Previous Linear Stability Analyses Research Articles

Investigation of Convective Heat Transfer and Stability on a Rotating Disk: A Novel Experimental Method and Thermal Modeling

Experimental and numerical investigations are conducted on a rotating disk from the perspective of convective heat transfer to understand the effect of heating on the stability of flow. A non-invasive approach with a thermal camera is employed to determine local Nusselt numbers for different rotational rates and perturbation parameters, i.e., the strength of the heat transfer. A novel transient temperature data extraction over the disk radius and an evaluation method are developed and applied for the first time for the air on a rotating disk. The evaluation method utilizes the lumped capacitance approach with a constant heat flux input. Nusselt number distributions from this experimental study show that there is a good agreement with the previous experimental correlations and linear stability analysis on the subject. A significant result of this approach is that by using the experimental setup and developed approach, it is possible to qualitatively show that instability in the flow starts earlier, i.e., an earlier departure from laminar behavior is observed at lower rotational Reynolds numbers with an increasing perturbation parameter, which is due to the strength of heating. Two experimental setups are modeled and simulated using a validated in-house Python code, featuring a three-dimensional thermal model of the disk. The thermal code was developed for the rotating disks and brake disks with a simplified geometry. Experimentally evaluated heat transfer coefficients are implemented and used as convective boundary conditions in the thermal code. Radial temperature distributions are compared with the experimental data, and there is good agreement between the experiment and the model. The model was used to evaluate the effect of radial conduction, which is neglected when using the lumped capacitance approach to determine heat transfer coefficients. It was observed that the radial conduction has a slight effect. The methodology and approach used in this experimental study, combined with the numerical model, can be used for further investigations on the subject.

Nonlinear Evolution of Instabilities in a Laminar Separation Bubble at Hypersonic Speeds

The development of both convective stationary perturbation and global instabilities in the vicinity of a laminar separation bubble above an axisymmetric compression corner in hypersonic flow was investigated using numerical simulations. The flow configuration of interest corresponded to the cone–cylinder–flare model used in experimental measurements in the Boeing/U.S. Air Force Office of Scientific Research Mach-6 Quiet Tunnel (BAMQT) at Purdue University. For a flare angle of 10 deg and unit Reynolds number of 11.5×106 m−1, their surface flow visualizations identified the presence of streamwise elongated thermal streaks near the reattachment location that had a dominant azimuthal wave number m=36. Previous linear stability analyses predicted that the amplification characteristics of small-amplitude, unsteady, and convective instabilities within this flow were consistent with the surface pressure fluctuations measured in the experiment. However, their accompanying investigation of global instabilities showed that the separation bubble was weakly unstable at the 10 deg flare angle, with the most unstable global mode corresponding to a stationary disturbance with m≈5–6 (i.e., well below the measured wave number of m=36. Besides characterizing the global instability for selected flare angles, the present numerical simulations quantify the details of the stationary equilibrium state associated with supercritical bifurcation resulting from the nonlinear saturation of the unstable global mode. Although velocity perturbations associated with the saturated global mode are dominated by the fundamental spanwise wavelength associated with the linear global instability, the surface heat flux downstream of the reattachment is dominated by m=36, in agreement with the experimental measurements.

Numerical study of instabilities of particle-laden fronts in continuously stratified environments

Numerical simulations are conducted to study instabilities and the associated convective motion of particle-laden layers settling in continuously stratified environments. We show that when the background density stratification is insignificant relative to the bulk excessive density of the particle-laden layer, the unstable motions of the particle-laden interface are mainly driven by Rayleigh–Taylor instability but become double-diffusive convection when the background stratification is relatively significant. Our results agree with theoretical prediction based on linear stability analysis. However, in the Rayleigh–Taylor instability regime, the motion of particle-laden plumes can be further suppressed by the background density stratification while the plumes reach the height of neutral buoyancy. This leads to the second stage of flow development, in which secondary instability occurs at the plumes' tip in the form of double-diffusive convection. Due to the change in the background density gradient within the plumes' head, the occurrence of secondary instability is accompanied by a shift of the dominant mode, which is particularly significant in cases with a high background Prandtl number (i.e., salinity-induced stratification). The theoretical argument on the mode shift is based on previous linear stability analysis for the two-layer structured background density gradient provided. The ratio between the particles' settling velocity and velocity scaling for the developed local density gradient at the plumes' tip then allows us to distinguish and predict whether the final convective motion is driven mainly by double-diffusive or settling-driven buoyancy-dominant convection.

