Thermocapillary Driven Turbulent Heat Transfer

A dimensionless number depending on the usual Prandtl and Marangoni numbers, Π_s ∼ Ma/(1 + Pr¹) = Ma Pr/(1 + Pr), is introduced for thermocapillary driven flows. Three heat transfer models are proposed in terms of Π_s. 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 ∼ (Π_S−Π_Sc)^1/2, Π_Sc corresponding to the critical value of Π_s for the marginal state. The third model on turbulent flow, introduces a thermal microscale η_θ ∼ (1 + Pr^-1)^1/4(να²/P_s)^1/4 = (1 + Pr)^1/4 (α³/P_s)^1/4, with ν and α, respectively, being kinematic and thermal diffusivities, and P_s the production rate of thermocapillary energy. The first expression relating η_θ 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 η_θ and an integral scale l, the model leads to Nu ∼ l/η_θ ∼ $Πs1/3$⁠. Recent experimental literature are interpreted by special cases of the foregoing models corresponding to Pr > 1.