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Dissipative heating in convective flows. (English) Zbl 0308.76072
The paper deals with the energetics of different types of convection in a compressible fluid with particular references to the convection in the Earth’s mantle and core. Integrating the entropy equation for a steadily convecting fluid layer the authors derive an upper bound to the total rate of the dissipative heating that is valid for any equation of state or stress-strain relations. Shear-stress heating in a liquid is investigated especially in the case that the coefficient of expansion a is small and the Prandtl number is high. The theoretical results are confirmed by numerical experiments using the Boussinesq approximation. Applying the results to the Earth’s mantle the principal uncertainty is in the value of a. Moreover, a convectively driven hydromagnetic dynamo is considered occupying a spherical region. If the radius of this sphere is greater than the temperature scale height, the ohmic heating rate may exceed the heat flux emerging from its surface since the ohmic heat is feed back into the internal energy of the liquid. Some geophysical problems indicate the need for the consistent non-Boussinesq calculation not yet attempted here.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76R05 Forced convection
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