Shear Banding Flow Research Articles

Study of the electroosmotic flow of a structured fluid with a new generalized rheological model

The electroosmotic flow of a viscoelastic fluid in a capillary system was investigated analytically. The rheology of the fluid was characterized by a novel generalized exponential model equation. The charge density obeys the Boltzmann distribution, which governs the electrical double-layer field and body force generated by the applied electrical field. Mathematically, this scenario can be modeled by the Poisson-Boltzmann partial differential equation, by assuming that the zeta potential is small, i.e., less than 25 mV (Debye-Hückel approximation). Considering a pulsating electric field, the shear viscosity and the alteration in the volumetric flow were presented as a function of the material parameters through the characteristic dimensionless numbers by using an exponential-type generalized rheological model. Thixotropy, shear thinning, yield stress mechanisms, and weight concentration were analyzed through numerical results. Finally, the flow properties and rheology were predicted using experimental data reported elsewhere for worm-like micellar solution of cetyl trimethyl ammonium tosilate (CTAT). The rheological equation of state proposed in this study describes the alterations in the structure resulting from applied forces (tangential and normal). These forces induced a structural evolution (kinetic model) due to the relaxation processes caused by shear strain. It is important to mention that in electroosmotic flows, complex behavior such as (i) thixotropy, (ii) rheopexy, and (iii) shear banding flow is scarcely explained in terms of the change in the structure of the fluid under flow.Graphical

Achieving strength-ductility synergy in metallic glasses via electric current-enhanced structural fluctuations

Structural applications of metallic glasses are limited by their room-temperature brittleness and strain-softening, associated with the extreme localization of plastic flow in shear bands. Herein, we alleviate this dilemma by the structural fluctuation induced by electric currents, achieving simultaneous improvement of strengths, compression plasticity, and strain hardening capacity. The electric current enhances the structural fluctuations at the atomic scale, introducing more soft zones while densifying their surroundings, which differs from previously reported rejuvenation strategies. This unique structural pattern featuring soft zones surrounded by hard zones leads to extensive sprouts and interactions of shear bands, capturing substantial atoms to participate in the deformation. The plausible mechanism behind the electric current-influenced dynamic evolution of amorphous clusters is proposed. Electric current introduces interatomic electrostatic forces between shell atoms of amorphous clusters through charge transfer. This force decreases the coordination of Zr- and Cu-centered clusters and increases the Al-centered icosahedra in the glassy matrix. Differential evolution of clusters under electric currents enhances atomic structural fluctuations and originates from the cohesion mechanism determined by electron distribution. These findings are suggestive of vanishing room-temperature brittleness and shed atomic-scale light on modulations of the structure and properties of metallic glasses by electric currents.

Shear banding predictions for wormlike micellar systems under a contraction–expansion complex flow

This study focuses on computational modeling of shear-banded wormlike micellar solutions (WLM) in a complex planar Couette flow, driven by a moving top plate over a rounded-corner 4:1:4 obstruction. The BMP+_τp model is used, which is constructed within an Oldroyd-B-like form, coupled with a thixotropic fluidity-based structure equation. Solute energy dissipation drives fluid–structure adjustment in a construction–destruction dynamics affected by viscoelasticity. This model reproduces conventional WLM features, such as shear thinning, extensional hardening/softening, viscoelasticity, apparent yield stress, and shear banding, with a bounded extensional viscosity and an N1Shear upturn at high deformation rates. The BMP+_τp characterization for shear banding is based on extremely low solvent fractions and appropriate shear-banding intensity parameters. Flow structure is analyzed through velocity, stress, and fluidity, whereupon banded and non-banded response is contrasted at appropriately selected flow rates. Solutions are obtained with our hybrid fe-fv algorithm, capturing essential shear-banded flow features reported experimentally. For a fluid exhibiting banding, banded solutions are generated at a flow rate within the flow curve unstable branch. In the fully developed simple shear flow regions, a split velocity profile is observed, with different viscosity bands at equal stress levels, enhanced with a shock-capture procedure. Non-banded solutions are derived for the lowest and highest flow rates sampled, located in the stable branches. Within the constriction zone, banded profiles are lost due to the mixed non-homogeneous deformation. Shear-banding fluids display less intense viscosity/stress features, correlated with their relatively stronger shear thinning response. The constriction resistance provokes a pressure-level adjustment, leading to fully developed Couette-like constant values upstream–downstream.

