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

  • Stephen M. Cox (a1) (a2)

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.

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References

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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. 134.
Barenblatt, G. I. & Zel'dovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285312.
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, 12321235.
Berzins, M. & Dew, P. M. 1990 Chebyshev polynomial software for elliptic-parabolic systems of PDEs. ACM Trans. Math. Software (to be published).
Brady, J. F. 1984 Flow development in a porous channel and tube. Phys. Fluids 27, 10611067.
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, 127150.
Brady, J. F. & Acrivos, A. 1982 Closed-cavity laminar flows at moderate Reynolds number. J. Fluid Mech. 115, 427442.
Bundy, R. D. & Weissberg, H. L. 1970 Experimental study of fully developed laminar flow in a porous pipe with wall injection. Phys. Fluids 13, 26132615.
Calogero, F. 1984 A solvable nonlinear wave equation. Stud. Appl. Maths 70, 189199.
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, 122.
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, 325328.
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).
Doering, C. R., Gibbon, J. D., Holm, D. D. & Nicolaenko, B. 1988 Low-dimensional behaviour in the complex Ginzburg-Landau equation. Nonlinearity 1, 279309.
Durlofsky, L. & Brady, J. F. 1984 The spatial stability of a class of similarity solutions. Phys. Fluids 27, 10681076.
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, 197226.
Glendinning, P. 1987 Asymmetric perturbations of Lorernz-like equations. Dyn. Stability Systems 2, 4353.
Glendinning, P. & Sparrow, C. 1984 Local and global behavior near homoclinic orbits. J. Statist. Phys. 35, 645696.
Goldshtik, M. A. & Javorsky, N. I. 1989 On the flow between a porous rotating disk and a plane. J. Fluid Mech. 207, 128.
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, 130141.
Proudman, I. 1960 An example of steady laminar flow at large Reynolds number. J. Fluid Mech. 9, 593602.
Proudman, I. & Johnson, K. 1962 Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161168.
Raithby, G. D. & Knudsen, D. C. 1974 Hydrodynamic development in a duct with suction and blowing. Trans. ASME E: J. Appl. Mech. 41, 896902.
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, 2340.
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, 233246.
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, 414432.
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, 535544.
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, 299310.
Terrill, R. M. 1965 Laminar flow in a uniformly porous channel with large injection. Aeronaut. Q. 16, 323332.
Terrill, R. M. 1967 Flow through a porous annulus. Appl. Sci. Res. 17, 204222.
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, 134144.
Terrill, R. M. & Shrestha, G. M. 1965 Laminar flow through parallel and uniformly porous walls of different permeability. Z. Angew. Math. Phys. 16, 470482.
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, 451485.
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, 151178.
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Two-dimensional flow of a viscous fluid in a channel with porous walls

  • Stephen M. Cox (a1) (a2)

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