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The motion of particles in the Hele-Shaw cell

Published online by Cambridge University Press:  26 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California at San Diego, La Jolla, CA 92093-0411, USA
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Abstract

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 261 , 25 February 1994 , pp. 199 - 222
Copyright
© 1994 Cambridge University Press

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References

Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic.
De Mestre, L. J. 1973 Low-Reynolds-number fall of slender cylinders near boundaries. J. Fluid Mech. 58, 641.Google Scholar
Dvinski, A. S. & Popel, A. S. 1987a Motion of a rigid cylinder between parallel plates in Stokes flow. Part 1: Motion in a quiescent fluid and sedimentation. Comput. Fluids 15, 391–404.Google Scholar
Dvinski, A. S. & Popel, A. S. 1987b Motion of a rigid cylinder between parallel plates in Stokes flow. Part 2: Poiseuille and Couette flow. Comput. Fluids 15, 405–419.Google Scholar
Fung, Y. C. 1984 Biodynamics, Circulation. Springer.
Ganatos, P., Weinbaum, S. & Pfeffer, R. 1980a A strong interaction theory for the creeping motion of a sphere between parallel boundaries. Part 1. Perpendicular motion. J. Fluid Mech. 99, 739–753.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1980b A strong interaction theory for the creeping motion of a sphere between parallel boundaries. Part 2. Parallel motion. J. Fluid Mech. 99, 755–783.Google Scholar
Ganatos, P., Weinbaum, S. & Pfeffer, R. 1982 Gravitational and zero-drag motion of a sphere of arbitrary size in an inclined channel at low Reynolds number. J. Fluid Mech. 124, 27–43.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall - I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall - II. Couette flow. Chem. Engng Sci. 22, 653–660.Google Scholar
Halpern, D. & Secomb, T. W. 1991 Viscous motion of disk-shaped particles through parallel-sided channels with near-minimal widths. J. Fluid Mech. 231, 545–560.Google Scholar
Halpern, D. & Secomb, T. W. 1992 The squeezing of red blood cells through parallel-sided channels with near-minimal widths. J. Fluid Mech. 244, 307–322.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hsu, R. & Ganatos, P. 1989 The motion of a rigid body in viscous fluid bounded by a plane wall. J. Fluid Mech. 207, 29–72.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics, Principles and Selected Applications. Butterworth-Heinemann.
Kucaba-Pietal, A. 1986 Nonaxisymmetric Stokes flow past a torus in the presence of a wall. Arch. Mech. 38, 647–663.Google Scholar
Lee, J. S. & Fung, Y. C. 1969 Stokes flow around a circular cylindrical post confined between two parallel plates. J. Fluid Mech. 37, 657–670.Google Scholar
Lee, J. S. & Fung, Y. C. 1969 Stokes flow around a circular cylindrical post confined between two parallel plates. J. Fluid Mech. 37, 657–670.Google Scholar
O'Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Pozrikidis, C. 1994 Shear flow over a plane wall with an axisymmetric cavity or a circular orifice of finite thickness. Phys. Fluids A, 6, 1–12.Google Scholar
Vrahopoulou, E. P. 1992 Flow distortions around particles between parallel walls with application to streak formation in slide-coating methods. Chem. Engng Sci. 47, 1027–1037.Google Scholar
Wakiya, S. 1959 Effect of a submerged object on a slow viscous flow V. Spheroid at an arbitrary angle of attach. Res. Rep. Fac. Engng Niigata Univ. Japan 8, 17–30 (in Japanese).
Zhou, H. & Pozrikidis, C. 1993 The flow of suspensions in channels: Single files of drops. Phys. Fluids A 5, 311–324.Google Scholar