Linear to turbulent Görtler instability transition

We present results from a highly resolved large-eddy simulation of a freely developing Blasius profile over a concave boundary in a large spanwise domain. Due to the large initial Reynolds and Görtler numbers (Reθ,0 = 1175, Gθ,0 = 75), we observe the onset of two dominant wavelengths: the first dominating in the linear/transition region, λ1, and the second dominating in the turbulent region, λ2. Extending previous linear stability analysis (LSA) to higher Görtler numbers and non-dimensional wavenumbers, both dominant wavelengths of the Görtler instability correspond to predictions of LSA, the latter comparable to laminar theory by replacing the kinematic viscosity with the turbulent viscosity in the definition of the Görtler number. The predicted spatial modes compare well with the computed profiles for both λ1 and λ2. The skin friction coefficient Cf is found heterogeneous in the spanwise direction according to the emerging wavelengths λ1 and λ2 of the Görtler instability. We report a smooth increase in Cf from the theoretical predictions of a laminar boundary layer to those for a turbulent boundary layer over a flat plate. The values only slightly overshoot these predictions in the domain of existence of the second dominant wavelength λ2, very different from that reported at lower Reynolds numbers.

Stabilization of a straight longitudinal dune under bimodal wind with large directional variation.

It has been observed that the direction in which a sand dune extends its crest line depends on seasonal variation of wind direction; when the variation is small, the crest line develops more or less perpendicularly to the mean wind direction to form a transverse dune with some undulation. In the case of bimodal wind with a large relative angle, however, the dune extends its crest along the mean wind direction and evolves into an almost straight longitudinal dune. Motivated by these observations, we investigate the dynamical stability of isolated dunes using the crest line model, where the dune dynamics is represented by its crest line motion. First, we extend the previous linear stability analysis under the unidirectional wind to the case with nonzero slant angle between the wind direction and the normal direction of the crest line, and show that the stability diagram does not depend on the slant angle. Second, we examine how the linear stability is affected by the seasonal changes of wind direction in the case of bimodal wind with equal strength and duration. For the transverse dune, we find that the stability is virtually the same with that for the unidirectional wind as long as the dune evolution during a season is small. On the other hand, in the case of the longitudinal dune, the dispersions of the growth rates for the perturbation are drastically different from those of the unidirectional wind, and we find that the largest growth rate is always located at k=0. This is because the growth of the perturbation with k≠0 is canceled by the alternating wind from opposite sides of the crest line even though it grows during each duration period of the bimodal wind. For a realistic parameter set, the system is in the wavy unstable regime of the stability diagram for the unidirectional wind, thus the straight transverse dune is unstable to develop undulation and eventually evolves into a string of barchans when the seasonal variation of wind direction is small, but the straight longitudinal dune is stabilized under the large variation of bimodal wind direction. We also perform numerical simulations on the crest line model, and find that the results are consistent with our linear analysis and the previous reports that show that the longitudinal dunes tend to have a straight ridge elongating over time.

Characterization of Viscous Fingering and Channeling for the Assessment of Polymer-Based Heavy Oil Displacements

The ability to determine a predictive stability criterion is of great practical importance for designing stable polymer-based displacements. Where one usually resorts to a limited number of core-scale experiments or coarse-scale reservoir simulations, the first ones are potentially impacted by length scale issues, while the second ones possibly smooth out sharp displacing fronts and physical instability due to numerical diffusion. This paper proposes a new hydrodynamical stability criterion based on the previous linear stability analysis results. This criterion is tested for 2D polymer oil displacement by performing high viscosity contrasts, high-resolution numerical experiments at pilot scale. We investigate mesh resolution issues and several perturbation ideas. Different factors are considered such as mobility ratios, polymer adsorption and degradation and heterogeneities. The analysis is based on a combination of reservoir simulation and image processing techniques. We show the development of viscous fingering in homogeneous porous media is driven by the shock mobility ratio defined as the ratio of the total fluids upstream mobility over the total fluids downstream mobility. This stability criterion proves to predict both the polymer upstream and polymer-free downstream saturation fronts stability, typical of a polymer displacement, whether polymer adsorbs on the rock or degradates, or not. The observed fingers dynamical behavior is in line with previous works addressing single-phase miscible flow or immiscible oil displacement in porous media: fingers transversally merge while growing in the flow direction. Time evolution of fingers spreading and number is linear. Investigation on porous media of variable heterogeneity distributions shows how viscous fingering couples with heterogeneity and leads to even more marked, distorted and unstable flow patterns. In those cases, flow patterns are not solely driven by the porous medium heterogeneity. The more unstable the flow is, the more sensitive it is to heterogeneity. In-depth fingers analysis shows a very specific time evolution behavior, quite different from viscous fingering in homogeneous media. Such a flow pattern is related to production data such as water and polymer breakthrough times and/or oil recovery profiles as a function of time, which can be used in turn to interpret displacement stability and porous medium heterogeneity features.

