Linear stability diagrams for the shear flow of an Oldroyd-B fluid with slip along the fixed wall

Abstract

We consider the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. Slip is allowed by means of a generic slip equation predicting that the shear stress is a non-monotonic function of the velocity at the wall. The complete one-dimensional stability analysis to one-dimensional disturbances is carried out and the corresponding neutral stability diagrams are constructed. Asymptotic results for large values of the elasticity number and finite element calculations are also presented. The instability regimes are within or coincide with the negative-slope regime of the slip equation. The numerical calculations agree with the linear stability results when the size of the initial perturbation is small. Large perturbations may destabilize a linearly stable steady state, leading to a periodic solution. The period and the amplitude of the periodic solutions increase with elasticity.