Abstract
Creeping flow through an array of spheres with small volume fraction φ is studied theoretically. It is observed that it can be described macroscopically by Brinkman's equation. A generalized version of the reciprocity relations is used to determine the viscous term up to O(φ2) for the case of random configuration and up to O(φ3) for the case of periodic, cubic configurations of the fixed bed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Sanchez-Palencia, Comportement local et macroscopique d'un type de milieux physiques hétérogènes. Intl. J. Engng. Sci. 12 (1974) 331-351.
J.B. Keller, Darcy's law for flows in porous media and the two-space method. In: R.L. Sternberg, A.J. Kalinowski and J.S. Papadakis (eds.) Nonlinear partial differential equations in engineering and applied science. New York: Marcel Dekker (1980) pp. 429-443.
P.M. Adler, Porous media. Boston: Butterworth-Heinemann (1992) 544 pp.
P.G. Saffman, On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50 (1971) 93-101.
S. Haber and R. Mauri, Boundary conditions for Darcy's flow through porous media. Int. J. Multiphase Flow 9 (1983) 561-574.
S. Kim and W.B. Russel, Modelling of porous media by renormalization of the Stokes equations. J. Fluid Mech. 154 (1985) 269-286.
G.S. Beavers and D.C. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967) 197-207.
E.J. Hinch, An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83 (1977) 695-720.
J. Rubinstein, Effective equations for flow in random porous media with a large number of scales. J. Fluid Mech. 170 (1986) 379-383.
J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres. J. Stat. Phys. 44 (1986) 849-863.
C.K.W. Tam, The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38 (1969) 537-546.
L. Durlofsky and J.F. Brady, Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30 (1987) 3329-3341.
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. Amsterdam: North-Holland (1978) 420 pp.
E. Sanchez-Palencia, Nonhomogeneous media and vibration theory. Berlin: Springer-Verlag (1980) 326 pp.
R. Mauri, Dispersion, convection and reaction in porous media. Phys. Fluids A 3 (1991) 743-756.
R. Mauri, Heat and mass transport in random velocity fields with application to dispersion in porous media. J. Eng. Math. 29 (1995) 77-89.
T.S. Lundgren, Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51 (1972) 273-299.
K.F. Freed and M. Muthukumar, On the Stokes problem for a suspension of spheres at finite concentrations. J. Chem. Phys. 68 (1978) 2088-2096.
S. Childress, Viscous flow past a random array of spheres. J. Chem. Phys. 56 (1972) 2527-2539.
I.D. Howells, Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64 (1974) 449-475.
H. Hasimoto, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (1959) 317-328.
A.A Zick and G.M. Homsy, Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115 (1982) 13-26.
A.S Sangani and A. Acrivos, Slow flow through a periodic array of spheres. Int. J. Multiphase Flow 8 (1982) 343-360.
M. Zuzovski, P.M. Adler and H. Brenner, Spatially periodic suspensions of convex particles in linear shear flows. Part III. Phys. Fluids A 26 (1983) 1714-1723.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mauri, R. A new application of the reciprocity relations to the study of fluid flows through fixed beds. Journal of Engineering Mathematics 33, 103–112 (1998). https://doi.org/10.1023/A:1004299402988
Issue Date:
DOI: https://doi.org/10.1023/A:1004299402988