Onset of Viscous Dissipation Instability in Plane Couette Flow with Temperature-Dependent Viscosity

Abstract

The conditions for the onset of dissipation thermal instability with temperature-dependent viscosity in the plane Couette flow of a Newtonian fluid are analyzed. The studied system consists of a horizontal fluid layer confined between an adiabatic (fixed) lower wall and an isothermal (moving) upper wall. Both the exponential and the linear fluidity models are considered in order to account for the thermodependency of the fluid’s viscosity. The linear stability analysis of the base solution with respect to arbitrarily oriented normal modes is carried out numerically by employing a shooting method. The most unstable disturbances are proven to be stationary longitudinal rolls, and their stability is governed by three dimensionless parameters: the viscous dissipation Rayleigh number, Prandtl number and a parameter that represents the variability of the viscosity with temperature. It is shown that the effect of the variation of the viscosity is to promote the stability of the base flow. As expected, the two viscosity models’ results diverge as the variability of the viscosity increases, and the exponential model is found to be more stable than the linear fluidity one. By considering the thermophysical properties of real fluids, it is shown that viscous dissipation thermal instability precedes hydrodynamic instability. An energy budget analysis is proposed to better understand both the stabilization effect of the thermal variability of the viscosity and differences with viscous dissipation hydrodynamic instability.