Instability Of Plane Couette Flow Research Articles

Viscoelasticity-Induced Instability in Plane Couette Flow at Very Low Reynolds Number

Elasto-inertial turbulence (EIT), a new turbulent state found in polymer solutions with viscoelastic properties, is associated with drag-reduced turbulence. However, the relationship between EIT and drag-reduced turbulence is not currently well-understood, and it is important to elucidate the mechanism of the transition to EIT. The instability of viscoelastic fluids has been studied in a canonical wall-bounded shear flow to investigate the transition process of EIT. In this study, we numerically deduced that an instability occurs in the linearly stable viscoelastic plane Couette flow for lower Reynolds numbers, at which a non-linear unstable solution exists. Under instability, the flow structure is elongated in the spanwise direction and regularly arranged in the streamwise direction, which is a characteristic structure of EIT. The regularity of the flow structure depends on the Weissenberg number, which represents the strength of elasticity; the structure becomes disordered under high Weissenberg numbers. In the energy spectrum of velocity fluctuations, a steep decay law of the structure’s scale towards a small scale is observed, and this can be recognized as a ubiquitous feature of EIT. The existence of instability in viscoelastic plane Couette flow supports the idea that the transitional path toward EIT may be mediated by subcritical instability.

An orbital instability of minimal plane Couette turbulence

While oblique stripes have been observed in both laboratory and numerical experiments, their mechanism has not yet been theoretically understood. We investigate an orbital instability of plane Couette flow to infinitesimal disturbances using Lyapunov analysis, assuming the main flow to be a tessellation by identical copies of a minimal turbulence which reduces the numerical cost. Because of the periodicity of the linearized equations in the streamwise and spanwise directions, Bloch’s theorem is applicable and an orbital instability of disturbances with scales larger than the minimal flow unit can be investigated by a calculation within the minimal flow unit. By comparing Lyapunov exponents, we find that the most unstable mode has the Bloch wavenumber parallel to the streamwise direction, which however does not correspond to oblique patterns observed in laboratory experiments and numerical simulations. Therefore the infinitesimal orbital instability of minimal plane Couette turbulence does not explain the obliqueness of the stripe pattern.

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.

Non-modal instability in plane Couette flow of a power-law fluid

In this paper, we study the linear stability of a plane Couette flow of a power-law fluid. The influence of shear-thinning effect on the stability is investigated using the classical eigenvalue analysis, the energy method and the non-modal stability theory. For the plane Couette flow, there is no stratification of viscosity. Thus, for the stability problem the stress tensor is anisotropic aligned with the strain rate perturbation. The results of the eigenvalue analysis and the energy method show that the shear-thinning effect is destabilizing. We focus on the effect of non-Newtonian viscosity on the transition from laminar flow towards turbulence in the framework of non-modal stability theory. Response to external excitations and initial conditions has been studied by examining the ε-pseudospectrum and the transient energy growth. For both Newtonian and non-Newtonian fluids, it is found that there can be a rather large transient growth even though the linear operator of the Couette flow has no unstable eigenvalue. The results show that shear-thinning significantly increases the amplitude of response to external excitations and initial conditions.

Hydromagnetic instability in plane Couette flow

We study the stability of a compressible magnetic plane Couette flow and show that compressibility profoundly alters the stability properties if the magnetic field has a component perpendicular to the direction of flow. The necessary condition of an instability can be satisfied in a wide variety of flows in laboratory and astrophysical conditions. The instability can operate even in a very strong magnetic field which entirely suppresses other magnetohydrodynamics (MHD) instabilities. The growth time of this instability can be rather short and reach approximately 10 shear time scales.

Stability of the viscous flow of a polymeric fluid past a flexible surface

The instability in plane Couette flow of a viscoelastic fluid past a deformable surface is examined using the temporal linear stability theory in the zero Reynolds number limit. The polymeric fluid is described using the Oldroyd-B model and the flexible wall is modeled as a linear viscoelastic solid surface. The analysis shows that the wall flexibility tends to reduce the decay rate of the stable discrete modes for the polymeric flow past a rigid wall, and one of the discrete modes becomes unstable when the wall deformability parameter Γ=Vη∕(GR) exceeds a certain critical value Γc. Here, V is the top-plate velocity, η is the zero shear viscosity of the polymeric fluid, G is the shear modulus of the wall, and R is the width of the fluid layer. The analysis reveals the presence of two classes of modes, the first of which becomes unstable for perturbations with wavelength comparable to the channel width (finite wavelength modes), and the second becomes unstable for perturbations with wavelength small compared to the channel width (short wave modes). The latter class of modes are found to be absent for the highly concentrated polymer solutions with β≲0.23, where β is the ratio of solvent-to-solution viscosity. We have mapped out the regions in the parameter space (W¯-H) where the finite wavelength and short wave modes are unstable, where W¯=(λG∕η), and λ is the relaxation time of the viscoelastic fluid. Fluid elasticity is found to have a stabilizing influence on the unstable mode, such that when the shortwave instability is absent for β≲0.23, the flow becomes stable for any Weissenberg number W¯>W¯max. Here, W¯max increases proportional to H for H≫1. However, when the shortwave instability is present, the instability persists for W¯≫1. The behavior of both classes of modes with respect to the parameters, like W¯, H, β, and the ratio of solid-to-fluid viscosity ηr, is examined.

