Couette–Poiseuille Flow in Semi-Elliptic Channels

Abstract

We present a novel analytical solution for Couette flows of incompressible Newtonian fluids in channels with a semi-elliptical cross section. The flow is steady, unidirectional, satisfies the no-slip condition at the boundaries, and is driven by the movement of the planar wall at constant velocity. The theoretical approach consists of a mapping function to rewrite the problem in an elliptical coordinate system coupled with Fourier's method for the solution of a Laplace equation with Dirichlet-type boundary conditions in the new domain. We then use our new solution together with available results for Poiseuille flows in a similar geometry (Alassar and Abushoshah, 2012, “Hagen–Poiseuille Flow in Semi-Elliptic Microchannels,” ASME J. Fluids Eng. 134(12), p. 124502) to study the so-called Couette–Poiseuille flows, where both pressure-driven and boundary-driven mechanisms act simultaneously. We present a detailed analysis of the flow field in Couette, Poiseuille, and Couette–Poiseuille flows in semi-elliptical channels with cross sections of different aspect ratios. For the latter case, we also determine the critical values of the axial pressure gradient that (i) increase the maximum flow velocity above that of the moving wall and shift its position toward the center of the channel, (ii) mark the onset of flow reversal with the emergence of a backflow region below the static wall, and (iii) eventually lead to a zero net flow rate through the channel.