Heat transfer in gravity-driven film flow of power-law fluids

Abstract

A mathematical model for the flow and heat transfer in an accelerating liquid film of a non-Newtonian power-law fluid is presented. The thermal boundary layer equation permits exact similarity solutions only in the particular case when the power-law index n is equal to unity, i.e. for Newtonian films. To this end, the heat transfer problem is solved by means of a local nonsimilarity approach with n and local Prandtl number Prx being the only parameters. A critical Prandtl number Pr ∗ x is introduced, which is a monotonically increasing function of n. The nonsimilar heat transfer problem is integrated numerically for several parameter combinations in the ranges 0.2 ⩽ n ⩽ 2.0 and 0.001 ⩽ Prx ⩽ 1000 and the calculations for n = 1 compared favourably with earlier results for Newtonian liquid films. For high Prandtl numbers, the temperature gradient at the wall is controlled by the wall gradient of the streamwise velocity component, which is practically independent of n for dilatant fluids (n > 1.0) but increases significantly with increasing pseudo-plasticity (n < 1.0) . For Prx ⪡ 1, on the other hand, the wall gradient of the temperature field increases slowly with n and this modest variation is ascribed to the displacement effect caused by the presence of the momentum boundary layer. Curve-fit formulas for the temperature gradient at the wall are provided in order to facilitate rapid and yet accurate estimates of the local heat transfer coefficient and the Nusselt number.