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Near-contact approach of two permeable spheres

Published online by Cambridge University Press:  19 August 2021

Rodrigo B. Reboucas Open the ORCID record for Rodrigo B. Reboucas [Opens in a new window]  and
Michael Loewenberg Open the ORCID record for Michael Loewenberg [Opens in a new window]
Rodrigo B. Reboucas
Affiliation:
Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520-8286, USA
Michael Loewenberg*
Affiliation:
Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520-8286, USA
*
†Email address for correspondence: [email protected]
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Abstract

An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit $K=k/a^{2} \ll 1$ is considered, where $k$ is the arithmetic mean permeability, and $a^{-1}=a^{-1}_{1}+a^{-1}_{2}$ is the reduced radius, and $a_1$ and $a_2$ are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact $F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$, where $\mu$ is the fluid viscosity, $W$ is the relative velocity and $\tilde {f}_c$ depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions, $\tilde {f}_c=0.7507$; for a permeable particle in near contact with a spherical drop, $\tilde {f}_c$ is reduced by a factor of $2^{-6/5}$.

JFM classification

Low-Reynolds-number flows: Porous media Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A1
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Auriault, J.L. 2009 On the domain of validity of Brinkman's equation. Transp. Porous Media 79, 215–223.CrossRefGoogle Scholar
Barnocky, G. & Davis, R.H. 1989 The lubrication force between spherical drops, bubbles and rigid particles in a viscous fluid. Intl J. Multiphase Flow 15, 627–638.CrossRefGoogle Scholar
Bars, M.L. & Woster, M.G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.CrossRefGoogle Scholar
Bäbler, M.U., Sefcik, J., Morbidelli, M. & Baldyga, J. 2006 Hydrodynamic interactions and orthokinetic collisions of porous aggregates in the Stokes regime. Phys. Fluids 18, 013302.CrossRefGoogle Scholar
Beavers, G.S. & Joseph, D.D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.CrossRefGoogle Scholar
Belfort, G., Davis, R.H. & Zydney, A.L. 1994 The behavior of suspensions and macromolecular solutions in crossflow microfiltration. J. Membr. Sci. 96, 1–58.CrossRefGoogle Scholar
Blawzdziewicz, J., Wajnryb, E. & Loewenberg, M. 1999 Hydrodynamic interactions and collision efficiencies of spherical drops covered with an incompressible surfactant film. J. Fluid Mech. 395, 29–59.CrossRefGoogle Scholar
Blue, L.E. & Jorgenson, J.W. 2015 $1.1\ \mathrm {\mu }\textrm {m}$ superficially porous particles for liquid chromatography: part II: column packing and chromatographic performance. J. Chromatogr. 1380, 71–80.CrossRefGoogle Scholar
Burganos, V.N., Michalopoulou, A.C., Dassios, G. & Payatakes, A.C. 1992 Creeping flow around and through a permeable sphere moving with constant velocity towards a solid wall: a revision. Chem. Engng Commun. 117, 85–88.CrossRefGoogle Scholar
Cao, Y., Gunzburger, M., Hua, F. & Wang, X. 2010 Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8, 1–25.CrossRefGoogle Scholar
Chen, S.B. 1998 Axisymmetric motion of multiple composite spheres: solid core with permeable shell, under creeping flow conditions. Phys. Fluids 10, 1550–1563.CrossRefGoogle Scholar
Chen, S.B. & Cai, A. 1999 Hydrodynamic interactions and mean settling velocity of porous particles in a dilute suspension. J. Colloid Interface Sci. 217, 328–340.CrossRefGoogle Scholar
Civan, F. 2007 Reservoir Formation Damage, 2nd edn, chap. 18. Elsevier Inc.Google Scholar
Da Cunha, F.R. & Hinch, E.J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211–223.CrossRefGoogle Scholar
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Dalmont.Google Scholar
Davis, A.M.J. 2001 Axisymmetric flow due to a porous sphere sedimenting towards a solid sphere or a solid wall: application to scavanging of small particles. Phys. Fluids 13, 3126–3133.CrossRefGoogle Scholar
Davis, R.H. & Stone, H.A. 1993 Flow through beds of porous particles. Chem. Engng Sci. 48 (23), 3993–4005.CrossRefGoogle Scholar
Debbech, A., Elasmi, L. & Feuillebois, F. 2010 The method of fundamental solution for the creeping flow around a sphere close to a membrane. Z. Angew. Math. Mech. 90 (12), 920–928.CrossRefGoogle Scholar
Gheorghitza, I. 1963 La formule de Stokes pour les enveloppes sphériques poreuses. Arch. Rat. Mech. Anal. 12, 52–57.CrossRefGoogle Scholar
