The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number

Abstract

A zero Reynolds number approximation to the Orr–Sommerfeld equation is used to assess the effects that viscosity stratification has on the stability of a very viscous flow on an incline when surface tension is negligible. Results indicate that for a two-layer system with uniform density, the flow is always unstable when the viscosity of the upper layer is greater than that of the lower layer, regardless of the thickness of the upper layer. The wavenumber of the fastest growing mode is on the order of the inverse of the thickness of the upper layer, implying that the instability is manifested in waves having finite wavelength, whereas previous research on this topic has focused on a long wavelength approximation. It is further shown that neutral stability is independent of the angle of inclination of the underlying slope, although the growth rate of any instability is not. The results suggest that the transverse surficial ridges, which commonly occur on the surfaces of rock glacier forms, may be the product of a flow instability arising from the differing viscosities of the layers that comprise such features.