The effect of variable viscosity on the interfacial stability of two-layer Poiseuille flow

Abstract

In this paper the linear stability analysis of the interface between two Newtonian liquids with temperature-dependent viscosity in plane Poiseuille flow is presented. A piecewise linear temperature profile is considered. The linearized equations describing the evolution of small, two-dimensional disturbances are derived and the stability problem is formulated as an eigenvalue problem for a set of ordinary differential equations. The continuous eigenvalue problem is solved numerically by a pseudospectral method based on Chebyshev polynomial expansions. The method leads to a generalized matrix eigenvalue problem, which is solved by the QZ algorithm. Results on the onset of instability are presented in the form of stability maps for a range of thickness ratios, disturbance wave numbers, imposed temperature differences, constant-temperature viscosity ratios, thermal conductivity ratios, and Reynolds numbers. Increasing the imposed temperature difference, constant-temperature viscosity ratio, or Reynolds number can have a stabilizing or destabilizing effect, depending on the flow configuration (thickness ratio) and disturbance wavelength. Increasing the thermal conductivity ratio has a destabilizing effect on the interface for all configurations and disturbance wavelengths.