Goren, S.L. 1979 The hydrodynamic force resisting the approach of a sphere to a plane permeable wall. J. Colloid Interface Sci. 69, 78–85.CrossRefGoogle Scholar
Hocking, L.M. 1973 The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres. J. Engng Maths 7, 207–221.CrossRefGoogle Scholar
Hwang, K.J. & Sz, P.Y. 2011 Membrane fouling mechanism and concentration effect in cross-flow microfiltration of BSA/dextran mixtures. Chem. Engng J. 166, 669–677.CrossRefGoogle Scholar
James, D.F. & Davis, A.M.J. 2001 Flow at the interface of a model fibrous porous medium. J. Fluid Mech. 426, 47–72.CrossRefGoogle Scholar
Jenkins, J.T. & Koenders, M.A. 2005 Hydrodynamic interaction of rough spheres. Granul. Matt. 7, 13–18.CrossRefGoogle Scholar
Jones, R.B. 1978 Hydrodynamic interactions of two permeable spheres I: the method of reflections. Physica A 92, 545–556.CrossRefGoogle Scholar
Joseph, D.D. & Tao, L.N. 1964 The effect of permeability on the slow motion of a porous sphere in a viscous fluid. Z. Angew. Math. Mech. 44, 361–364.CrossRefGoogle Scholar
Khabthani, S., Sellier, A. & Feuillebois, F. 2019 Lubricating motion of a sphere towards a thin porous slab with Saffman slip condition. J. Fluid Mech. 867, 949–968.CrossRefGoogle Scholar
Knox, D.J., Duffy, B.R., McKee, S. & Wilson, S.K. 2017 Squeeze-film flow between a curved impermeable bearing and a flat porous bed. Phys. Fluids 29, 023101.CrossRefGoogle Scholar
Le-Clech, P., Chen, V. & Fane, T.A.G. 2006 Fouling in membrane bioreactors used in wastewater treatment. J. Membr. Sci. 284, 17–53.CrossRefGoogle Scholar
Liapis, A.I. & McCoy, M.A. 1994 Perfusion chromatography: effect of micropore diffusion on column performance in systems utilizing perfusive adsorbent particles with a bidisperse porous structure. J. Chromatogr. 660 (1), 85–96, 17th International Symposium on Column Liquid Chromatography.CrossRefGoogle Scholar
Michalopoulou, A.C., Burganos, V.N. & Payatakes, A.C. 1992 Creeping axisymmetric flow around a solid particle near a permeable obstacle. AIChE J. 38, 1213–1228.CrossRefGoogle Scholar
Michalopoulou, A.C., Burganos, V.N. & Payatakes, A.C. 1993 Hydrodynamic interactions of two permeable particles moving slowly along their centerline. Chem. Engng Sci. 48, 2889–2900.CrossRefGoogle Scholar
Neale, G. & Nader, W. 1974 Practical significance of Brinkman's extension of Darcy's law: coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Engng 52, 475–478.CrossRefGoogle Scholar
Nield, D.A. & Bejan, A. 2013 Convection in Porous Media, 4th edn. Springer.CrossRefGoogle Scholar
Nir, A. 1981 On the departure of a sphere from contact with a permeable membrane. J. Engng Maths 15, 65–75.CrossRefGoogle Scholar
Ochoa-Tapia, J.A. & Whitaker, S. 1995 Momentum transfer at the boundary between a porous medium and a homogeneous fluid – I. Theoretical development. Intl J. Heat Mass Transfer 38, 2635–2646.CrossRefGoogle Scholar
O'Neill, M.E. & Bhatt, D.S. 1991 Slow motion of a solid sphere in the presence of a naturally permeable surface. Q. J. Mech. Appl. Maths 44, 91–104.CrossRefGoogle Scholar
Payatakes, A.C. & Dassios, G. 1987 Creeping flow around and through a permeable sphere moving with constant velocity towards a solid wall. Chem. Engng Commun. 58, 119–138.CrossRefGoogle Scholar
Ramon, G.Z. & Hoek, E.M.V. 2012 On the enhanced drag force induced by permeation through a filtration membrane. J. Membr. Sci. 392–393, 1–8.CrossRefGoogle Scholar
Ramon, G.Z., Huppert, H.E., Lister, J.R. & Stone, H.A. 2013 On the hydrodynamic interaction between a particle and a permeable surface. Phys. Fluids 25, 073103.CrossRefGoogle Scholar
Rodrigues, A.E., Ahn, B.J. & Zoulalian, A. 1982 Intraparticle-forced convection effect in catalyst diffusivity measurements and reactor design. AIChE J. 28 (4), 541–546.CrossRefGoogle Scholar
Saffman, P.G. 1971 On the boundary condition at the surface of a porous medium. Stud. Appl. Maths 50, 93–101.CrossRefGoogle Scholar
Sherwood, J.D. 1988 The force on a sphere pulled away from a permeable half-space. Physico-Chem. Hydrodyn. 10, 3–12.Google Scholar
Smart, J.R. & Leighton, D.T. 1989 Measurement of the hydrodynamic surface roughness of noncolloidal spheres. Phys. Fluids A 1 (1), 52–60.CrossRefGoogle Scholar
Wang, J., Cahyadi, A., Wu, B., Pee, W., Fane, A.G. & Chew, J.W. 2020 The roles of particles in enhancing membrane filtration: a review. J. Membr. Sci. 595, 117570.CrossRefGoogle Scholar
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