Modulation of the turbulence regeneration cycle by inertial particles in planar Couette flow

Two-way coupled direct numerical simulations are used to investigate the effects of inertial particles on self-sustained, turbulent coherent structures (i.e. the so-called regeneration cycle) in plane Couette flow at low Reynolds number just above the onset of transition. Tests show two limiting behaviours with increasing particle inertia, similar to the results from previous linear stability analyses: low-inertia particles trigger the laminar-to-turbulent instability whereas high-inertia particles tend to stabilize turbulence due to the extra dissipation induced by particle–fluid coupling. Furthermore, it is found that the streamwise coupling between phases is the dominant factor in damping of the turbulence and is highly related to the spatial distribution of the particles. The presence of particles in different turbulent coherent structures (large-scale vortices or large-scale streaks) determines the turbulent kinetic energy of particulate phase, which is related to the particle response time scaled by the turnover time of large-scale vortices. By quantitatively investigating the periodic character of the whole regeneration cycle and the phase difference between linked sub-steps, we show that the presence of inertial particles does not alter the periodic nature of the cycle or the relative length of each of the sub-steps. Instead, high-inertia particles greatly weaken the large-scale vortices as well as the streamwise vorticity stretching and lift-up effects, thereby suppressing the fluctuating amplitude of the large-scale streaks. The primary influence of low-inertia particles, however, is to strengthen the large-scale vortices, which fosters the cycle and ultimately reduces the critical Reynolds number.

Numerical investigation of the saturation process in an incompressible cavity flow

A numerical study of the saturation process inside a rectangular open cavity is presented. Previous experiments and linear stability analysis of the problem completely described the flow in its onset, as well as in a saturated regime, characterized by three-dimensional centrifugal modes. The morphology of the modes found in the experiments matched the ones predicted by linear analysis, but with a shift in frequencies for the oscillating modes. A three-dimensional incompressible direct numerical simulation (DNS) is employed for a detailed investigation of the saturation process inside a cavity with dimensions similar to the one used in the experiments, to further explain the behaviour of these modes. In this work, periodic boundary conditions are first imposed to better understand the effect of the saturation process far from the walls. Then, the effects of spanwise solid wall boundary conditions are investigated with a DNS reproducing the full dynamics of the experiments. The main flow structures are identified using the dynamic mode decomposition technique and compared with previous experimental and linear stability analysis results. The main reason for the aforementioned shift in frequency is explained in this paper, as it is a function of the velocity of the main recirculating vortex.

E × B flow shear drive of the linear low-n modes of EHO in the QH-mode regime

A new model for the edge harmonic oscillations (EHOs) in the quiescent H-mode regime has been developed, which successfully reproduces the recent observations in the DIII-D tokamak. In particular, at high E × B flow shear only a few low-n kink modes remain unstable at the plasma edge, consistent with the EHO behavior, while at low E × B flow shear, the unstable mode spectrum is significantly broadened, consistent with the low-n broadband electromagnetic turbulence behavior. The model is based on a new mechanism for destabilizing low-n kink/peeling modes by the E × B flow shear, which underlies the EHOs, separately from the previously found Kelvin–Helmholtz drive. We find that the differential advection of mode vorticity by sheared E × B flows modifies the 2D pattern of mode electrostatic potential perpendicular to the magnetic field lines, which in turn causes a radial expansion of the mode structure, an increase of field line bending away from the mode rational surface, and a reduction of inertial stabilization. This enhances the kink drive as the parallel wavenumber increases significantly away from the rational surface at the plasma edge where the magnetic shear is also strong. This destabilization is also shown to be independent of the sign of the flow shear, as observed experimentally, and has not been taken into account in previous pedestal linear stability analyses. Verification of the veracity of this EHO mechanism will require analysis of the nonlinear evolution of low-n kink/peeling modes so destabilized in the linear regime.

