Inertial effect on the stability of viscoelastic cone-and-plate flow

Abstract

The stability of axially symmetric cone-and-plate flow of an Oldroyd-B fluid at non-zero Reynolds number is analysed. We show that stability is controlled by two parameters: [Escr ]₁≡DeWe and [Escr ]₂≡Re/We, where De, We, and Re are the Deborah, Weissenberg and Reynolds numbers respectively. The linear stability problem is solved by a perturbation method for [Escr ]₂ small and by a Galerkin method when [Escr ]₂=O(1). Our results show that for all values of the retardation parameter β and for all values of [Escr ]₂ considered the base viscometric flow is stable if [Escr ]₁ is sufficiently small. As [Escr ]₁ increases past a critical value the flow becomes unstable as a pair of complex-conjugate eigenvalues crosses the imaginary axis into the right half-plane. The critical value of [Escr ]₁ decreases as [Escr ]₂ increases indicating that increasing inertia destabilizes the flow. For the range of values considered the critical wavenumber k_c first decreases and then increases as [Escr ]₂ increases. The wave speed on the other hand decreases monotonically with [Escr ]₂. The critical mode at the onset of instability corresponds to a travelling wave propagating inward towards the apex of the cone with infinitely many logarithmically spaced toroidal roll cells.