Influence of precipitate distributions on adiabatic shear band formation and flow behavior of heat-treated 7050 aluminum alloy

Influence of precipitate distributions on adiabatic shear band formation and flow behavior of heat-treated 7050 aluminum alloy

Continuum modeling of soft glassy materials under shear

Soft Glassy Materials (SGM) consist in dense amorphous assemblies of colloidal particles of multiple shapes, elasticity, and interactions, which confer upon them solid-like properties at rest. They are ubiquitously encountered in modern engineering, including additive manufacturing, semi-solid flow cells, dip coating, adhesive locomotion, where they are subjected to complex mechanical histories. Such processes often include a solid-to-liquid transition induced by large enough shear, which results in complex transient phenomena such as non-monotonic stress responses, i.e., stress overshoot, and spatially heterogeneous flows, e.g., shear banding or brittle failure. In the present article, we propose a pedagogical introduction to a continuum model based on a spatially resolved fluidity approach that we recently introduced to rationalize shear-induced yielding in SGMs. Our model, which relies upon non-local effects, quantitatively captures salient features associated with such complex flows, including the rate dependence of the stress overshoot, as well as transient shear-banded flows together with non-trivial scaling laws for fluidization times. This approach offers a versatile framework to account for subtle effects, such as avalanche-like phenomena, or the impact of boundary conditions, which we illustrate by including in our model the elasto-hydrodynamic slippage of soft particles compressed against solid surfaces.

Microstructural evolution and reverse flow in shear-banding of entangled polymer melts.

The temporal and spatial evolution of shear banding under startup of shear flow was simulated for highly entangled, linear, monodisperse polyethylene melts of differing molecular weight, C750H1502, C1200H2402, and C3000H6002, using a high-fidelity coarse-grained dissipative particle dynamics method. It was determined that shear stress was dominated by segmental orientation of entangled strands at low shear rates, but at a critical shear rate below the reciprocal of the Rouse time, flow-induced disentanglement resulted in the onset of chain tumbling that reduced the average degree of orientation, leading to a regime of decreasing shear stress, with a commensurate onset of increasing average chain extension imposed by the strong flow kinematics that ultimately drove the steady-state shear stress higher. During startup of shear flow, shear band development began immediately after the maximum in the first normal stress difference, where distinct fast and slow bands formed. The slow bands consisted of relatively entangled and coiled molecules, whereas the fast bands consisted of more disentangled and extended chains that experienced quasiperiodic rotation/retraction cycles. The simulation results often exhibited a generation of temporary reverse flow, in which the local fluid velocity was temporarily opposite to that of the bulk flow direction, at the onset of the shear-banding phenomena; this effect was consistent with earlier experiments and theoretical results. The physical mechanism for the generation of reverse flow during shear-band formation was investigated and found to be related to the recoil of the molecules comprising the slow band. Overall, the phenomenon of shear banding appeared to arise due to flow-induced disentanglement from orientational ordering and segmental stretching that affected individual chains to different degrees, ultimately resulting in regions of relatively coiled and entangled chains that evolved into a slow band, whereas the locally disentangled chains, experiencing quasiperiodic stretch-rotation cycles, formed a fast band. The transitional period resulted in a kinematic instability that generated the temporary reverse-flow phenomenon.

Role of micellar entanglements on kinetics of shear banding flow formation

We investigate the effects of micellar entanglement number on the kinetics of shear banding flow formation in a Taylor–Couette flow. Three sets of wormlike micellar solutions, each set with a similar fluid elasticity and zero-shear-rate viscosity, but with varying entanglement densities, are studied under the startup of steady shear. Our experiments indicate that in the set with low fluid elasticity, the transient shear banding flow is characterized by the formation of a transient flow reversal in a range of entanglement densities. Outside of this range, the transient flow reversal is not observed. For the sets of medium and high elasticities, the transient flow reversals exist for relatively small entanglement densities and disappear for large entanglement densities. Our analysis shows that wall slip and elastic instabilities do not affect the transient flow feature. We identify a correlation between micellar entanglement number, the width of the stress plateau, and the extent of the transient flow reversal. As the micellar entanglement number increases, the width of the stress plateau first increases; then, at a higher micellar entanglement number, the plateau width decreases. Therefore, we hypothesize that the transient flow reversal is connected to the micellar entanglement number through the width of the stress plateau.