Linear and weakly nonlinear analyses of cylindrical Couette flow with axial and radial flows

Extending previous linear stability analyses of the instabilities developing in permeable Taylor–Couette–Poiseuille flows where axial and radial throughflows are superimposed on the usual Taylor–Couette flow, we further examine the linear behaviour and expand the analysis to consider the weakly nonlinear behaviour of convective-type instabilities by means of the derivation of the fifth-order amplitude equation together with direct numerical simulations. Special attention is paid to the influence of the radius ratio $\unicode[STIX]{x1D702}=r_{in}/r_{out}$, and particularly to wide gaps (small $\unicode[STIX]{x1D702}$) and how they magnify the effects of the radial flow. The instabilities take the form of pairs of counter-rotating toroidal vortices superseded by helical ones as the axial flow is increased. Increasing the radial inflow draws these vortices near the inner cylinder, where they shrink relative to the annular gap, when this gap is wide. Strong axial and radial flows in a narrow annular gap lead to a very large azimuthal wavenumber with steeply sloped helical vortices. Strong radial outflow in a wide annular gap results in very large helical vortices. The analytical and numerical saturated vortices match quite well. In addition, radial inflows or outflows can turn the usually supercritical bifurcation from laminar to vortical flow into a subcritical one. The radial flow above which this change occurs decreases as the radius ratio $\unicode[STIX]{x1D702}$ decreases. A practical motivation for this weakly nonlinear analysis is found in modelling dynamic filtration devices, which rely on vortical instabilities to reduce the processes of accumulation on their membranes.

Discretized modeling of beads-on-a-string morphology from electrically driven, conducting, and viscoelastic polymer jets

In this paper, we provide a theoretical investigation of axisymmetric instabilities observed during electrospinning, which lead to beads-on-a-string morphology. We used a discretized method to model the instability phenomena observed in the jet. We considered the fluid to be analogous to a bead-spring model. The motion of these beads is governed by the electrical, viscoelastic, surface tension, aerodynamic drag, and gravitational forces. The bead is perturbed at the nozzle, and the growth of the instability is observed over time, and along the length of the jet. We considered both lower electrical conducting polyisobutylene (PIB)-based Boger fluids and highly electrical conducting, polyethylene oxide (PEO)/water systems. In PIB fluids, the onset of the axisymmetric instability is predominantly based on the capillary mode, and the growth rate of the instability is decreased with the viscoelasticity of the jet. However, in the PEO/water system, the instability is electrically driven, and a significant increase in the growth rate of the instability is observed with the increase in the voltage. Our predictions from the discretized model are in good agreement with the previous linear stability analysis and experimental results. Our results also revealed the non-stationary behavior of the disturbance, where the amplitude of the perturbation is observed to be oscillating. Furthermore, we showed that the discretized model is also used to observe the non-axisymmetric behavior of the jet, which can be further used to study the bending instability in electrospinning.

Non-Boussinesq stability analysis of natural-convection gaseous flow on inclined hot plates

The buoyancy-driven boundary-layer flow that develops over a semi-infinite inclined hot plate is known to become unstable at a finite distance from the leading edge, characterized by a critical value of the Grashof number Gr based on the local boundary-layer thickness. The nature of the resulting instability depends on the inclination angle ϕ, measured from the vertical direction. For values of ϕ below a critical value ϕc the instability is characterized by the appearance of spanwise traveling waves, whereas for ϕ>ϕc the bifurcated flow displays Görtler-like streamwise vortices. The Boussinesq approximation, employed in previous linear stability analyses, ceases to be valid for gaseous flow when the wall-to-ambient temperature ratio Θw is not close to unity. The corresponding non-Boussinesq analysis is presented here, accounting also for the variation with temperature of the different transport properties. A temporal stability analysis including nonparallel effects of the base flow is used to determine curves of neutral stability, which are then employed to delineate the dependences of the critical Grashof number and of its associated wave length on the inclination angle ϕ and on the temperature ratio Θw for the two instability modes, giving quantitative information of interest for configurations with Θw-1∼1. The analysis provides in particular the predicted dependence of the crossover inclination angle ϕc on Θw, indicating that for gaseous flow with Θw-1∼1 spanwise traveling waves are predominant over a range of inclination angles 0⩽ϕ⩽ϕc that is significantly wider than that predicted in the Boussinesq approximation.

