Linear hydrodynamical stability of two-layer inclined flow with viscosity and density stratifications

Abstract

The linear hydrodynamical stability of two superimposed fluids of different densities and viscosities flowing down an inclined plane at the limit of zero Reynolds number is studied. There are two modes of disturbances at zero Reynolds number, one is the free surface mode and the other is the interfacial mode. The free surface mode is shown to be always stable. The stability of the flow system is governed by the interfacial mode, and stability properties depend on the values of the ratios of densities, viscosities, and depths of the two fluid layers. Asymptotic properties of stability for the limiting cases of long waves and small depth ratio are examined and the corresponding neutral stability curves for the governing interfacial mode are provided. Growth rate curves against wavenumber for various sets of parameter values are presented. The authors conclude with some remarks on how to obtain a more stable two-layer fluid system.