Similarity solutions of heat transfer in shear flow over an otherwise still fluid

Abstract

Thermal energy and momentum transport in two-fluid systems are investigated. The hydrodynamic part of the problem was considered by Wang [“The boundary layers due to shear flow over a still fluid,” Phys. Fluids A 4, 1304–1306 (1992)] and Andersson and Mukhopadhyay [“Boundary layers due to shear flow over a still fluid: A direct integration approach,” Appl. Math. Comput. 242, 856–862 (2014)], whereas the accompanying heat transfer analysis is new and fills a gap in the literature. Momentum and thermal boundary layers form on both sides of the planar interface between a shear-driven upper fluid and an underlying still fluid. Similarity transformations reduce the governing partial differential equations to sets of ordinary differential equations for each fluid, coupled by boundary conditions at the interface. The thermal part of the transformed two-fluid problem depends explicitly on the ratios between the kinematic viscosities, the thermal conductivities, the Prandtl numbers, and the temperatures of the lower and upper fluids. Moreover, an implicit dependence on the density ratio appears due to the fluid velocities. Similarity solutions obtained by means of a direct integration approach are presented for six different two-fluid systems. The computed results show that the temperature and velocity fields in the upper and lower fluid depend crucially on the mechanical and thermophysical properties of the two fluids, as does the heat flux through the interface between the two immiscible fluids.