Stability of a Homogeneous Fluid Cooled Uniformly from Above

The stability of an initially homogeneous layer of fluid which is cooled uniformly from above is examined. A simplified mathematical model in which the velocity and temperature perturbations vanish at the top and bottom boundaries but which takes into account nonlinear, time-dependent temperature profiles is developed. Solution of the stability equation is achieved by expanding the velocity perturbation in a Fourier sine series with time dependent coefficients and solving the resulting system of coupled differential equations numerically. The initial behavior of the system is found to depend upon the Rayleigh and Prandtl numbers. For large Rayleigh numbers the time needed for the system to manifest convective behavior is independent of the depth of the fluid layer but depends on the rate and manner of cooling and on the properties of the fluid. For large Rayleigh numbers, the horizontal wavenumber of disturbances which are amplified the most when convective behavior is manifest in the system also is independent of the depth of the fluid layer but depends on the rate and manner of cooling and on the properties of the fluid.