Flow of wormlike micelles: From shear banding to elastic turbulence

Shear banding is a ubiquitous phenomenon in complex fluids flows. Most often it corresponds to a flow state in which a fluid splits into layers of differing apparent viscosities supporting different local shear rates. Shear banding has been observed, either transiently or at steady state, in different classes of complex systems including polymeric fluids and soft glassy materials. Its description and understanding is particularly advanced in surfactant wormlike micelles, for which the experimental evidence of its mechanical signature dates back to the 1990s. Over the past 10 years a strong effort was made to combine global rheology with local time-resolved techniques to probe both the velocity field and the structural properties in various macro- and micro-fluidic geometries. Overall, complex flows have been observed to emerge on top of the shear-banding flow and have been connected to the development of viscoelastic instabilities.In this talk I will try to describe the complete picture of the shear-banding transition in wormlike micelles flowing in Taylor-Couette geometry, from the onset of banding to the development of elastic turbulence. The response of shear-banding wormlike to time-dependent flow protocols such as step stress and shear startup will be examined. The mechanical signatures of the onset of banding will be discussed and compared with general criteria derived by Moorcroft and Fielding. Then I will describe the spatio-temporal dynamics over longer time scales showing the development of secondary flows on top of shear banding. I will also address the transition towards a disordered flow state reminiscent of elastic turbulence in regular polymers. Finally, we will see that non-shear banding wormlike micelles also exhibit elastic instabilities and turbulence, providing a way to explore a large range of elasticity numbers.

Shear banding and serrated flow behaviors of high toughness Zr61Ti2Cu25Al12 bulk metallic glass under bending

By interrupted three-point bending experiments, the detailed evolutions of shear-banding and serrated flow as well as their correlation of high toughness Z61Ti2Cu25Al12 bulk metallic glass (BMG) under bending were systematically studied. Three stages can be classified during plastic bending of Z61Ti2Cu25Al12 BMG based on the different characteristics of shear-banding and serrated flow behaviors. Moreover, there has a strong correlation between shear-banding and serrated flow at different stages of plastic bending. It is noteworthy that the driving force for shear bands propagation of BMG beam under bending depends on the bending moment, instead of the applied load. Thus, remarkable positive correlations among bending moment, shear band zone height and average load drop of serrations were found during plastic bending. During the progressive propagation of multiple shear bands, intense interactions and competitions among these shear bands have occurred, which manifested by the self-organized criticality behavior from serrated flow dynamics, as well as the very finer irregular striation patterns and uneven fluctuant morphologies which presented on the shear offset surface. The sizeable plastic zone of Z61Ti2Cu25Al12 BMG is very favorable to achieve the greater macroscopic plasticity and lower the risk of fast catastrophic fracture. Thus, Z61Ti2Cu25Al12 BMG is a promising engineering material for structure elements which are naturally subjected to bending load, such as compliant mechanisms and microelectromechanical systems.

Kinetics of shear banding flow formation in linear and branched wormlike micelles.

We investigate the flow evolution of a linear and a branched wormlike micellar solution with matched rheology in a Taylor-Couette (TC) cell using a combination of particle-tracking velocimetry, birefringence, and turbidity measurements. Both solutions exhibit a stress plateau within a range of shear rates. Under startup of a steady shear rate flow within the stress plateau, both linear and branched samples exhibit strong transient shear thinning flow profiles. However, while the flow of the linear solution evolves to a banded structure at longer times, the flow of the branched solution transitions to a curved velocity profile with no evidence of shear banding. Flow-induced birefringence measurements indicate transient birefringence banding with strong micellar alignment in the high shear band for the linear solution. The transient flow-induced birefringence is stronger for the branched system at an otherwise identical Wi. At longer times, the birefringence bands are replaced by a chaotic flow reminiscent of elastic instabilities. Visualization of the flow-induced turbidity in the velocity gradient-vorticity plane reveals quasi-steady banding with a turbidity contrast between high and low shear bands in the linear solution. However, the turbidity evolves uniformly within the gap of the TC cell for the branched solution, corroborating the non-banded quasi-steady velocimetry results. Finally, we show that while elastic instabilities in the linear solution emerge in the high shear band, the flow of branched solution at high Wi becomes unstable due to end effects, with growing end regions that ultimately span the entire axial length of the TC cell.

