Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow

Abstract

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.