ONSET OF CONVECTION IN A HORIZONTAL WATER LAYER WITH MAXIMUM DENSITY EFFECTS

The onset of convection in a horizontal water layer with regard to the density anomaly (near 4°C) is studied using linear stability analysis. The resulting perturbation equations are solved with the aid of Galerkin's method. With the choice of reasonable test functions, it is shown that seven terms are sufficient for an approach close to 1%. The results are represented in stability diagrams, Ra = Ra(T1, T2) where T1 and T2 are the temperatures of the lower and upper wall, respectively. The nonlinear density-temperature relation is approximated by three different polynomials having 2, 3 and 5 terms. Assuming the fifth order polynomial to be exact, the critical Rayleigh numbers calculated with the simple parabolic relation are about 10% too large. This discrepancy is reduced to approximately +3% by adding a cubic term to the density-temperature relation. © 1979.