The Standing Accretion Shock Instability: Enhanced Growth in Rotating Progenitors

Abstract We investigate the effect of progenitor rotation on the standing accretion shock instability (SASI) using two- and three-dimensional hydrodynamic simulations. We find that the growth rate of the SASI is a near-linearly increasing function of the specific angular momentum in the accreting gas. Both the growth rate and the angular frequency in the two-dimensional model with cylindrical geometry agree well with previous linear stability analyses. When excited by very small random perturbations, a one-armed spiral mode dominates the small rotation rates predicted by current stellar evolution models, while progressively higher-order modes are seen as the specific angular momentum increases.

Numerical simulation of the Rayleigh–Taylor instability of a miscible slice in a porous medium

Convective instability in miscible slices is important in understanding the contaminant spreading in groundwater and sample dispersion in a chromatographic column. In the present study, the temporal evolution of the Rayleigh–Taylor instability of a miscible high-density slice in a porous medium is analyzed theoretically using nonlinear numerical simulations. Nonlinear governing equations are derived and solved with the Fourier spectral method. To connect the previous linear stability analysis and the present nonlinear simulation, the most unstable disturbance which was identified in the linear analysis is employed as an initial condition for the nonlinear study. In contrast to the fingering between two semi-infinite regions, the nonlinear fingering of a finite slice is influenced by the depth of the high-density region. The present nonlinear analysis shows that there exists a critical depth below which the system is linearly unstable, but nonlinear phenomena cannot be expected. The nonlinear simulation results show that nonlinear competition yields a series of cells of slightly different widths and amplitudes. Also, it is found that the upper stable region hinders the development of instability motions and stabilizes the system.

Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

Baroclinic instability in the Venus atmosphere simulated by GCM

Baroclinic instability in the super-rotation of Venus is investigated by a newly developed atmospheric general circulation model. First, we adopt an idealized super-rotation, i.e., solid-body rotating flow in a weakly stratified layer at cloud level, as an initial basic state in a nominal case. With the evolution of time, baroclinic instability occurs in a weakly stratified layer with large vertical shear of the basic zonal flow. Horizontal wind associated with the baroclinic instability modes is of a few m s−1. The initial structure of the unstable modes is similar to those obtained in previous linear stability analyses. However, it is modified by nonlinear interactions in the later stage, reaching a quasi-steady state. Meridional transport of momentum and heat by these unstable modes accelerates the super-rotation by ~ 0.05 m s−1 day−1 at midlatitudes. Furthermore, the dependence of baroclinic instability on the basic state, i.e., the meridional profiles of zonal flow and the vertical profiles of static stability, are subsequently investigated. For the super-rotation with midlatitude jets at cloud level, the modes are modified from baroclinic to barotropic in the later stage. Typically, their horizontal wind is of O(10) m s−1. Their amplitude is maintained by energy conversion from zonal-mean available potential energy associated with the baroclinic basic state. In the case where static stability is smaller than that in the nominal case, the baroclinic modes transfer angular momentum from midlatitude to the equator near a 70 km level and accelerate the super-rotation by more than 10 m s−1 in the equatorial region.

3-D PTV measurement of Marangoni convection in liquid bridge in space experiment

Microgravity experiments have been conducted on the International Space Station in order to clarify the transition processes of the Marangoni convection in liquid bridges of high Prandtl number fluid. The use of microgravity allows us to generate large liquid bridges, 30 mm in diameter and up to 60 mm in length. Three-dimensional particle tracking velocimetry (3-D PTV) is used to reveal complex flow patterns that appear after the transition of the flow field to oscillatory states. It is found that a standing-wave oscillation having an azimuthal mode number equal to one appears in the long liquid bridges. For the liquid bridge 45 mm in length, the oscillation of the flow field is observed in a meridional plane of the liquid bridge, and the flow field exhibits the presence of multiple vortical structures traveling from the heated disk toward the cooled disk. Such flow behaviors are shown to be associated with the propagation of surface temperature fluctuations visualized with an IR camera. These results indicate that the oscillation of the flow and temperature field is due to the propagation of the hydrothermal waves. Their characteristics are discussed in comparison with some previous results with long liquid bridges. It is shown that the axial wavelength of the hydrothermal wave observed presently is comparable to the length of the liquid bridge and that this result disagrees with the previous linear stability analysis for an infinitely long liquid bridge.

