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confidence: 99%

Interface Instability in Shear-Banding Flow

2006

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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. For a system showing a shear-thinning behaviour, the mechanical signature, in the measured flow curve (σ = f (γ)), of this type of phenomena is the stress plateau separating low and high viscosity branches. This stress plateau is at the origin of a shear-banding transition from a flow at first homogeneous towards a stationary state where two layers supporting different shear rates coexist. A change of the applied shear rate only affects the relative proportion of each layer, the width of the induced band increasing linearly with the macroscopic shear rate [6]. Among the complex fluids with stress plateau, the systems of reversible giant micelles have been the object of the most intensive surveys (see [1] for a review). Very recently, the use of dynamic light scattering to measure velocity profiles brought confirmation of the very simple shear-banding scenario previously mentioned for the widely studied CPCl/NaSal solution [7]. However, the authors mention the existence of temporal fluctuations in the highly sheared band. More complex pictures have also emerged with the development of time-and spatially- * Corresponding author ; Electronic address: slerouge@ccr.jussieu.fr resolved techniques such as rapid nuclear magnetic resonance (NMR) velocimetry [8] and ultrasonic velocimetry [9], with for example, the observation of periodic or quasirandom oscillations of the interface position driven by wall slip. Moreover, the flow birefringence, technique sensitive to molecular alignment, revealed complex kinetics of banding formation and strongly differing banding organizations from one sample to another [10]. Some of these behaviors are reproduced, at least qualitatively, by several theoretical works, which predict fluctuating or chaotic flows, taking into account a coupling between flow and microstruture via the concentration or the micellar length [11]. They also motivated a recent theoretical work, in which the author, using the nonlocal Johnson-Segalman model, studies the stability of planar shear banded flow with respect to small perturbations with wavevectors in the interfacial plane [12]. An interfacial instability is predicted in the plateau domain for almost all shear rates, the interface profile being then modulated by a wave with wavevectors confined along the flow direction. In the present letter, we report on the formation kinetics of the banding structure and follow particularly the spatio-temporal dynamics of the interface between bands for a sa...

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“…For smaller ℓ/L x , however, interfacial undulations are predicted by the linear analysis of Refs. [16] to become unstable. Our numerics successfully reproduce this instability during the initial evolution away from IC1: the eigenvectors and growth rates ω(q x ) match the analytical results of Refs.…”

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confidence: 99%

“…A 1D (y) calcula-tion then predicts separation into bands of shear rateṡ γ 1 = 0.66,γ 2 = 7.09, at a selected shear stress T * xy = 0.506. Recent analysis [16] showed this stationary 1D banded state to be linearly unstable to 2D (x, y) perturbations corresponding to undulations of the interface with wavevector q = q xx . The most unstable mode has q x L ≈ 2π, and the instability involves feedback of the normal stress with velocity fluctuations across the interface.…”

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Nonlinear Dynamics of an Interface between Shear Bands

Fielding

Olmsted

2006

Phys. Rev. Lett.

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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 ℓ to cell length Lx. The instability saturates in finite amplitude interfacial fluctuations. For decreasing ℓ/Lx these undergo a non equilibrium transition from simple travelling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.PACS numbers: 47.50.+d, 47.20.-k, 36.20.-r. Complex fluids such as polymers, liquid crystals and surfactant solutions have mesoscopic structure that is readily perturbed by flow [1]. For example, wormlike surfactant micelles with lengths of the order of microns can be induced to stretch, disentangle and entangle, and break or increase in length. Their mechanical response is therefore highly non-Newtonian, with shear flows inducing normal stresses (e.g. σ xx − σ yy ), and with the shear stress σ xy being a nonlinear function of applied shear ratė γ. Recent work on such fluids has led to a fairly consistent picture of "shear banding": coexistence in shear flow of viscously thicker (nascent) and thinner (flow-induced) bands of material flowing at different local shear rates, for an overall average imposed shear rate. This phenomenon can be described by constitutive models for which the shear stress is a non-monotonic function of shear rate for homogeneous flow. This leads to a separation into bands of differing shear rate that coexist at a common shear stress [2,3]. Although most studies have assumed a flat interface between the bands and (with a few exceptions [4]) predicted a timeindependent banded state, an accumulating body of data has demonstrated that the average shear stress, and the banding interface, can fluctuate [5,6,7,8,9,10,11,19]. An important question is then whether these fluctuations resemble small amplitude capillary waves stabilised by surface tension, or whether they arise from an underlying instability, stabilised at large amplitude by nonlinearities. In this Letter we give strong evidence supporting the latter scenario, via the first theoretical study of the nonlinear dynamics of a shear banding interface.The model -The generalised Navier Stokes equation for a viscoelastic material in a Newtonian solvent of viscosity η and density ρ is:where v(r) is the velocity field. The pressure P is determined by incompressibility, ∇ · v = 0. The viscoelastic stress Σ(r) evolves according to the non-local ("diffusive") Johnson Segalman (DJS) model [12,13] (with plateau modulus G and relaxation time τ . D and Ω are the symmetric and antisymmetric parts of the velocity gradient tensor, (∇v) αβ ≡ ∂ α v β . For a = 1 and ℓ = 0 this model reduces t...

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Linear Instability of Planar Shear Banded Flow

Cited by 77 publications

References 34 publications

Minimal model for chaotic shear banding in shear thickening fluids

Minimal model for chaotic shear banding in shear thickening fluids

Interface Instability in Shear-Banding Flow

Nonlinear Dynamics of an Interface between Shear Bands

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