Skip to main content
×
Home
    • Aa
    • Aa

Steady streaming confined between three-dimensional wavy surfaces

  • ROMAIN GUIBERT (a1), FRANCK PLOURABOUÉ (a1) and ALAIN BERGEON (a1)
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.

Copyright
Corresponding author
Email address for correspondence: plourab@imft.fr
References
Hide All
P. W. Duck & F. T. Smith 1979 Steady streaming induced between oscillating cylinders. J. Fluid Mech. 91, 93110.

M. Firdaouss , J.-L. Guermond & P. Le Quéré 1997 Non-linear corrections to Darcy's law at low Reynolds numbers. J. Fluid Mech. 343, 331350.

J. B. Grotberg 1984 Volume-cycled oscillatory flow in a tapered channel. J. Fluid Mech. 141, 249264.

S. Gupta , D. Poulikakos & V. Kurtcuoglu 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.

P. Hall 1974 Unsteady viscous flow in a pipe of slowly varying cross-section. J. Fluid Mech. 64, 209226.

D. Lo Jacono , F. Plouraboué & A. Bergeon 2005 Weak-inertial flow between two rough surfaces. Phys. Fluids 17, 063602.

M. J. Manton 1971 Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451459.

C. C. Mei & J.-L. Auriault 1991 The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647663.

T. Nishimura , S. Arakawa , M. Shinichiro & Y. Kawamura 1989 Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Engng Sci. 44, 21372148.

N. Padmanabhan & T. J. Pedley 1987 Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section. J. Fluid Mech. 178, 325343.

C. Pozrikidis 1987 Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495514.

M. E. Ralph 1986 Oscillatory flows in wavy-walled tubes. J. Fluid Mech. 168, 515540.

K. P. Selderov & H. A. Stone 2001 Peristaltically driven channel flows with applications toward micromixing. Phys. Fluids 13, 18371855.

I. J. Sobey 1980 a On the flow furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96 (1), 126.

I. J. Sobey 1980 b On the flow furrowed channels. Part 2. Observed flow patterns. J. Fluid Mech. 96 (1), 2732.

I. J. Sobey 1985 Dispersion caused by separation during oscillatory flow through a furrowed channel. Chem. Engng Sci. 40, 21292134.

A. D. Stroock , S. K. W. Dertinger , A. Ajdari , I. Mezić , H. A. Stone & G. M. Whitesides 2002 Chaotic mixer for microchannels. Science 295, 647651.

S. L. Waters 2001 Solute uptake through the walls of a pulsating channel. J. Fluid Mech. 433, 193208.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 75 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.