SHORELINE SAND WAVES AND BEACH NOURISHMENTS

The effects of the feedback between the changing coastal morphology and the wavefield on the generation and propagation of large scale (O(1-10 km)) shoreline sand waves is examined with a quasi-2D morphodynamic model. Traditional shoreline change models do not include this feedback and are only able to describe diffusion of shoreline sand waves and furthermore they are unable to describe migration. It is found with the present model that if there is a dominant littoral drift, the feedback causes downdrift migration of coastline features no matter if they grow or decay. Consistently with previous studies, simulations show that a rectilinear coastline becomes unstable and sand waves tend to grow spontaneously from random perturbations, if the wave incidence angle is larger then about 42o (θc) at the depth of closure (high angle wave instability). The initial wavelengths at which the sand waves develop are 2-3 km and this is similar to previous linear stability analysis. The implications of high angle wave instability for beach nourishments are investigated. The nourished shoreline retreats initially due to cross-shore transport because the nourished profile is steeper than the equilibrium profile. When a dominant littoral drift is present, the nourishment also migrates downdrift. If the wave angle at the depth of closure is below θc the alongshore transport contributes to the diffusion of the nourishment. However, if the angle is above θc (constant high angle wave conditions) the diffusion is reversed and the nourishment can trigger the formation of a shoreline sand wave train. Numerical experiments changing the proportion of 'high angle waves' and 'low angle waves' in the wave climate show that relatively small proportions of low angle waves slow down the growth of sand waves. These simulations with more realistic wave climates show shoreline sand waves that migrate downdrift maintaining more or less the same amplitude for years.

Simulation of transient Rayleigh–Bénard–Marangoni convection induced by evaporation

Numerical simulation of thermal convection induced by solvent evaporation in an initially isothermal fluid is considered. Both thermocapillarity and buoyancy driving forces are taken into account, and a criterium based on the Peclet number is used to analyze the stability of this transient problem. Critical Marangoni and Rayleigh numbers are obtained for a large range of Biot and Prandtl numbers. Results of the non-linear simulations are compared with a previous linear transient stability analysis based on a non-normal approach and with visualizations performed during polyisobutylene (PIB)/toluene solutions drying experiments. A scaling analysis is developed for the Marangoni problem and correlations are derived to predict the order of magnitude of temperature and velocity as a function of Bi, Ma and Pr numbers.

Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers

The finite amplitude stability of a plane Couette flow over a deformable solid medium is analyzed with emphasis on the class of high Reynolds number (Re) modes, referred to as the wall modes, for which the viscous stresses are confined to a thin layer adjacent to the fluid-solid interface with thickness O(Re−1/3) times the channel width in the limit Re⪢1. Here, the Reynolds number is defined in terms of the top plate velocity V and the channel width R. Previous linear stability analyses have shown that the wall modes are unstable for Newtonian flow past a linear viscoelastic solid. In the present study, the analysis is extended to examine the weakly nonlinear stability of these unstable wall modes in order to determine the nature of bifurcation of the transition point to finite amplitude states. To account for the finite strain deformations, the flexible solid medium is described by a neo-Hookean elastic model which is a generalization of the commonly used linear constitutive model. The linear stability analysis provides the critical shear rate Γc and the critical wavenumber in the axial direction αc, where the dimensionless shear rate is defined as Γ=ρV2/G, where ρ is the fluid density and G is the shear modulus of the elastic solid. The critical parameter Γc for the neo-Hookean solid is found to be close to Γc for the linear elastic solid analyzed in the previous studies. The first Landau constant s(1), which is the finite amplitude correction to the linear growth rate, is evaluated in the weakly nonlinear stability analysis using both the numerical technique and the high Re asymptotic analysis. The real part of the Landau constant, sr(1), is negative for the wall mode instability in the limit Re⪢1 for a wide range of dimensionless solid thickness H, indicating that there is a supercritical bifurcation of the wall mode instability. The amplitude of the supercritically bifurcated equilibrium state is derived in the vicinity of the critical point. The equilibrium amplitude, in the form A1e2/(Γ−Γc), is found to scale as Re−1/3 in the limit Re⪢1 and is proportional to H2.3 for H⪢1 in the same limit.