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Steady streaming in a channel with permeable walls

Published online by Cambridge University Press:  28 August 2013

KONSTANTIN ILIN
KONSTANTIN ILIN*
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK email: konstantin.ilin@york.ac.uk
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Abstract

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) injection/suction of a viscous fluid at the walls. We obtain an asymptotic expansion of the solution of the Navier–Stokes equations in the limit when the amplitude of normal displacements of fluid particles near the walls is much smaller than both the width of the channel and the thickness of the Stokes layer. It is shown that the steady part of the flow in this problem is much stronger than the steady flow produced by vibrations of impermeable boundaries. Another interesting feature of this problem is that the direction of the steady flow is opposite to what one would expect if the flow was produced by vibrations of impermeable walls.

Keywords

Boundary-layer theorySteady streamingAsymptotic methodsOscillating viscous flowsNavier-Stokes equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 1 , February 2014 , pp. 65 - 82
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Carlsson, F., Sen, M. & Löfdahl, H. A. (2005) Fluid mixing induced by vibrating walls. Eur. J. Mech. B Fluids 24, 366–378.CrossRefGoogle Scholar
[2]Fung, Y. C. & Yih, C. S. (1968) Peristaltic transport. J. Appl. Mech. 35, 669–675.Google Scholar
[3]Hœpffner, J. & Fukagata, K. (2009) Pumping or drag reduction? J. Fluid Mech. 635, 171–187.CrossRefGoogle Scholar
[4]Ilin, K. (2008) Viscous boundary layers in flows through a domain with permeable boundary. Eur. J. Mech. B Fluids 27, 514–538.CrossRefGoogle Scholar
[5]Ilin, K. & Morgulis, A. (2011) Steady streaming between two vibrating planes at high Reynolds numbers. E-print: arXiv:1108.2710v1 [physics.flu-dyn].Google Scholar
[6]Ilin, K. & Sadiq, M. A. (2010) Steady viscous flows in an annulus between two cylinders produced by vibrations of the inner cylinder. E-print: arXiv:1008.4704v2 [physics.flu-dyn].Google Scholar
[7]Jaffrin, M. Y. & Shapiro, A. H. (1971) Peristaltic pumping. Ann. Rev. Fluid Mech. 3, 13–37.Google Scholar
[8]Lighthill, J. (1978) Acoustic streaming. J. Sound Vib. 61, 391–418.Google Scholar
[9]Longuet-Higgins, M. S. (1953) Mass transport in water waves. Philos. Trans. Roy. Soc. London. A. Math. Phy. Sci. 245, 535–581.Google Scholar
[10]Nayfeh, A. H. (1973) Perturbation Methods, John Wiley, New York, NY.Google Scholar
[11]Riley, N. (1967) Oscillatory viscous flows. Review and extension. J. Inst. Maths. Applics. 3, 419–434.Google Scholar
[12]Riley, N. (2001) Steady streaming. Ann. Rev. Fluid Mech. 33, 43–65.CrossRefGoogle Scholar
[13]Sadiq, M. A. (2011) Steady streaming due to the vibrating wall in an infinite viscous incompressible fluid. J. Phys. Soc. Jpn. 80, 034404 (9 pages).Google Scholar
[14]Selverov, K. P. & Stone, H. A. (2001) Peristaltically driven channel flows with applications towards micromixing. Phys. Fluids 13, 1838–1859.Google Scholar
[15]Trenogin, V. A. (1970) The development and applications of the Lyusternik–Vishik asymptotic method. Uspehi Mat. Nauk 25 (4), 123–156.Google Scholar
[16]Vladimirov, V. A. (2008) Viscous flows in a half space caused by tangential vibrations on its boundary. Stud. Appl. Math. 121, 337–367.Google Scholar
[17]Wilson, D. E. & Panton, R. L. (1979) Peristaltic transport due to finite amplitude bending and contraction waves. J. Fluid Mech. 90, 145–159.Google Scholar
[18]Yi, M., Bau, H. H. & Hu, H. (2002) Peristaltically induced motion in a closed cavity with two vibrating walls. Phys. Fluids 14, 184–197.Google Scholar