Two-dimensional flow of a viscous fluid in a channel with porous walls

Abstract

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Imperfect homoclinic bifurcations

“…Just as the standard imperfection theory for the bifurcations of stationary points [12] allows one to describe the effect of asymmetry in terms of µ and ǫ, our aim here is to give an analogous description for general global bifurcations. We note that this is in the spirit of the work of Glendinning [14] and Cox [13] for the particular case of Lorenz-like bifurcations.…”

“…Whilst the effect of small symmetry-breaking terms on the bifurcations of stationary solutions has a long history (the imperfection theory of Golubitsky and Schaeffer [10,11,12]) there appears to have been no systematic attempt to describe the equivalent modifications of global bifurcations (although see [13,14] for a special case). Our aim here is to provide the foundations for such an approach.…”

Unsteady flow and heat transfer of UCM fluid in a porous channel with variable thermal conductivity and ion slip effects

“…Terrill and Shrestha (1966) discussed the problem of steady incompressible two dimensional viscous fluid flow through a channel with uniformly porous walls and obtained a solution for large Reynolds number. Later the more general problem of symmetric and asymmetric suction driven flow through porous parallel plates is analyzed by Cox (1991). Ramana Murthy et al (2007) examined the viscous fluid flow between porous parallel plates due to periodic suction and injection by retaining non linear convective terms and the analysis is carried out up to second order flow.…”

Homotopy Analysis Method to Walter’s B fluid in a vertical channel with porous wall

“…He has solved the NavierStokes equations by using a perturbation method for very low cross-flow Reynolds numbers. After his pioneering work, this problem has been studied by many researchers considering various variations in the problem [2,3].…”