# Average pressure and velocity fields in non-uniform suspensions of spheres in Stokes flow

## Abstract

A widely used method for the approximate numerical simulation of the bulk behavior of particle suspensions consists in filling the entire space with copies of a fundamental cell in which *N* particles are arranged according to some probability distribution. Until now this method has only been used for suspensions that are spatially uniform in the mean. The case of spatially non-uniform systems, on the other hand, has not been considered. Here the average velocity and pressure fields for such a non-uniform suspension of identical rigid spheres in Stokes flow are calculated, and analytic solutions expressed in terms of multipole coefficients are presented. The results match and extend others obtained by the authors in parallel work using a completely different approach. In particular, the definition of a quantity to be identified with the mixture pressure is fully supported by the present results. An explicit result for the structure of the viscous stress in the suspension is also found. It is shown that, for spatially non-uniform systems, the stress contains a non-symmetric contribution analogous to a baroclinic source of vorticity.

As a byproduct of the analysis, certain integrals of two periodic functions introduced by Hasimoto are calculated. These integrals would arise in similar problems, *e.g.* the electric field produced by electric multipoles in a periodic cubic structure.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.G.K. Batchelor, Sedimentation in a dilute dispersion of spheres.
*J. Fluid Mech.*52 (1972) 245–268.Google Scholar - 2.G.K. Batchelor and J.T. Green, The determination of the bulk stress in a suspension of spherical particles to order
*c*^{2}.*J. Fluid Mech.*56 (1972) 401–427.Google Scholar - 3.H. Brenner, Rheology of a dilute suspension of dipolar spherical particles in an external field.
*J. Colloid Interface Sci.*32 (1970) 141–158.Google Scholar - 4.H. Brenner, Antisymmetric stress induced by the rigid-body rotation of dipolar suspensions.
*Int. J. Engng. Sci.*22 (1984) 645–682.Google Scholar - 5.A.J.C. Ladd, Hydrodynamic interactions in a suspension of spherical particles.
*J. Chem. Phys.*88 (1988) 5051–5063.Google Scholar - 6.A.J.C. Ladd, Dynamical simulations of sedimenting spheres.
*Phys. Fluids*5 (1993) 299–310.Google Scholar - 7.A.J.C. Ladd, Sedimentation of homogeneous suspensions of non-Brownian spheres.
*Phys. Fluids*9 (1997) 491–499.Google Scholar - 8.G. Mo and A.S. Sangani, A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles.
*Phys. Fluids*6 (1994) 1637–1652.Google Scholar - 9.A.S. Sangani and G.B. Mo, An
*O(N)*for Stokes and Laplace interactions of spheres.*Phys. Fluids*8 (1996) 1990–2010.Google Scholar - 10.R.H. Davis and A. Acrivos, Sedimentation of non-colloidal particles at low Reynolds numbers.
*Ann. Rev. Fluid Mech.*17 (1985) 91–118.Google Scholar - 11.A. Acrivos, G.K. Batchelor, E.J. Hinch, D.L. Koch and R. Mauri, Longitudinal shear-induced diffusion of spheres in a dilute suspension.
*J. Fluid Mech.*240 (1992) 651–657.Google Scholar - 12.B. Cichocki, B.U. Felderhof, K. Hinsen, E. Wajnryb and J. Blawzdziewicz, Friction and mobility of many spheres in Stokes flow.
*J. Chem. Phys.*100 (1994) 3780–3790.Google Scholar - 13.M. Zuzovsky, P.M. Adler and H. Brenner, Spatially periodic suspensions of convex particles in linear shear flows. III. Dilute arrays of spheres suspended in Newtonian fluids.
*Phys. Fluids*26 (1983) 1714–1723.Google Scholar - 14.A. Acrivos, Shear-induced particle diffusion in concentrated suspensions of non-colloidal particles.
*J. Rheol.*39 (1995) 813–826.Google Scholar - 15.A.S. Sangani and C. Yao, Bulk conductivity of composites with spherical inclusions.
*J. Appl. Phys.*63 (1988) 1334–1341.Google Scholar - 16.M. Marchioro, M. Tanksley and A. Prosperetti, Flow of spatially non-uniform suspensions. Part I: Phenomenology.
