Criteria for Shear Banding in Time-Dependent Flows of Complex Fluids

Abstract

We study theoretically the onset of shear banding in the three most common time-dependent rheological protocols: step stress, finite strain ramp (a limit of which gives a step strain), and shear startup. By means of a linear stability analysis we provide a fluid-universal criterion for the onset of banding for each protocol, which depends only on the shape of the experimentally measured timedependent rheological response function, independent of the constitutive law and internal state variables of the particular fluid in question. Our predictions thus have the same highly general status, in these time-dependent flows, as the widely known criterion for banding in steady state (of negatively sloping shear stress vs. shear rate). We illustrate them with simulations of the rolie-poly model of polymer flows, and the soft glassy rheology model of disordered soft solids. [4, 5], and (possibly) bio-active fluids [6]. At a fundamental level shear banding can be viewed as a non-equilibrium, flow-induced phase transition, or equivalently as a hydrodynamic instability of viscoelastic origin. In practical terms it drastically alters the rheology (flow response) of these materials and thus impacts industrially in plastics, foodstuffs, well-bore fluids, etc.In steady state, the criterion for shear banding is (usually [7]) that the underlying constitutive relation between shear stress Σ and shear rateγ for homogeneous flow has negative slope, dΣ/dγ < 0. However most practical flows involve a strong time-dependence, whether perpetually or during a startup process in which a steady flow is established from an initial rest-state. Data in polymers [8][9][10][11][12][13][14][15], surfactants [16][17][18], soft glasses [19][20][21][22], and simulations [23][24][25][26][27][28][29][30] reveals that shear bands often also arise during these time-dependent flows, and can be sufficiently long lived to represent the ultimate flow response of the material for practical purposes, even if the constitutive curve is monotonic, dΣ/dγ > 0.In view of these widespread observations, crucially lacking is any known criterion for the onset of banding in time-dependent flows. This Letter provides such criteria, with the same fluid-universal status as the criterion given above in steady state: independent of the internal constitutive properties of the particular fluid in question, and depending only on the shape of the experimentally measured rheological response function. It does so for each of the three most common time-dependent experimental protocols: step stress, finite strain ramp, and shear startup. Our aim is thereby to develop a unified understanding of experimental observations of time-dependent shear banding, and to facilitate the design of flow protocols that optimally enhance or mitigate it as desired.The criteria are derived via a linear stability analysis performed within a highly general framework that encompasses most widely used models for the rheology of polymeric fluids (polymers solutions, melts and wormlike micelles) and soft glas...

Mean-Field Scenario for the Athermal Creep Dynamics of Yield-Stress Fluids

“…In spite of the irrelevance of thermal fluctuations, the creep dynamics will depend on the initial condition determined by the preparation process and the subsequent waiting period, during which slow processes such as coarsening or compaction can alter the level of relaxation. Our approach contrasts previous attempts addressing systems in which thermal fluctuations are important, based on the soft glassy rheology model [13][14][15][16] or mode-coupling theory [17,18], Our description is based on a mean-field version of the elasto-plastic scenario, that describes the flow as resulting from interactions among local plastic events triggered by the external driving, and accounts for the flow properties of athermal YSF [19][20][21][22][23][24][25]. It extends a previous formulation for imposed shear-rate [26] to a system subjected to an imposed stress, allowing us to address complex protocols, in particular typical creep experiments.…”

Modeling the relaxation of polymer glasses under shear and elongational loads

“…This is because the polymeric stress contribution increases monotonically with strain (there is negligible polymer relaxation) which decreases the severity of the stress decline after the overshoot in the total stress. Because transient shear banding is triggered by the declining stress, 54 it is mitigated by the polymeric contribution.…”

“…40,54,57 These instabilities typically occur within the region where stress is decreasing and strain increasing, which can crudely be viewed as negative differential elastic constant (although this picture is certainly oversimplified 54 ). We therefore repeated the simulations allowing the flow to become nonuniform and indeed find transient strain rate variations with position that rise to a maximum near the stress peak and then decay only very slowly; these results are also in Fig.…”

“…A theoretical criterion for shear banding in step-stress has recently been derived; this requires that the underlying homogeneous creep curve γ (t), differentiated to givė γ (t), obeys ∂ 2 tγ /∂ tγ > 0. 54 We have explored this question numerically using the same methods as explained in Sec. VIII below for shear startup.…”

Poiseuille flow of soft glasses in narrow channels: From quiescence to steady state

“…Their response to inhomogeneous stress fields is, however, not well studied. The main focus has so far been on studying the response to spatially uniform stress fields [3][4][5][6][7][8][9].…”