The linear stability of flat-plate boundary-layer flow of fluid with temperature-dependent viscosity

The linear stability of boundary-layer flow of fluid with temperature-dependent viscosity over a heated or cooled flat-plate is investigated. Decomposition of the disturbance into normal temporal modes leads to a sixth-order modified eigenvalue problem. Making the additional ad hoc assumption of parallel flow leads to a simpler sixth-order parallel eigenvalue problem which, unlike the modified problem, reduces to the classical Orr-Sommerfeld problem in the isothermal case. Two viscosity models are considered, and for both models numerically-calculated stability results for both the modified and parallel eigenvalue problems are obtained. For both viscosity models it is, perhaps surprisingly, found that for both eigenvalue problems a non-uniform decrease in viscosity across the layer stabilizes the flow while a non-uniform increase in viscosity across the layer destabilizes the flow. Results for the two eigenvalue problems are shown to be quantitatively similar with, however, the parallel problem always over-predicting the critical Reynolds number in comparison to the modified problem. Finally, we discuss the physical interpretation of our results in terms of velocity-profile shape and thin-layer effects.