Shear-banding fluid(s) under time-dependent shear flows. Part I: Spatiotemporal dynamics

We report on the response of semidilute shear-banding wormlike micelles to time-dependent flow protocols such as step stress and shear startup. We use global rheology coupled with direct optical visualizations in the Taylor–Couette flow to provide a detailed 2D description of the spatiotemporal dynamics of the shear banding flow under imposed stress. We compare these dynamics to their extensively studied counterpart under an imposed shear rate. We show that, for a given asymptotic state, the onset of banding is delayed and the transients are longer when the stress is used as a control parameter. The way the banding structure is built up strongly depends on the control parameter. In the two cases, secondary flows develop on top of the shear-banding instability: after shear startup, the interface between bands is formed in the bulk of the system and migrates inward to its stationary position in the gap of the Taylor–Couette device, before being destabilized due to elastic instability; under creep flow, the interface between bands forms at the inner wall and becomes wavy while it is still migrating outward to its steady position, leading to permanent rearrangements of the flow structure until a steady state is reached, potentially explaining longer transients. We also outline the possible mechanical signature of the onset of shear banding in terms of shape of the bulk mechanical signals, which is the subject of Part II of this work [Briole et al., J. Rheol., 65, 1201–1217 (2021)].

Modelling the Failure Behaviour of Mudstones under High Pressures

Understanding the failure behaviour of rocks is very important for many engineering applications. While conceptual models to predict failure behaviour of rocks have been proposed, very few models exist to quantitatively predict the brittle–ductile characteristics of mudstones at high confining pressures. In this study, we performed triaxial compression tests at confining pressures up to 130 MPa on dry indurated mudstones from the Naparima Hill Formation, Trinidad. Experimental results including stress–strain curves, failure modes, and strength parameters of the mudstones were obtained. Analysing the stress–strain curves, shows that the mudstones experienced brittle, brittle–ductile transition, and ductile failure behaviours as the confining pressure increases. The failure behaviour at confining pressures less than 50 MPa is accompanied by axial splitting fractures and shear fractures. At confining pressures greater than 50 MPa, shear band and ductile flow fractures are observed in the mudstones. A failure behaviour model was developed using the failure behaviour and failure strength of the mudstones. This model predicts the brittle, brittle–transition and ductile zones for mudstones at high confining pressures. The brittle to ductile transition is not sudden, but a gradual process that is controlled by the confining pressure and failure strength. The model indicates a narrow brittle-ductile transition zone for weak mudstones as compared to strong mudstones.

Dynamics of DNA-Bridged Dumbbells in Concentrated, Shear-Banding Polymer Solutions

Although experimental evidence for shear-banding flows in concentrated polymer solutions has accumulated over the last 20 years, the origin of such shear-banding flows is still under heated debate....

Rheology of Cohesive Granular Media: Shear Banding, Hysteresis, and Nonlocal Effects

Powders or cohesive granular materials are widely handled in industries. However, our understanding of the rheology of these materials is limited. Here, we provide a comprehensive analysis of the rheology of a cohesive granular medium, sheared in a normal-stress-imposed plane shear cell over a wide range of shear rate, employing numerical simulations. At high imposed shear rates, the flow is homogeneous, and the rheology is well described by the existing scaling laws, involving the inertial number and the effective cohesion number [S. Mandalet al., Insights into the Rheology of Cohesive Granular Media, Proc. Natl. Acad. Sci. U.S.A. 117, 8366 (2020)]. However, at low imposed shear rates, the flow is inhomogeneous, exhibiting the coexistence of flowing and nonflowing regions in the material, popularly known as shear banding. We thoroughly analyze the crucial features of this shear-banded flow regime and discuss striking similarities between the shear banding for granular media and other complex fluids. We reveal that the occurrence of shear banding is related to the existence of a nonmonotonic intrinsic rheological curve and that increasing adhesion increases the nonmonotonicity and the tendency toward shear localization. A simple theoretical model based on a nonlocal rheological model coupled with a nonmonotonic flow curve is proposed and is shown to successfully reproduce all the key features of the shear banding observed in the numerical simulations. The results have important implications for the handling of powders in industries.

