Instability of plane Couette flow under competing effects of viscoelasticity and vertical throughflow

Abstract

The linear instability of plane laminar flow of a Navier–Stokes–Voigt fluid between parallel plates in relative motion with a uniform vertical throughflow is analysed. The validity of Squire’s theorem is established, thereby justifying restriction to two-dimensional spanwise-independent perturbations. The resulting continuous eigenvalue problem is discretized using the Chebyshev collocation method, yielding a generalized matrix eigenvalue problem that is solved numerically to determine the stability thresholds. The critical Reynolds number marking the onset of instability is computed for different values of the throughflow-dependent Reynolds number and the Kelvin–Voigt viscoelastic parameter. A key finding of the analysis is the appearance of bimodal neutral stability curves within limited parameter ranges, accompanied by branch-to-branch transitions in which the dominant instability changes identity—a feature absent when either mechanism acts in isolation. The results further reveal that throughflow may be stabilizing, destabilizing, or even produce mixed behavior, depending strongly on the strength of viscoelastic effects. Streamline visualizations at criticality uncover the formation of complex, spatially structured flow fields, highlighting the intricate dynamics driven by the interplay between viscoelasticity and throughflow.