Stability analysis of natural convection evolving along a vertical heated plate

Natural convection evolving along a vertical fiat plate is considered under (i) isothermal and (ii) uniform-heat-flux plate boundary conditions. Basic flow is assumed to be governed by steady boundary-layer equations. Linear and weakly-nonlinear stability analyses of the basic flow are made with the aid of the Galerkin method in which the field variables are expanded in te rms of Chebyshev polynomials. The Stewartson-Stuart equation for the two-di mensional propagating disturbance is derived and its stability is examined for the cases of the Prandtl number Pr = 0.733 (air) and Pr = 6.7 (water). The main results are such that (i) the motion is supercritical over almost all the linearly unstable region except for a narrow boundary region define d by small wavenumbers; (ii) Huerre's criterion shows that the motion is co nvectively unstable along the most highly amplified path over the whole com puted range of the Reynolds number; (iii) Newell's criterion shows that the motion tends to be modulationally unstable at lower values of the Reynolds number especially in air.