Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-27T17:20:44.664Z Has data issue: false hasContentIssue false

Steady streaming confined between three-dimensional wavy surfaces

Published online by Cambridge University Press:  03 August 2010

ROMAIN GUIBERT
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
FRANCK PLOURABOUÉ*
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
ALAIN BERGEON
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
*
Email address for correspondence: plourab@imft.fr

Abstract

We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σh where σ ~ O(1), but long-wavelength variations λ, with h/λ ≪ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel's wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Balédent, O., Ambarkia, K., Kongolob, G., Bouzerara, R., Gondry-Jouetc, C. & Meyer, M. E. 2007 Cerebral flow modeling using electrical analogue: MRI velocimetry validation. Med. Nucl. 31 (1), 1628.Google Scholar
Duck, P. W. & Smith, F. T. 1979 Steady streaming induced between oscillating cylinders. J. Fluid Mech. 91, 93110.CrossRefGoogle Scholar
Firdaouss, M., Guermond, J.-L. & Le Quéré, P. 1997 Non-linear corrections to Darcy's law at low Reynolds numbers. J. Fluid Mech. 343, 331350.CrossRefGoogle Scholar
Grotberg, J. B. 1984 Volume-cycled oscillatory flow in a tapered channel. J. Fluid Mech. 141, 249264.CrossRefGoogle Scholar
Gupta, S., Poulikakos, D. & Kurtcuoglu, V. 2008 Analytical solution for pulsatile viscous flow in a straight elliptic annulus and application to the motion of the cerebrospinal fluid. Phys. Fluids 20, 093607.Google Scholar
Hall, P. 1974 Unsteady viscous flow in a pipe of slowly varying cross-section. J. Fluid Mech. 64, 209226.CrossRefGoogle Scholar
Leal, L. G. 1992 Laminar Flow and Convective Transport Processes. Butterworth–Heinemann Series in Chemical Engineering.Google Scholar
Lo Jacono, D., Plouraboué, F. & Bergeon, A. 2005 Weak-inertial flow between two rough surfaces. Phys. Fluids 17, 063602.CrossRefGoogle Scholar
Manton, M. J. 1971 Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451459.CrossRefGoogle Scholar
Mei, C. C. & Auriault, J.-L. 1991 The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647663.CrossRefGoogle Scholar
Nelissen, R. M. 2008 Fluid mechanics of intrathecal drug delivery. PhD thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.Google Scholar
Nishimura, T., Arakawa, S., Shinichiro, M. & Kawamura, Y. 1989 Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Engng Sci. 44, 21372148.CrossRefGoogle Scholar
Padmanabhan, N. & Pedley, T. J. 1987 Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section. J. Fluid Mech. 178, 325343.CrossRefGoogle Scholar
Pozrikidis, C. 1987 Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495514.CrossRefGoogle Scholar
Ralph, M. E. 1986 Oscillatory flows in wavy-walled tubes. J. Fluid Mech. 168, 515540.CrossRefGoogle Scholar
Ramachandra Rao, A. & Devanathan, R. 1973 Pulsatile flow in tubes of varying cross-sections. Z. Angew. Math. Phys. 24, 203213.Google Scholar
Selderov, K. P. & Stone, H. A. 2001 Peristaltically driven channel flows with applications toward micromixing. Phys. Fluids 13, 18371855.CrossRefGoogle Scholar
Sobey, I. J. 1980 a On the flow furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96 (1), 126.Google Scholar
Sobey, I. J. 1980 b On the flow furrowed channels. Part 2. Observed flow patterns. J. Fluid Mech. 96 (1), 2732.CrossRefGoogle Scholar
Sobey, I. J. 1985 Dispersion caused by separation during oscillatory flow through a furrowed channel. Chem. Engng Sci. 40, 21292134.CrossRefGoogle Scholar
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezić, I., Stone, H. A. & Whitesides, G. M. 2002 Chaotic mixer for microchannels. Science 295, 647651.CrossRefGoogle ScholarPubMed
Waters, S. L. 2001 Solute uptake through the walls of a pulsating channel. J. Fluid Mech. 433, 193208.CrossRefGoogle Scholar