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Dynamic simulation of hydrodynamically interacting spheres in a quiescent second-order fluid

Published online by Cambridge University Press:  26 April 2006

Ronald J. Phillips
Ronald J. Phillips
Affiliation:
Department of Chemical Engineering and Materials Science, University of California, Davis, Davis, CA 95616, USA
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Abstract

A method is described for calculating the motion of N spherical particles suspended in a quiescent second-order fluid. The method requires calculation of only the low-Reynolds-number Newtonian velocity profile. This profile is used in conjunction with what has been called the ‘Reciprocal theorem method’ to evaluate particle velocities accurate to leading order in the Deborah number. If the Newtonian velocity field is found by a multipole moment expansion, then it is shown that the method can be integrated neatly into the Stokesian dynamics method of simulating Newtonian suspensions. Simulation results involving two, three, four and six particles are reported as illustrative examples, and are compared with corresponding results for particles in Newtonian fluids and with experimental results found in the literature. In addition, simulations of sedimenting suspensions are performed by using periodic boundary conditions to model an unbounded system, and the observed formation of clusters in the sedimenting system is shown to be in qualitative agreement with experimental observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 315 , 25 May 1996 , pp. 345 - 365
Copyright
© 1996 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Allen, E. & Uhlherr, P. H. T. 1989 Nonhomogeneous sedimentation in viscoelastic fluids. J. Rheol. 33, 627.Google Scholar
Bartram, E., Goldsmith, H. L. & Mason, S. G. 1975 Particle motions in non-Newtonian media. III. Further observations in elasticoviscous fluids. Rheol. Acta 14, 776.Google Scholar
Bird, R. B. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. J. Wiley and Sons.
Bisgaard, C. 1983 Velocity fields around spheres and bubbles investigated by laser-doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian Dynamics. Ann. Rev. Fluid Mech. 20, 111.Google Scholar
Brady, J. F., Phillips, R. J., Lester, J. C. & Bossis, G. 1988 Dynamic simulation of hydrodynamically interacting suspensions. J. Fluid Mech. 195, 257.Google Scholar
Brunn, P. 1976 The slow motion of a sphere in a second-order fluid. Rheol. Acta 15, 163.Google Scholar
Brunn, P. 1977a The slow motion of a rigid particle in a second-order fluid. J. Fluid Mech. 82, 529.Google Scholar
Brunn, P. 1977b Interaction of spheres in a viscoelastic fluid. Rheol. Acta 16, 461.Google Scholar
Brunn, P. 1980 The motion of rigid particles in viscoelastic fluids. J. Non-Newtonian Fluid Mech, 7, 271.Google Scholar
Bush, M. B. & Phan-Thien, N. 1984 Drag force on a sphere in creeping motion through a Carreau model fluid. J. Non-Newtonian Fluid Mech. 16, 303.Google Scholar
Chan, P. C.-H. & Leal, L. G. 1977 A note on the motion of a spherical particle in a general quadratic flow of a second-order fluid. J. Fluid Mech. 82, 549.Google Scholar
Chang, C. Y. & Powell, R. L. 1994 Self-diffusion of bimodal suspensions of hydrodynamically interacting spherical particles in shearing flow. J. Fluid Mech. 281, 51.Google Scholar
Chhabra, R. P. & Uhlherr, P. H. T. 1980 Creeping motion of spheres through shear thinning elastic fluids described by the Carreau viscosity equation. Rheol. Acta 19, 187.Google Scholar
Chiba, K., Song, K.-W. & Horikawa, A. 1986 Motion of a slender body in quiescent polymer solutions. Rheol. Acta 25, 380.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1978 A numerical-solution technique for three-dimensional Stokes flows, with application to the motion of strongly interacting spheres in a plane. J. Fluid Mech. 84, 79.Google Scholar
Gauthier, G., Goldsmith, H. L. & Mason, S. G. 1971 Particle motions in non-Newtonian media. I. Couette flow. Rheol. Acta 10, 344.Google Scholar
Happel, J. & Brenner, H. 1986 Low Reynolds Number Hydrodynamics. Martinus-Nijhoff.
Highgate, D. J. & Whorlow, R. W. 1969 End effects and particle migration effects in concentric cylinder rheometry. Rheol. Acta 8, 142.Google Scholar
Hocking, L. M. 1964 The behaviour of clusters of spheres falling in a viscous fluid. Part 2. Slow motion theory. J. Fluid Mech. 20, 129.Google Scholar
Joseph, D. D., Liu, Y. J., Poletto, M. & Feng, J. 1994 Aggregation and dispersion of spheres falling in viscoelastic liquids. J. Non-Newtonian Fluid Mech. 54, 45.Google Scholar
Joseph, D. D., Nelson, J., Hu, H. H. & Liu, Y. J. 1992 Competition between inertial pressures and normal stresses in the flow induced anisotropy of solid particles. In Theoretical and Applied Rheology (ed. P. Moldenaers & R. Keunings). Elsevier.
Karnis, A. & Mason, S. G. 1967 Particle motions in sheared suspensions. XIX. Viscoelastic media. Trans. Soc. Rheol. 10, 571.Google Scholar
Kim, S. 1986 The motion of ellipsoids in a second order fluid. J. Non-Newtonian Fluid Mech. 21, 255.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics. Butterworth-Heinemann.
Leal, L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69, 305.Google Scholar
Leal, L. G. 1979 The motion of small particles in non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 5, 33.Google Scholar
Liu, Y. J. & Joseph, D. D. 1993 Sedimentation of particles in polymer solutions. J. Fluid Mech. 255, 565.Google Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561.Google Scholar
Lunsmann, W. J., Genieser, L., Armstrong, R. C. & Brown, R. A. 1993 Finite element analysis of steady viscoelastic flow around a sphere in a tube – calculations with constant viscosity models. J. Non-Newtonian Fluid Mech. 48, 63.Google Scholar
McKinley, G. H., Armstrong, R. C. & Brown, R. A. 1993 The wake instability in viscoelastic flow past confined cylinders. Phil. Trans. R. Soc. Lond. A 344, 265.Google Scholar
Michele, J., Patzold, R. & Donis, R. 1977 Alignment and aggregation effects in suspensions of spheres in non-Newtonian media. Rheol. Acta 16, 317.Google Scholar
Petit, L. & Noetinger, B. 1988 Shear-induced structures in macroscopic dispersions. Rheol. Acta 27, 437.Google Scholar
Phillips, R. J., Brady, J. F. & Bossis, G. 1988a Hydrodynamic transport properties of hard-sphere dispersions. I. Suspensions of freely mobile particles. Phys. Fluids 31, 3462.Google Scholar
Phillips, R. J., Brady, J. F. & Bossis, G. 1988b Hydrodynamic transport properties of hard-sphere dispersions. II. Porous media. Phys. Fluids 31, 3473.Google Scholar
Riddle, M. J., Narvaez, C. & Bird, R. B. 1977 Interactions between two spheres falling along their line of centers in a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 2, 23.Google Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.
Satrape, J. V. & Crochet, M. J. 1994 Numerical simulation of the motion of a sphere in a Boger fluid. J. Non-Newtonian Fluid Mech. 55, 91.Google Scholar
Weinbaum, S., Ganatos, P. & Zong-Yi, Y. 1990 Numerical multipole and boundary integral equation techniques in Stokes flow. Ann. Rev. Fluid Mech. 22, 275.Google Scholar