Localisation of optimal perturbations in variable viscosity channel flow

We discuss how a variable fluid viscosity affects the nonmodal stability characteristics of the pressure driven flow between two parallel walls maintained at different temperatures. In this work, we specify the fluid viscosity to be a function of the fluid temperature. We employ an Arrhenius model to model the viscosity of water, and Sutherland's law to model the viscosity of air. We impose a stable density stratification, and find that strong density stratification can suppress optimal transient growth regardless of how strong the viscosity variation is. Some studies have been inclined to neglect viscosity stratification, since the changes in levels of optimal growth, when compared to the uniform viscosity case, are often not too significant. In this article, we show significant localisation of optimal perturbation energy in the less viscous region, a feature that is not observed in uniform viscosity flows. This can have a bearing on the route to turbulence in these systems.