Molecular dynamics of shear banding flow in associative polymers solutions: effect of sticker groups location

Non-equilibrium molecular dynamics are used in a coarse-grained scale to simulate the simple shear flow of telechelic and non-telechelic associative polymer solutions. The work includes the study of inception of flow and stress relaxation in transient flows and the constitutive curve at steady state. The instabilities (plateau stress, stress maxima) and the formation of bands with different velocity gradient and concentration depend on the location of the hydrophobic sites (sticker sites, SS) in the chain. Results allow to shed light on the origin of banded flow within the transient state, related to depletion of bridges between aggregates. This situation induces instabilities in the velocity profiles and concentration inhomogeneities along the velocity gradient during the stress overshoot. As the steady state is reached, instabilities disappear for systems where the SS are in the middle of the chain. The conformation relaxation occurs during a time scale larger than that of the stress, but this is not sufficient condition for the presence of banded flow in telechelic polymer solutions, as it holds in polymer melts. The difference in concentration among the bands suggests the existence of a shear-gradient concentration coupling mechanism under banded flow conditions, such as spatial non-homogeneities in the network density, induces chain diffusivity in the gradient direction and increasing normal stresses, especially in telechelic chains. The role of normal stresses on the flow-concentration coupling instability is analyzed in detail.

The physics of fault friction: insights from experiments on simulated gouges at low shearing velocities

Abstract. The strength properties of fault rocks at shearing rates spanning the transition from crystal–plastic flow to frictional slip play a central role in determining the distribution of crustal stress, strain, and seismicity in tectonically active regions. We review experimental and microphysical modelling work, which is aimed at elucidating the processes that control the transition from pervasive ductile flow of fault rock to rate-and-state-dependent frictional (RSF) slip and to runaway rupture, carried out at Utrecht University in the past 2 decades or so. We address shear experiments on simulated gouges composed of calcite, halite–phyllosilicate mixtures, and phyllosilicate–quartz mixtures performed under laboratory conditions spanning the brittle–ductile transition. With increasing shear rate (or decreasing temperature), the results consistently show transitions from (1) stable velocity-strengthening (v-strengthening) behaviour, to potentially unstable v-weakening behaviour, and (2) back to v strengthening. Sample microstructures show that the first transition seen at low shear rates and/or high temperatures represents a switch from pervasive, fully ductile deformation to frictional sliding involving dilatant granular flow in localized shear bands where intergranular slip is incompletely accommodated by creep of individual mineral grains. A recent microphysical model, which treats fault rock deformation as controlled by competition between rate-sensitive (diffusional or crystal–plastic) deformation of individual grains and rate-insensitive sliding interactions between grains (granular flow), predicts both transitions well. Unlike classical RSF approaches, this model quantitatively reproduces a wide range of (transient) frictional behaviours using input parameters with direct physical meaning, with the latest progress focusing on incorporation of dynamic weakening processes characterizing co-seismic fault rupture. When implemented in numerical codes for crustal fault slip, the model offers a single unified framework for understanding slip patch nucleation and growth to critical (seismogenic) dimensions, as well as for simulating the entire seismic cycle.

Effects of elasticity and flow ramp up on kinetics of shear banding flow formation in wormlike micellar fluids

We report experiments on spatiotemporal evolution of the velocity profiles in shear-banding wormlike micelles upon inception of the flow in a Taylor–Couette (TC) cell. Both moderately entangled and highly entangled solutions are considered over a broad range of fluid elasticity E. Fluid elasticity, E = Wi/Re, characterizes the relative importance of the elastic to inertial effects. For both moderately and highly entangled solutions, upon inception of the shear, and during the stress decay period, fluid moves in the opposite direction to that of the imposed motion in a subset of the gap beyond critical thresholds of elasticity and flow ramp up rate, which depend on the fluid entanglement density. Surprisingly, beyond a second critical threshold of the fluid elasticity, the transient backflow disappears in moderately entangled solutions, highlighting the importance of the micellar entanglement on transient evolution of the flow in shear banding systems. More interestingly, we report the formation of multibanded quasisteady velocity profiles under certain conditions of fluid elasticity and flow ramp up rate. The multibanded profiles are characterized by a low shear band near the inner cylinder, a high shear band in the middle of the TC gap, and another low shear band near the outer cylinder. Finally, we show that the apparent wall slip at the inner cylinder of the TC cell is more pronounced for highly entangled solutions and decreases as the fluid elasticity increases. Experimental observations are compared with the existing simulations of the Vasquez–Cook–McKinley model, and several suggestions are made for future simulations.

