Limiting angular dependencies of heat transfer and stratification in a heat-generating fluid

A theoretical study is carried out for the distribution of heat flux to the boundary, as well as of the temperature and flow velocity in the lower part of the volume taken up by a one-component heat-generating fluid. The treatment is based on the analysis of the converging boundary layer in view of the conditions of joining its characteristics with those of the fluid in the stably stratified region of the volume. It is found that the dependence of the heat flux at the boundary, q, on the polar angle θ at θ∗≪θ≪1 (where θ∗ is some boundary angle), and the dependence of the temperature in the volume on the ratio of the height z to the characteristic size of the volume R, are power dependences, q∼θa, Tb∼z/Rb. The exponents for the cases of laminar and turbulent boundary layers are a=2, b=4/5 and a=24/13, b=9/13, respectively. The heat flux at θ<θ∗ weakly depends on θ and assumes the minimum value at θ=0. The ratio of the minimum heat flux qm to its average value 〈q〉, as well as the boundary angle θ∗ as a function of the modified Rayleigh number, are given by the estimates qm/〈q〉∼RaI−1/6, θ∗∼RaI−1/12. The results are in quite satisfactory agreement with experiment.