Stability analysis of two-layer immiscible viscous fluids in an inclined closed tube

The present investigation is concerned with the role of bottom slope in the stability properties of two-dimensional disturbances in sheared two-layer immiscible fluids with stepwise densities. A linear analysis of normal mode instability of the interface in an inclined closed tube is carried out using full Navier-Stokes equation using Chebyshev-tau method. It has been observed that Reynolds number and the tilt angle both have destabilizing effect on the interface eigenvalue problem of Orr-Sommerfeld equation. As a special example, Stokes flow is proven to be stable for any angles. A weakly nonlinear model equation, which is Kuramoto-Sivashinsky type, for the interface is given using multi-scale method, and the long wave dispersion relation gives a sufficient condition for triggering the instability.