Stability of stratified flows. II - Viscous two-layer flow

Flow stability to small perturbations is examined for different velocity profiles arising in a two-layer viscous flow in the presence of density and viscosity stratification at various Reynolds numbers and as a function of the direction of perturbed motion. A system of equations of motion for such flows is solved and transformed from differential to algebraic form, so that an analytical expression can be formulated for the neutral stability curve. Neutral stability criteria are obtained for various flow velocity profiles, Reynolds numbers, viscosity- and density-stratification conditions, and as a function of the perturbation direction. In particular, it is shown that a Couette flow in the presence of stable (unstable) density stratification is absolutely stable (unstable) to small perturbations for any Reynolds number, wave number, and perturbation direction. Stratified flow stability is found to increase with increasing upper-layer viscosity or decreasing Reynolds number in the cases of both stable and unstable density stratification.