The stability of buoyancy-driven gaseous boundary layers over inclined semi-infinite hot plates

The free-convective boundary-layer flow that develops over a semi-infinite inclined hot plate is known to become unstable at a finite distance from the leading edge, characterized by a critical value of the Grashof number $Gr_\delta$ based on the local boundary-layer thickness $\delta$. The character of the instability depends on the inclination angle $\phi$, measured from the vertical direction. For values of $\phi$ below a critical value $\phi_c$ the instability is characterized by the appearance of spanwise vortices, whereas for $\phi>\phi_c$ the bifurcated flow displays G\"ortler-like streamwise vortices. The Boussinesq approximation, employed in previous linear stability analyses, ceases to be valid for gaseous flow when the wall-to-ambient temperature ratio $\theta_w=T_w/T_\infty$ is not close to unity. The corresponding non-Boussinesq analysis is presented here, accounting also for the variation with temperature of the different transport properties. The base-flow profiles are used in a parallel-flow temporal stability analysis to delineate the dependence of the critical Grashof numbers $Gr_\delta$ on the inclination angles $\phi$ and on the temperature ratio $\theta_w$. The analysis provides in particular the values of the crossover inclination angles $\phi_c(\theta_w)$.