Numerical Study of Shear Banding in Flows of Fluids Governed by the Rolie-Poly Two-Fluid Model via Stabilized Finite Volume Methods

The flow of viscoelastic fluids may, under certain conditions, exhibit shear-banding characteristics that result from their susceptibility to unusual flow instabilities. In this work, we explore both the existing shear banding mechanisms in the literature, namely; constitutive instabilities and flow-induced inhomogeneities. Shear banding due to constitutive instabilities is modelled via either the Johnson–Segalman or the Giesekus constitutive models. Shear banding due to flow-induced inhomogeneities is modelled via the Rolie–Poly constitutive model. The Rolie–Poly constitutive equation is especially chosen because it expresses, precisely, the shear rheometry of polymer solutions for a large number of strain rates. For the Rolie–Poly approach, we use the two-fluid model wherein the stress dynamics are coupled with concentration equations. We follow a computational analysis approach via an efficient and versatile numerical algorithm. The numerical algorithm is based on the Finite Volume Method (FVM) and it is implemented in the open-source software package, OpenFOAM. The efficiency of our numerical algorithms is enhanced via two possible stabilization techniques, namely; the Log-Conformation Reformulation (LCR) and the Discrete Elastic Viscous Stress Splitting (DEVSS) methodologies. We demonstrate that our stabilized numerical algorithms accurately simulate these complex (shear banded) flows of complex (viscoelastic) fluids. Verification of the shear-banding results via both the Giesekus and Johnson-Segalman models show good agreement with existing literature using the DEVSS technique. A comparison of the Rolie–Poly two-fluid model results with existing literature for the concentration and velocity profiles is also in good agreement.

Strain-hardening and suppression of shear-banding in rejuvenated bulk metallic glass.

Strain-hardening (the increase of flow stress with plastic strain) is the most important phenomenon in the mechanical behaviour of engineering alloys because it ensures that flow is delocalized, enhances tensile ductility and inhibits catastrophic mechanical failure1,2. Metallic glasses (MGs) lack the crystallinity of conventional engineering alloys, and some of their properties-such as higher yield stress and elastic strain limit3-are greatly improved relative to their crystalline counterparts. MGs can have high fracture toughness and have the highest known 'damage tolerance' (defined as the product of yield stress and fracture toughness)4 among all structural materials. However, the use of MGs in structural applications is largely limited by the fact that they show strain-softening instead of strain-hardening; this leads to extreme localization of plastic flow in shear bands, and is associated with early catastrophic failure in tension. Although rejuvenation of an MG (raising its energy to values that are typical of glass formation at a higher cooling rate) lowers its yield stress, which might enable strain-hardening5, it is unclear whether sufficient rejuvenation can be achieved in bulk samples while retaining their glassy structure. Here we show that plastic deformation under triaxial compression at room temperature can rejuvenate bulk MG samples sufficiently to enable strain-hardening through a mechanism that has not been previously observed in the metallic state. This transformed behaviour suppresses shear-banding in bulk samples in normal uniaxial (tensile or compressive) tests, prevents catastrophic failure and leads to higher ultimate flow stress. The rejuvenated MGs are stable at room temperature and show exceptionally efficient strain-hardening, greatly increasing their potential use in structural applications.

Non-local Effects in Shear Banding of Polymeric Flows

Many fluids undergo shear banding, in which two states of different apparent viscosity coexist for a given shear rate (or for a given stress). In the idealized case of an infinite gap between shearing plates the selection of the conditions for shear banding has been shown to depend on the spatial structure and shape of the interface between shear bands. With the advent of microfluidic design for processing and additive manufacturing, the processing of many complex fluids often occurs in situations where this idealized limit doesn't apply, and the gap between walls, in either shearing flow or more often for pressure driven flow, is no longer `infinite' compared to the structural scales. It is increasingly clear that the effective rheology and structure of flowing fluids in these conditions requires information about the entire sample size, i.e. that the rheology is intrinsically non-local. In this review we discuss some recent attempts (both theoretical and experimental) to address non-local rheology and its implications for shear banding flows of polymeric fluids. This manifests itself in rheology extracted from velocity profiles, as well as the dependence of shear banding conditions on the position of the interface between shear bands, as well as the system size.