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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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References

Afraimovich, V. S. & Shil'nikov, L. P. 1983 Strange attractors and quasiattractors. In Nonlinear Dynamics and Turbulence (ed. G. I. Barenblatt, G. Iooss & D. D. Pitman), pp. 1–34.
Barenblatt, G. I. & Zel'dovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Berman, A. S. 1953 Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232–1235.Google Scholar
Berzins, M. & Dew, P. M. 1990 Chebyshev polynomial software for elliptic-parabolic systems of PDEs. ACM Trans. Math. Software (to be published).Google Scholar
Brady, J. F. 1984 Flow development in a porous channel and tube. Phys. Fluids 27, 1061–1067.Google Scholar
Brady, J. F. & Acrivos, A. 1981 Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow. J. Fluid Mech. 112, 127–150.Google Scholar
Brady, J. F. & Acrivos, A. 1982 Closed-cavity laminar flows at moderate Reynolds number. J. Fluid Mech. 115, 427–442.Google Scholar
Bundy, R. D. & Weissberg, H. L. 1970 Experimental study of fully developed laminar flow in a porous pipe with wall injection. Phys. Fluids 13, 2613–2615.Google Scholar
Calogero, F. 1984 A solvable nonlinear wave equation. Stud. Appl. Maths 70, 189–199.Google Scholar
Childress, S., Ierley, G. R., Spiegel, E. A. & Young, W. R. 1989 Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form. J. Fluid Mech. 203, 1–22.Google Scholar
Cox, S. M. 1989 A similarity solution of the Navier-Stokes equations for two-dimensional flow in a porous-walled channel. Ph.D. thesis, University of Bristol.
Cox, S. M. 1990 The transition to chaos in an asymmetric perturbation of the Lorenz system.. Phys. Lett. A 144, 325–328.Google Scholar
Cox, S. M. 1991 Analysis of steady flow in a channel with one porous wall, or with accelerating walls. SIAM J. Appl. Maths (in press).Google Scholar
Doering, C. R., Gibbon, J. D., Holm, D. D. & Nicolaenko, B. 1988 Low-dimensional behaviour in the complex Ginzburg-Landau equation. Nonlinearity 1, 279–309.Google Scholar
Durlofsky, L. & Brady, J. F. 1984 The spatial stability of a class of similarity solutions. Phys. Fluids 27, 1068–1076.Google Scholar
Foias, C., Nicolaenko, B., Sell, G. R. & Temam, R. 1988 Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. J. Maths Pures Appl. 67, 197–226.Google Scholar
Glendinning, P. 1987 Asymmetric perturbations of Lorernz-like equations. Dyn. Stability Systems 2, 43–53.Google Scholar
Glendinning, P. & Sparrow, C. 1984 Local and global behavior near homoclinic orbits. J. Statist. Phys. 35, 645–696.Google Scholar
Goldshtik, M. A. & Javorsky, N. I. 1989 On the flow between a porous rotating disk and a plane. J. Fluid Mech. 207, 1–28.Google Scholar
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Proudman, I. 1960 An example of steady laminar flow at large Reynolds number. J. Fluid Mech. 9, 593–602.Google Scholar
Proudman, I. & Johnson, K. 1962 Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168.Google Scholar
Raithby, G. D. & Knudsen, D. C. 1974 Hydrodynamic development in a duct with suction and blowing. Trans. ASME E: J. Appl. Mech. 41, 896–902.Google Scholar
Robinson, W. A. 1976 The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls. J. Engng Maths 10, 23–40.Google Scholar
Shrestha, G. M. 1967 Singular perturbation problems of laminar flow in a uniformly porous channel in the presence of a transverse magnetic field. Q. J. Mech. Appl. Maths 20, 233–246.Google Scholar
Shrestha, G. M. & Terrill, R. M. 1968 Laminar flow with large injection through parallel and uniformly porous walls of different permeability. Q. J. Mech. Appl. Maths 21, 414–432.Google Scholar
Skalak, F. M., Wang, C.-Y. 1978 On the nonunique solutions of laminar flow through a porous tube or channel. SIAM J. Appl. Maths 34, 535–544.Google Scholar
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer.
Stuart, J. T. 1988 Euler partial differential equations: singularities in their solution. In Symposium to Honor Professor C. C. Lin (ed. D. J. Benney, F. H. Shu & C. Yuan). World Scientific.
Tabor, M. 1989 Chaos and Integrability in Nonlinear Dynamics. An Introduction. Wiley.
Terrill, R. M. 1964 Laminar flow in a uniformly porous channel. Aeronaut. Q. 15, 299–310.Google Scholar
Terrill, R. M. 1965 Laminar flow in a uniformly porous channel with large injection. Aeronaut. Q. 16, 323–332.Google Scholar
Terrill, R. M. 1967 Flow through a porous annulus. Appl. Sci. Res. 17, 204–222.Google Scholar
Terrill, R. M. & Shrestha, G. M. 1964 Laminar flow through channels with porous walls and with an applied transverse magnetic field. Appl. Sci. Res. 11, 134–144.Google Scholar
Terrill, R. M. & Shrestha, G. M. 1965 Laminar flow through parallel and uniformly porous walls of different permeability. Z. Angew. Math. Phys. 16, 470–482.Google Scholar
Watson, E. B. B., Banks, W. H. H., Zaturska, M. B. & Drazin, P. G. 1990 On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls. J. Fluid Mech. 212, 451–485.Google Scholar
Zaturska, M. B., Drazin, P. G. & Banks, W. H. H. 1988 On the flow of a viscous fluid driven along a channel by suction at porous walls. Fluid Dyn. Res. 4, 151–178.Google Scholar