Convection–dissipation instability in the horizontal plane Couette flow of a highly viscous fluid

The linear stability of the plane Couette flow against thermoconvective rolls is studied. The case of a flow without a boundary-imposed temperature gradient is investigated. The non-uniform, possibly unstable, basic temperature distribution is caused by the effect of the internal viscous heating. Asymmetric thermal boundary conditions are considered: the bottom boundary is adiabatic, while the top boundary is isothermal. The focus is on a fluid with a large, mathematically infinite, Prandtl number, although the two-dimensional transverse roll instability is discussed also for a finite Prandtl number. The transition to the instability is described through the governing parameter GePe2, where Ge is the Gebhart number and Pe is the Péclet number. The response of the basic Couette flow to arbitrarily oriented oblique rolls is tested, so that a complete set of disturbance modes is taken into account. It is shown that the Couette flow is more unstable to longitudinal rolls than to any other oblique roll mode.