Stability Of Plane Couette Flow Research Articles

Stability of plane Couette flow with constant wall transpiration

Plane Couette flow with constant wall transpiration, i.e., constant blowing from below and constant suction from above, is a solution of the Navier-Stokes equations in terms of exponentials. It is characterized by two Reynolds numbers Re and ReV, where Re parametrizes the moving wall and ReV describes the influence of the transpiration. For this flow the modified Orr-Sommerfeld equation admits one of the very few exact solutions in terms of hypergeometric functions. Based on this, we solve the stability problem, though for the main part focus on temporal stability. For small wall-transpiration rates up to ReV≃6.71, the flow remains unconditionally stable, analogous to the classical Couette flow for arbitrary Reynolds numbers Re. By further increasing ReV an instability sets in and a minimum critical Reynolds number of Re=668350.491 is reached at ReV=9.799. Hence, around this point, the destabilizing effect of blowing outweighs the stabilizing effect of suction. By further increasing the transpiration rate beyond this point, the corresponding critical Reynolds number Re increases again and continues to grow. This limiting case is accompanied by the development of a strong boundary-layer-like velocity profile near the upper wall and the flow transitions to the asymptotic suction boundary layer (ASBL). Thus, the present analysis comprises the whole range from classical Couette flows extended by transpiration to the ASBL, which is known to have a strongly stabilizing effect. Published by the American Physical Society 2024

Instability in strongly stratified plane Couette flow, with application to supercritical fluids.

This paper addresses the stability of plane Couette flow in the presence of strong density and viscosity stratifications. It demonstrates the existence of a generalised inflection point that satisfies the generalised Fjørtoft's criterion of instability when a minimum of kinematic viscosity is present in the base flow. The characteristic scales associated with this minimum are identified as the primary controlling parameters of the associated instability, regardless of the type of stratification. To support this finding, analytical stability models are derived in the long wave approximation using piecewise linear base flows. Numerical stability calculations are carried out to validate these models and to provide further information on the production of disturbance vorticity. All instabilities are interpreted as arising from the interaction between two vorticity waves. Depending on the type of stratification, these two waves are produced by different physical mechanisms. When both strong density and viscosity stratifications are present, we show that they result from the concurrent action of shear and inertial baroclinic effects. The stability models developed for simple fluid models ultimately shed light on a recently observed unstable mode in supercritical fluids (Ren et al., J. Fluid Mech., vol. 871, 2019, pp. 831-864), providing a quantitative prediction of the stability diagram and identifying the dominant mechanisms at play. Furthermore, our study suggests that the minimum of kinematic viscosity reached at the Widom line in these fluids is the leading cause of their instability. The existence of similar instabilities in different fluids and flows (e.g., miscible fluids) is finally discussed.

A self-sustaining mechanism for plane Couette flow with a flexible lower boundary

Abstract The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First, it is shown that a three-dimensional (3D) linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the 3D wave. Governing nonlinear vortex-wave interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.

The linear stability of plane Couette flow with a compliant boundary

The linear stability of plane Couette flow subject to one rigid boundary and one flexible boundary is considered at both finite and asymptotically large Reynolds number. The wall flexibility is modelled using a very simple Hooke-type law involving a spring constant K and is incorporated into a boundary condition on the appropriate Orr–Sommerfeld eigenvalue problem. This problem is analyzed at large Reynolds number by the method of matched asymptotic expansions and eigenrelations are derived that demonstrate the existence of neutral modes at finite spring stiffness, propagating with speeds close to that of the rigid wall and possessing wavelengths comparable to the channel width. A large critical value of K is identified at which a new short wavelength asymptotic structure comes into play that describes the entirety of the linear neutral curve. The asymptotic theories compare well with finite Reynolds number Orr–Sommerfeld calculations and demonstrate that only the tiniest amount of wall flexibility is required to destabilize the flow, with the linear neutral curve for the instability emerging as a bifurcation from infinity.