*Int. J. Multiphase Flow*26 (1999) 783–831.Google Scholar - 17.M. Marchioro, M. Tanksley and A. Prosperetti, Flow of spatially non-uniform suspensions. Part II: An approach to the closure of the averaged equations.
*Int. J. Multiphase Flow*27 (2001) 237–276.Google Scholar - 18.W. Wang and A. Prosperetti, Flow of spatially non-uniform suspensions. Part III: Closure relations for porous media and spinning particles.
*Int. J. Multiphase Flow*, submitted (2001)Google Scholar - 19.M. Marchioro and A. Prosperetti, Conduction in non-uniform composites.
*Proc. R. Soc. London*A455 (1999) 1483–1508.Google Scholar - 20.M. Marchioro, M. Tanksley and A. Prosperetti, Mixture pressure and stress in diperse two-phase flow.
*Int. J. Multiphase Flow*25 (1999) 1395–1429.Google Scholar - 21.F. Feuillebois, Sedimentation in a dispersion with vertical inhomogeneities.
*J. Fluid Fluids*139 (1984) 145–171.Google Scholar - 22.D. Lhuillier, Ensemble averaging in slightly non-uniform suspensions.
*Eur. J. Mech. B/Fluids*11 (1992) 649–661.Google Scholar - 23.J.W. Park, D.A. Drew and R.T. Jr. Lahey, The analysis of void wave propagation in adiabatic monodispersed bubbly two-phase flows using an ensemble-averaged two-fluid model.
*Int. J. Multiphase Flow*24 (1998) 1205–1244.Google Scholar - 24.J.P. Hansen, Molecular-dynamics simulations of Coulomb systems in two and three dimensions. In: G. Ciccotti and W.G. Hoover (eds.),
*Molecular-Dynamics Simulation of Statistical-Mechanical Systems*Amsterdam: North-Holland (1986) pp. 89–129.Google Scholar - 25.J. Koplik and J.R. Banavar, Continuum deductions from molecular simulations.
*Ann. Rev. Fluid Mech.*27 (1995) 257–292.Google Scholar - 26.C.S. Campbell and C.E. Brennen, Computer simulations of granular shear flows.
*J. Fluid Mech.*151 (1985) 167–188.Google Scholar - 27.A.J.C. Ladd, Hydrodynamic transport coefficients of random dispersions of hard spheres.
*J. Chem. Phys.*93 (1990) 3484–3494.Google Scholar - 28.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.Google Scholar - 29.
- 30.S. Kim and S.J. Karrila,
*Microhydrodynamics*. Boston: Butterworth-Heinemann (1991) 507 pp.Google Scholar - 31.A. Prosperetti and A.V. Jones, Pressure forces in disperse two-phase flows.
*Int. J. Multiphase Flow*10 (1984) 425–440.Google Scholar - 32.R.C. Givler, An interpretation of the solid phase pressure in slow fluid-particle flows.
*Int. J. Multiphase flow*13 (1993) 717–722.Google Scholar - 33.D.A. Drew and R.T. Jr. Lahey, Analytical modeling of multiphase flow. In: Roco M.C. (ed.),
*Particulate Two-Phase Flow*Boston: Butterworth-Heinemann (1993) pp. 509–566.Google Scholar - 34.G.J. Hwang and H.H. Shen, Modeling the solid phase stress in a fluid-solid mixture.
*Int. J. Multiphase Flow*15 (1989) 257–268.Google Scholar - 35.J. Bouré, On the form of the pressure terms in the momentum and energy equations for two-phase flow models.
*Int. J. Multiphase Flow*5 (1979) 159–164.Google Scholar - 36.T.B. Anderson and R. Jackson, A fluid mechanical description of fluidized beds.
*I. & EC Fundamentals*6 (1967) 527–539.Google Scholar - 37.M. Ishii,
*Thermo-Fluid Dynamic Theory of Two-Phase Flow*. Paris: Eyrolles (1975) 248 pp.Google Scholar - 38.D.A. Drew, Mathematical modeling of two-phase flow.
*Ann. Rev. Fluid Mech.*15 (1983) 261–291.Google Scholar - 39.G.S. Arnold, D.A. Drew and R.T. Jr. Lahey, Derivation of constitutive equations for interfacial force and Reynolds stress for a suspension of spheres using ensemble cell averaging.
*Chem. Eng. Comm.*86 (1989) 43–54.Google Scholar - 40.G.K. Batchelor, The stress system in a suspension of force-free particles.
*J. Fluid Mech.*41 (1970) 545–570.Google Scholar - 41.D.Z. Zhang and A. Prosperetti, Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions.
*Int. J. Multiphase Flow*23 (1997) 425–453.Google Scholar - 42.E.W. Hobson,
*The Theory of Spherical and Ellipsoidal Harmonics*. Cambridge: C.U.P. (1931) reprinted by Chelsea, New York (1965) 500 pp.Google Scholar