Thermal instability of laminar natural convection flow on inclined isothermal plates

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Abstract
An analysis is made to deteimine the conditions marking the onset of longitudinal vortices in the laminar natural convection flow over a heated isothermal inclined plate. A linear stability theory based on Boussinesq approximation and without the conventional parallel‐flow assumption for basic velocity and temperature profiles is employed in the derivation of perturbation equations. An iterative procedure employing a fourth order Runge‐Kutta method is applied in the solution of the perturbation equations. A comparison between the neutral stability resuts obtained with and without the commonly used parallel‐flow approximation in the stability theory of small disturbances shows that the parallel‐flow assumption is an invalid one for the present problem. Solutions are obtained for Pr = 0.72, 1, 2, 10, 100, and ∞ and the critical Rayleigh number marking the onset of longitudinal vortices is found to be a rather weak function of Prandtl number. A comparison of the present neutral stability results for the inclination angles α = 20° ∼ 60° with experimental data reported in the literature reveals that the theory predicts critical values which are generally two orders of magnitude lower than the experimental data.