Stability of plane Couette flow past an initially stressed solid

In this paper, we carry out linear stability analysis of a coupled solid–fluid system in a plane Couette flow past an initially stressed neo-Hookean solid at creeping flow limit (Re=0) and arbitrary Reynolds number (Re), as well as for different solid thicknesses. In biological and engineering systems, we often encounter flows past initially stressed deformable solids. For these cases, initial strain or initial stress appears as an additional symmetry representing tensor in the constitutive relation. Consequently, initial stress induces anisotropy (transverse isotropy in the present case) and affects material properties. We employ a hyperelastic model for the initially stressed deformable solid to study the effects of both uni-axial and equi-biaxial initial stresses on flow instability. This study also presents a case where anisotropy in the solid interferes with the stability of the solid–fluid coupled system. In contrast with the previous formulations with two or three state Eulerian–Lagrangian formulations, the present formulation requires four configurations for the initially stressed solid, viz. stress-free, initially stressed, deformed, and the perturbed state. This formulation applies to any fluid–solid coupled problems where the solid contains initial/residual stress. Our results show that initial tensile stress stabilizes the flow, while compressive initial stress destabilizes the same. Fortuitously, this observation of coupled fluid–solid stability is in agreement with the stability of solid column-like structures, where compressive axial stress leads to buckling. We report similar results for uni-axial and equi-biaxial initial stress, for different wave modes at creeping flow limit and their extensions at higher Re, and various solid thicknesses. Two modes of instability, viz., short-wave and finite-wave modes present in the creeping-flow limit are significantly affected by tensile/compressive initial stress for both uni-axial and equi-biaxial cases. For uni-axial tensile initial stress, the finite-wave mode does not extend to high Re. Instead of them, we observe multiple upstream modes to become unstable for higher Re, which show the wall mode scaling as O(Re−1/3). These upstream modes are not observed for stress-free solid. This behavior is triggered by the additional coupling terms between initial stress and perturbed variable in the governing equations of initially stressed solids. However, for the equi-biaxial case, the finite-wave mode does extend to higher Re to show wall mode scaling. This mode further stabilizes/destabilizes depending on the tensile/compressive nature of the initial stress present in the solid.

Benchmark solution for the stability of plane Couette flow with net throughflow

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow

A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

Fractal sets of neutral curves for stably stratified plane Couette flow

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces.

Consistent formulations for stability of fluid flow through deformable channels and tubes

In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney–Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen–Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number.

Stability of plane Couette flow of a power-law fluid past a neo-Hookean solid at arbitrary Reynolds number

The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the scaling of wall modes is not observed in viscoelastic (modelled as upper-convected Maxwell or Oldroyd-B fluids) plane Couette flow past a deformable solid. We show that the difference in scaling exponents can be explained by postulating that the wall modes in a power-law fluid are determined by the actual viscosity corresponding to the shear rate of the laminar flow denoted by ηapp. A non-dimensional shear rate based on this viscosity Γapp=VηappGR can be defined, and we show that the postulate Γapp∼Re−1∕3 (motivated by the wall-mode scaling in a Newtonian fluid) captures all the numerically observed scalings for Γ vs Re for different values of n>0.3, which is found to be Γ∼Re−12n+1. Further, we numerically evaluated the wall layer thickness and this agreed with the theoretical scaling of δ∼Re−n2n+1. Interestingly, the theoretical and numerical prediction of wall modes is found to be valid for power-law index, n≥ 0.3. For n≤ 0.3, there is a marked departure from the wall-mode scalings, and our results show a scaling of Γ ∼ Re−1 corresponding to inviscid modes. The variation of the power-law index (n) can stabilise/destabilise the inviscid mode when compared with Newtonian fluid, and this result is observed only in the power-law model and is not seen in the flow of viscoelastic fluid past deformable surfaces. Thus, the present study shows that the shear-rate dependence of viscosity has a significant impact on both the qualitative and quantitative aspects of stability of non-Newtonian fluid flow past deformable surfaces.

