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We derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr in laminar vertical convection (VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: Nu∼Pr^1/4Ra^1/4, Re∼Pr^-1/2Ra^1/2 for Pr≪1 and Nu∼Pr^0Ra^1/4, Re∼Pr^-1Ra^1/2 for Pr≫1. These theoretical results are in excellent agreement with direct numerical simulations for Ra from 10^5 to 10^10 and Pr from 10^-2 to 30. The transition between the regimes takes place for Pr around 10^-1.
Olga Shishkina (Mon,) studied this question.