Thermocapillary Driven Turbulent Heat Transfer

A dimensionless number depending on the usual Prandtl and Marangoni numbers, Π s ∼ Ma/(1 + Pr -1 ) = 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 ∼ Π 1/4 S , 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 (να 2 /P s ) 1/4 = (I + Pr) 1/4 (α 3 /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 → ∞, η θ B ∼ (να 2 /∈) 1/4 , which is a Batchelor scale, and the second one explicitly includes its limit for Pr → 0, η θ C ∼ (α 3 /∈) 1/4 , which is an Oboukhov-Corrsin scale