Linear stability of plane Poiseuille flow of a Bingham fluid in a channel with the presence of wall slip

Abstract

In the present paper, the effects of a channel symmetric wall slip are investigated on the stability of plane Poiseuille flow of a Bingham fluid, based on the modal and non-modal linear stability analysis approaches. Both streamwise and spanwise slip conditions are considered and their results are compared with those of the no-slip condition, in terms of the flow stability. The results are obtained for different dimensionless groups that govern the stability picture, for example the Bingham number, the Reynolds number and the streamwise/spanwise slip numbers. The linearized perturbation equations are obtained and the corresponding eigenvalue problem is solved based on the Chebyshev collocation method at the Gauss-Lobatto grid points, using the QZ algorithm. In the modal analysis, the focus is on the changes made by the channel wall slip on the eigenvalue spectra. The flow is shown to be linearly stable for no-slip and streamwise slip conditions. However, for the spanwise slip condition, an unstable eigenvalue is found for large values of the spanwise slip number. The non-modal stability analysis is based on the transient energy growth of the perturbations, which is calculated using the eigenfunctions of the eigenvalue problem representing the amplitude of the perturbations. Increasing the streamwise slip decreases the maximum energy growth, eventually leading to a flow for which the transient energy only decays in time. Thus, two distinct flows showing the energy growth and energy decay regimes can be identified, for a range of Bingham, Reynolds and streamwise slip numbers.