Floquet stability analysis of a wall-bounded oscillatory flow of a viscoelastic fluid

Abstract

Oscillatory fluid flows play an important role in fluid mechanics for their long history and numerous applications. In this work we will start off from Stokes’ second problem of the boundary layer adjacent to an oscillatory wall in order to study the wall-bounded zero-mean oscillatory flow of a viscoelastic fluid placed between two parallel plates oscillating synchronously. From related experiments on the oscillatory flow of a viscoelastic solution in a vertical tube we know that the rectilinear flow at small forcing amplitudes gives rise to a secondary flow with toroidal vortices at larger amplitudes. Our purpose is to provide a theoretical understanding of this instability in a simpler setup. We analytically solve the governing equations of the periodic base flow, and carry out a Floquet linear stability analysis of the stress and velocity fields. We apply the Galerkin spectral method to numerically solve the corresponding generalized eigenvalue problem, and provide instability thresholds in forcing amplitude for both resonant and non-resonant forcing frequencies