On the existence of two-dimensional nonlinear steady states in plane Couette flow

TL;DR

This paper investigates the existence of two-dimensional nonlinear steady states in plane Couette flow, revealing that previous solutions are not reproducible and the problem remains unresolved, highlighting challenges in understanding flow stability.

Contribution

The study critically examines prior claims of steady solutions in plane Couette flow and demonstrates their non-reproducibility using homotopy methods, leaving the existence question open.

Findings

Previous solutions are not reproducible using homotopy methods.

Modified models influence the steady solutions obtained.

The existence of 2D steady states in plane Couette flow remains unresolved.

Abstract

The problem of two-dimensional steady nonlinear dynamics in plane Couette flow is revisited using homotopy from either plane Poiseuille flow or from plane Couette flow perturbed by a small symmetry-preserving identity operator. Our results show that it is not possible to obtain the nonlinear plane Couette flow solutions reported by Cherhabili and Ehrenstein [Eur. J. Mech. B/Fluids, 14, 667 (1995)] using their Poiseuille-Couette homotopy. We also demonstrate that the steady solutions obtained by Mehta and Healey [Phys. Fluids, 17, 4108 (2005)] for small symmetry-preserving perturbations are influenced by an artefact of the modified system of equations used in their paper. However, using a modified version of their model does not help to find plane Couette flow solution in the limit of vanishing symmetry-preserving perturbations either. The issue of the existence of two-dimensional nonlinear steady states in plane Couette flow remains unsettled.