Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

Martin Lellep,Moritz Linkmann,Alexander Morozov,Bruno Eckhardt

doi:10.1017/jfm.2021.840

Abstract

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

Highlights

Stability of parallel shear flows of dilute polymer solutions is an outstanding problem of mechanics of complex fluids
We study unidirectional laminar profiles v = {U(x2), 0, 0}, where U is chosen to resemble the shape and symmetry of either plane Poiseuille flow or plane Couette flow, while satisfying the boundary conditions (2.2): U(x2) =
The no-slip plane Couette flow (pCF) and plane Poiseuille flow (pPF) of Oldroyd-B fluids are linearly stable for a wide range of β and Wi (Gorodtsov & Leonov 1967; Wilson et al 1999)

Summary

Introduction

Stability of parallel shear flows of dilute polymer solutions is an outstanding problem of mechanics of complex fluids. The studies from the first class seek to approximate microscopically sound boundary conditions and address the question of linear stability of experimentally realisable flows Such an approach has indicated, for instance, that the classical two-dimensional Tollmien–Schlichting transition in plane channel flow of Newtonian fluids is significantly suppressed by the presence of wall slip (Gersting 1974; Lauga & Cossu 2005; Min & Kim 2005), the non-normal energy growth mechanism was shown to be less affected (Lauga & Cossu 2005). Waleffe (1997) used free-slip boundary conditions to construct exact coherent states in parallel shear flows of Newtonian fluids that revolutionised our understanding of the transition to turbulence in those flows (Eckhardt et al 2007; Tuckerman, Chantry & Barkley 2020; Graham & Floryan 2021). We observe that the main features of no-slip flows are preserved under the free-slip conditions, and we discuss how these instabilities can be employed to gain insight into the nature of elastic turbulence in parallel geometries

Problem set-up

Results

Conclusions