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

Published online by Cambridge University Press:  26 April 2006

Stephen M. Cox
Stephen M. Cox
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK Present address: Sibley School of Mechanical & Aerospace Engineering, Upson Hall, Cornell University, Ithaca, NY 14853, USA.
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 1 - 33
Copyright
© 1991 Cambridge University Press

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