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

What is it about?

This article extends the previous approach devised by the authors for the hydrodynamic part of the shear flow to a case involving heat transfer. The article considers a subset of the cases considered by Wang (1992) [The boundary layers due to shear flow over a still fluid, Phys. Fluids A 4, 304–1306 (1992)] and uses direct integration from Andersson & Mukhopadhyay (2014) [Boundary layers due to shear flow over a still fluid: A direct integration approach, Appl. Math. Comp. 242, 856 – 862 (2014)] employing the adapted formulation. The authors of the this article carried out a theoretical study on the heat transfer in the shear flow over a still fluid. The two-fluid problem was tackled using a similarity approach which is well-known for boundary layer problems to reduce the equations governing a 2-fluid immiscible system in which the lower fluid is originally quiescent and the top fluid is shear driven. The density of the lower fluid is higher than that of the upper fluid so that the two-fluid system is stable with respect to the formation of interfacial waves. The governing equations are coupled at the interface, and, subsequently, a heat transfer and momentum analyses are conducted. The transformation results in a reduction of these equations into coupled sets of Ordinary Differential Equations. A direct integration approach (DIA) was applied to evaluate the hydrodynamic and thermal equations numerically. The resulting boundary layers of different two-fluid configurations were plotted and analyzed in conjunction with the relative ratios of Prandtl number of the two fluids utilized. Analyses of the results are mainly on the characteristics of the hydrodynamic and thermal boundary layers for different combinations of two fluids.

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Photo by engin akyurt on Unsplash

Photo by engin akyurt on Unsplash

Why is it important?

In several practical situations, both in nature and engineering, two different immiscible fluids moving almost parallel to each other at different speeds can be found. Occurrence of two-fluid problems is found also in microfluidics which has wider applications in laminar two-stream scenarios due to the small velocity and length scales involved.