Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls

Abstract

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer and Majdalani [Phys. Fluids 15, 1485 (2003)] by taking the wall expansion ratio alpha to be time invariant and then proceed to reduce the Navier-Stokes equations into a fourth order ordinary differential equation with four boundary conditions. Using the homotopy analysis method (HAM), an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska [Fluid Dyn. Res. 4, 151 (1988)]. Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier-Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution of alpha(t) from an initial alpha(0) to a final state alpha(1). In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored. (C) 2010 American Institute of Physics. [doi:10.1063/1.3392770]