Stability of plane Couette flow for the compressible Navier–Stokes equations with Navier-slip boundary

This paper is devoted to the stability analysis of the plane Couette flow for the 3D compressible Navier–Stokes equations with Navier-slip boundary condition at the bottom boundary. It is shown that the plane Couette flow is asymptotically stable for small perturbation provided that the slip length, Reynolds and Mach numbers satisfy 3(1+ν˜)αγ2(ν+α)γ0≤1 and 2αν(ν+α)≤1 for some constant γ0>0. In particular, the Reynolds number ν−1 can be large if the slip length α is suitably small. This means that the constraint required in [11] on the Reynolds number to guarantee the stability of the plane Couette flow can be relaxed and improved so long as the slip effect at the boundary is involved.

The resolvent technique for investigating the stability of plane Couette flow: Bounds for the pressure terms

We derive bounds for the pressure terms of the operator used to prove nonlinear stability of plane Couette flow through the resolvent technique.

Stress modes in linear stability of viscoelastic flows

Although parallel shear flows of the upper convected Maxwell fluid are stable, recent articles [R. Kupferman, On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation, J. Non-Newt. Fluid Mech. 127 (2005) 169–190; C.R. Doering, B. Eckhardt, J. Schumacher, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers, J. Non-Newt. Fluid Mech. 135 (2006) 92–96] have exhibited transient growth at high Weissenberg number which is analogous to that observed in high Reynolds number Newtonian shear flows. The perturbations involved are pure stress perturbations with zero velocities. In this paper, we extend the analysis of such stress modes in plane Couette flow to the three-dimensional case. We also exhibit a class of “stress pressure modes” in plane Poiseuille flow. Finally, we discuss pure stress modes in elongational flow.

The stability of plane Couette flow of a power-law fluid with viscous heating

Linear stability analysis is carried out to examine the effect of shear thinning and shear thickening on the stability of plane Couette flow with viscous heating for a power-law fluid that obeys Arrhenius-type law. The relation between shear stress and shear rate is plotted, and the maximum shear stress that must be applied to the system is calculated for different values of the power-law index n. The results showed that the S-shaped curve characterizing the fluid flow that obeys Arrhenius-type law appears at smaller values of the activation energy parameter β, in contrast to the Newtonian case, as a result of shear thinning effect. The resulting eigenvalues are calculated using the Chebyshev collocation method with the QZ algorithm {QZ algorithm is used in solving the singular generalized eigenvalue problem [SIAM J. Numer. Anal. 10, 241 (1973)]}. It is found, for shear thinning/thickening fluid, that the instability occurs at lower/higher values of the Brinkman number Br than in the Newtonian case. Also, the results indicate, for both Newtonian and power-law fluids, that two modes of the instability occur: one termed an inviscid mode, and the second a coupled mode, not a viscous mode as conjectured in C. S. Yueh and C. I. Weng, “Linear stability analysis of plane Couette flow with viscous heating,” Phys. Fluids 8, 1802 (1996).

Hydromagnetic stability of plane Couette flow of an upper convected Maxwell fluid

Hydrodynamic stability of plane Couette flow of an upper convected Maxwell fluid is investigated in presence of a transverse magnetic field assuming that the magnetic Prandtl number is sufficiently small. The resulting equation is a modified Orr-Sommerfeld equation. The equations of stability are solved numerically using Chebyshev collocation method with QZ algorithm. The critical values of Reynolds number, wave number and wave speed are computed and the results are shown through the neutral curves. By increasing the amount of elasticity to a certain value, it is shown that, as the Hartmann number increases, the minimum critical Reynolds number decreases and it does not increase again in contrast to the Newtonian case.

Stability of creeping Couette flow of a power-law fluid past a deformable solid

The stability of plane Couette flow of a power-law fluid past a deformable solid of finite thickness is considered in this work. The solid is a neo-Hookean or linear elastic material which is incompressible and impermeable to the fluid, and linear stability analysis is applied in the creeping-flow limit. Four key dimensionless parameters govern the problem: an imposed shear rate, a solid-to-fluid thickness ratio, an interfacial tension, and a power-law index. The neo-Hookean solid exhibits a first normal stress difference, not present in linear elastic solids, which is strongly coupled to the imposed shear rate and the power-law index. For large thickness ratios, H ≫ 1 , the shear rate necessary to induce an instability, γ c , scales as γ c ∼ H − 1 / n , where n is the power-law index. This scaling can be understood in terms of a simple balance between viscous shear stresses in the fluid and elastic shear stresses in the solid. For small thickness ratios, shear-thinning has a stabilizing effect, in contrast to what is observed for thick solids. Whereas a shortwave instability is always observed with Newtonian fluids and neo-Hookean solids when interfacial tension is absent, it can be suppressed with power-law fluids for certain values of n. These results are potentially of interest for enhancing mixing in microfluidic devices and understanding the rheology of worm-like micelle solutions.

