Thermocapillary Driven Turbulent Heat Transfer

Abstract

A dimensionless number depending on the usual Prandtl and Marangoni numbers, Πs ∼ Ma/(1 + Pr1) = Ma Pr/(1 + Pr), is introduced for thermocapillary driven flows. Three heat transfer models are proposed in terms of Πs. The first model on laminar flow, using some dimensional arguments with a flow scale and the boundary layer concept, leads to Nu ∼ Πs1/4, Nu being the usual Nusselt number. The second model on transition flow, extending Landau’s original idea on the amplitude of disturbances past marginal stability of isothermal flow, leads to Nu − 1 ∼ (ΠS−ΠSc)1/2, ΠSc corresponding to the critical value of Πs for the marginal state. The third model on turbulent flow, introduces a thermal microscale ηθ ∼ (1 + Pr-1)1/4(να2/Ps)1/4 = (1 + Pr)1/4 (α3/Ps)1/4, with ν and α, respectively, being kinematic and thermal diffusivities, and Ps the production rate of thermocapillary energy. The first expression relating ηθ to Prandtl number explicitly includes its limit for Pr → ∞, ηθB ∼ (να2/ε)1/4, which is a Batchelor scale, and the second one explicitly includes its limit for Pr → 0, ηθC ∼ (α3/ε)1/4, which is an Oboukhov-Corrsin scale. In terms of ηθ and an integral scale l, the model leads to Nu ∼ l/ηθ ∼ Πs1/3. Recent experimental literature are interpreted by special cases of the foregoing models corresponding to Pr > 1.