The linear stability of channel flow of fluid with temperature-dependent viscosity

The classical fourth-order Orr-Sommerfeld problem which arises from the study of the linear stability of channel flow of a viscous fluid is generalized to include the effects of a temperature-dependent fluid viscosity and heating of the channel walls. The resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosity models. It is found that temperature effects can have a significant influence on the stability of the flow. For all the viscosity models considered a non-uniform increase of the viscosity in the channel always stabilizes the flow whereas a non-uniform decrease of the viscosity in the channel may either destabilize or, more unexpectedly, stabilize the flow. In all the cases investigated the stability of the flow is found to be only weakly dependent on the value of the Péclet number. We discuss our results in terms of three physical effects, namely bulk effects, velocity-profile shape effects and thin-layer effects.