The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions

Linear stability of plane Couette flow of dilute polymeric solutions modeled using the FENE-P constitutive equation has been examined. Specifically, the spurious, discrete and continuous spectra are identified as a function of the ratio of the solvent to the total viscosity β, maximum chain length L, disturbance wavenumber α and the Weissenberg number ( We). It is observed that reducing L shifts the entire spectrum to the left (in the eigenspectrum plane) by a factor directly related to the steady-state value of the Peterlin function f. Additionally, decreasing the value of L causes the splitting of the regular continuous spectrum, initially located at −1/ We, into two branches. Overall, the plane Couette flow of a FENE-P model is found to be unconditionally stable. Specifically, decreasing L increases the decay rate of the most dangerous disturbance while increasing the number of discrete and spurious modes. However, increasing both β and α decreases the number of discrete and spurious modes whereas increasing We has no influence on the number of spurious modes but results in an increase in the number of discrete modes.

On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation

It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non-modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their “passive” transport by shearing streamlines generates large cross-stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings.

Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number

The linear stability of plane Couette flow of an upper convected Maxwell (UCM) fluid of viscosity η , density ρ , relaxation time τ R and thickness R flowing past a linear viscoelastic solid of shear modulus G and thickness HR is analyzed using a combination of asymptotic analysis (at low Reynolds number) and numerical solution (at finite Reynolds number). The asymptotic analysis is used to analyze the effect of wall deformability on a class of high frequency modes in the UCM fluid with c ∼ O ( R e − 1 / 2 ) at R e ≪ 1 first studied by Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310–319; abbreviated GL here], who showed that these modes are stable in a rigid channel. Here, c is the wavespeed of perturbations (nondimensionalised by V), R e = ρ V R / η is the Reynolds number, V is the velocity of the moving plate. Our asymptotic results show that all the high frequency–low Re modes of GL are rendered unstable by solid wall deformability. The variation of the growth rate with the nondimensional solid elasticity parameter Γ = V η / ( G R ) shows an oscillatory behavior alternating between stable and unstable regions, and the variation for upstream traveling waves is completely antiphase with that for downstream waves. The parameter Γ is shown to be proportional to R e 1 / 4 at R e ≪ 1 for neutrally stable downstream waves. Numerical continuation shows that the instability at low Re continues to finite Re, and typically the instability ceases to exist above a critical Re. This critical Re increases with an increase in the Weissenberg number W = τ R V / R . In some cases, however, the instability continues to very high Re, and Γ ∝ R e − 1 in that limit for neutrally stable modes. Neutral stability curves in the Γ – W plane show that the predicted instability of the high frequency GL modes exists only at finite and large W, and is absent in the Newtonian fluid ( W → 0 ) limit. The asymptotic analysis for the Oldroyd-B model shows that the ratio of solvent viscosity to total viscosity of the solution should be O ( R e 1 / 2 ) in order for the instability to exist at low Re, meaning the instability is absent for realistic values of solvent viscosity at low Re. However, numerical results show that the instability of the high-frequency GL modes is present at finite Re in an Oldroyd-B fluid, and increasing solvent viscosity ratio has a stabilizing effect on the instability. Similarly, the solid to fluid viscosity ratio η r also has a stabilizing effect at finite Re, but in the limit of low Re, η r ∼ O ( R e 1 / 2 ) in order for the instability to exist. Our study thus demonstrates the presence of an instability in viscoelastic plane Couette flow past a deformable wall which is primarily due to the viscoelastic nature of the fluid, and is absent in Newtonian fluids.