Effect of wall slip on the stability of viscoelastic plane shear flow

We show that for shear flow of an upper convected Maxwell fluid with small but nonzero slip velocity, an increasing dependence of the slip velocity on the elastic normal stress in the flow direction leads to short wavelength flow instability at sufficiently high Weissenberg number (≳10). Pressure-dependent slip can also lead to instability, but only at unrealistically large Weissenberg number. If the slip velocity depends only on shear stress, then the flow is always stable. These analytical results are valid in a specific asymptotic limit, but are independent of the specific form of the model for slip. Numerical results for specific, phenomenological slip models and the Phan-Thien–Tanner bulk constitutive model show that the results are robust in the presence of nonlinear viscoelasticity. The scaling of the critical shear stress for instability with modulus and molecular weight and of the distortion period with polymer relaxation time are qualitatively consistent with experimental observations of the sharkskin instability in linear polyethylenes. The results may also have some relationship to the recent experimental observation of short wavelength instability in plane Couette flow of an entangled solution with wall slip.

Instability of plane Couette flow of two-phase liquids

The stability of plane Couette flow of a disperse liquid (gaseous or liquid suspension) is investigated. It is shown that in contrast to the flow of a pure liquid (gas), above a certain threshold particle mass density, this flow begins to become unstable to infinitesimal perturbations.

Instability of plane Couette flow of viscoelastic liquids

A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The first part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability. In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.

Thermal instability in plane couette flow heated from below

The conditions marking the onset of natural convection due to buoyant forces are investigated in the thermal entrance region of horizontal plane Couette flow. The base temperature profile produced by pure forced convection is approximated as a fourth order polynomial using the integral method. With this approximate base-temperature profile stability criteria are obtained by means of the local stability analysis, its modification, and the natural-amplification analysis. The last one takes into consideration the axial amplification rate of disturbances at the onset of thermal instability. This new concept is tested here for the first time. The consideration of axial amplification of disturbances makes the system more stable. It is shown that the results of the natural-amplification analysis agree well with the existing experimental data.

A simple model of the mutual interaction of parallel flow and cellular motion in a shear layer applied to the finite amplitude instability of plane Couette flow

The disturbing motion of plane Couette and Poiseuille flow is described using three parameters: two amplitudes corresponding to the disturbance of the parallel flow and the cellular motion, respectively, and the angle ϕ0 which defines the orientation of the vortex blobs with respect to the parallel flow. Equations of motion for these parameters are obtained using a Ritz-Galerkin method. For Reynolds numbers above a critical value sufficiently big disturbances will grow until a steady finite amplitude state is achieved. The energy of the disturbance remains finite, in spite of the highly truncated field representation using only three parameters. This is possible because of the nonlinear dependence of the field functions on ϕ0. The critical values of Reynolds number, above which finite amplitude states exist, are computed for the plane Couette flow and the Poiseuille channel flow.

Finite amplitude instability of plane Couette flow

The Orr–Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the generalized Rayleigh quotient iteration. Accurate results for large values of the parameters are obtained, and these further verify the belief that plane Couette flow is stable to infinitesimal disturbances. For finite disturbances, a formal expansion based on the method of Stuart and Watson as modified by Reynolds & Potter is used. This method shows a transition to instability for a large enough amplitude.

Instability of Three-Layer Viscous Stratified Flow

Instability of plane Couette flow of three superposed layers of fluids of different viscosity between two horizontal planes is investigated. It is found that there are two modes of disturbance dominated, respectively, by the two interfaces. The flow is found to be unstable in the zero-order approximation of wavenumber α for certain values of the depth and viscosity ratios. This is owing to a sort of resonance which prevails when a free wave at the lower interface forces a free wave at the upper interface. As is known from results previously obtained, the existence of a single surface of viscosity discontinuity will cause instability. The presence of an additional surface of discontinuity may or may not be stabilizing, depending upon the values of the depth and viscosity ratios at the additional interface.

Nonlinear Instability of Plane Couette Flow

On the basis of the Meksyn‐Stuart method, nonlinear instability of the plane Couette flow is investigated. The perturbed stream function and the mean flow field are determined by the Galerkin method. A neutral surface is obtained in a three‐dimensional space of the Reynolds number, wave‐number, and amplitude. The critical Reynolds number based on the width and the relative velocity is estimated to be about 190 000.

A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe

It is found that only a small change in either of the undisturbed velocity profiles concerned is required to change them from stable profiles to unstable profiles. The change must be such as to produce a local maximum in the magnitude of the vorticity, or in the case of the pipe, in the magnitude of the vorticity divided by the radius. The actual change in the vorticity (or vorticity/radius) need only be small, but the gradient of the vorticity (or vorticity/radius) must be finite. Viscosity will tend to damp out the distortion in the mean flow that is responsible for the instability, so that if the flow is to become turbulent, non-linear effects must become important before the distortion of the mean flow is reduced to an ineffective level. This requirement leads to the determination of critical Reynolds numbers which depend on the initial (small) distortion of the mean flow and the initial (smaller) amplitude of periodic disturbances. These critical Reynolds numbers are large.