Long-wavelength instabilities in three-layer flow down an incline

Abstract

This paper reports a long-wavelength instability which has not previously been identified for three-layer free-surface flow down an inclined plane. The instability is identified in the zeroth-order asymptotic solution in wave number, indicating that neither inertial nor finite wavelength effects are necessary to induce instabilities in three-layer systems. Various neutral stability boundaries are presented which demonstrate the effect of viscosity stratification, density stratification, and layer thicknesses. It is found that destabilization occurs in cases where the middle-layer viscosity (for equal densities in each layer) or density (for equal viscosities in each layer) is smaller than those of the adjacent layers. The regions of instability afford a smooth transition between neutrally stable regions of the parameter space where the in-phase and out-of-phase characteristics of the interfaces differ.