Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T00:38:21.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Faxén formulas for particles of arbitrary shape and material composition

Published online by Cambridge University Press:  11 January 2021

Benjamin E. Dolata Open the ORCID record for Benjamin E. Dolata [Opens in a new window]  and
Roseanna N. Zia Open the ORCID record for Roseanna N. Zia [Opens in a new window]
Benjamin E. Dolata
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA94305, USA
Roseanna N. Zia*
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA94305, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a duality between the functional forms of the Faxén formulas associated with a particle of a given shape and material composition and the corresponding singularity solutions for the velocity disturbances induced by that particle, and extend it to the case of systems with coupled transport processes, enabling the solution of a large family of problems via Faxén methods. Prior approaches to constructing proofs of duality of Faxén formulas and Stokes-flow singularities relied on knowledge of all boundary conditions on all particle surfaces, viz. the Lorentz reciprocal theorem approach. We recognized that, in order to bypass the complexity of boundary conditions one can instead invoke energy methods that give reciprocity between operators rather than between specific stress and velocity fields. We derive reciprocal relations between operators, from which we demonstrate that the Faxén/singularity duality is a consequence of a generalized reciprocal relation between conjugate thermodynamic variables. We use our reciprocal relations to derive expressions for the hydrodynamic force on a sphere of arbitrary composition, the hydrodynamic stresslet exerted on a deformable droplet in an arbitrary velocity field, the phoretic force exerted on a rigid particle in the thin double-layer limit in response to arbitrary externally imposed field and the total stresslet on a charged particle in an arbitrary velocity field, i.e. an electroviscous Faxén law.

JFM classification

Low-Reynolds-number flows: Stokesian dynamics Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A22
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aponte-Rivera, C. & Zia, R. N. 2016 Simulation of hydrodynamically interacting particles confined by a spherical cavity. Phys. Rev. Fluids 1 (2), 023301.CrossRefGoogle Scholar
Bafaluy, J., Pagonabarraga, I., Rubi, J. M. & Bedeaux, D. 1995 Thermocapillary motion of a drop in a fluid under external gradients. Faxén theorem. Physica A 213 (3), 277–292.CrossRefGoogle Scholar
Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.CrossRefGoogle Scholar
Batchelor, G. K. & Green, J. T. 1972 The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (2), 375–400.CrossRefGoogle Scholar
Beris, A. N. & Edwards, B. J. 1994 Thermodynamics of Flowing Systems: With Internal Microstructure. Oxford University Press.CrossRefGoogle Scholar
Bike, S. G. & Prieve, D. C. 1992 Electrohydrodynamics of thin double layers: a model for the streaming potential profile. J. Colloid Interface Sci. 154 (1), 87–96.CrossRefGoogle Scholar
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.CrossRefGoogle Scholar
Bonnecaze, R. T. & Brady, J. F. 1990 A method for determining the effective conductivity of dispersions of particles. Proc. R. Soc. Lond. A 430 (1879), 285–313.Google Scholar
Brunet, E. & Ajdari, A. 2004 Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries. Phys. Rev. E 69 (1), 016306.CrossRefGoogle ScholarPubMed
Casimir, H. B. G. 1945 On Onsager's principle of microscopic reversibility. Rev. Mod. Phys. 17 (2–3), 343.CrossRefGoogle Scholar
Chen, S. B. & Ye, X. 2000 Faxen's laws of a composite sphere under creeping flow conditions. J. Colloid Interface Sci. 221 (1), 50–57.CrossRefGoogle ScholarPubMed
Danov, K. D. 2001 On the viscosity of dilute emulsions. J. Colloid Interface Sci. 235 (1), 144–149.CrossRefGoogle ScholarPubMed
Dolata, B. E. & Zia, R. N. 2020 Heterogeneous dispersions as microcontinuum fluids. J. Fluid Mech. 888, A28.CrossRefGoogle Scholar
Faxén, H. 1922 Der widerstand gegen die bewegung einer starren kugel in einer zähen flüssigkeit, die zwischen zwei parallelen ebenen wänden eingeschlossen ist. Ann. Phys. 373 (10), 89–119.CrossRefGoogle Scholar
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44 (1), 65–78.CrossRefGoogle Scholar
Haj-Hariri, H., Nadim, A. & Borhan, A. 1993 Recriprocal theorem for concentric compound drops in arbitrary stokes flows. J. Fluid Mech. 252, 265–277.CrossRefGoogle Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83 (4), 695–720.CrossRefGoogle Scholar
Jackson, N. E. & Tucker, C. L. 2003 A model for large deformation of an ellipsoidal droplet with interfacial tension. J. Rheol. 47 (3), 659–682.CrossRefGoogle Scholar
Jeffrey, D. J., Morris, J. F. & Brady, J. F. 1993 The pressure moments for two rigid spheres in low-Reynolds-number flow. Phys. Fluids A 5 (10), 2317–2325.CrossRefGoogle Scholar
Kim, S. 1985 A note on Faxen laws for nonspherical particles. Intl J. Multiphase Flow 11 (5), 713–719.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
Kim, S. & Lu, S.-Y. 1987 The functional similarity between Faxén relations and singularity solutions for fluid-fluid, fluid-solid and solid-solid dispersions. Intl J. Multiphase Flow 13 (6), 837–844.CrossRefGoogle Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow, vol. 2. Gordon and Breach.Google Scholar
Liron, N. & Mochon, S. 1976 Stokes flow for a Stokeslet between two parallel flat plates. J. Engng Maths 10, 287–303.CrossRefGoogle Scholar
Masoud, H., Stone, H. A. & Shelley, M. J. 2013 On the rotation of porous ellipsoids in simple shear flows. J. Fluid Mech. 733, R6.CrossRefGoogle Scholar
Mohan, A. & Brenner, H. 2005 An extension of Faxen's laws for nonisothermal flow around a sphere. Phys. Fluids 17 (3), 038107.CrossRefGoogle Scholar
Morrison, F. A. Jr. 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34 (2), 210–214.CrossRefGoogle Scholar
Nadim, A. 1996 A concise introduction to surface rheology with application to dilute emulsions of viscous drops. Chem. Engng Commun. 148 (1), 391–407.CrossRefGoogle Scholar
Ohshima, H. 2006 Primary electroviscous effect in a dilute suspension of charged mercury drops. Langmuir 22 (6), 2863–2869.CrossRefGoogle Scholar
Onsager, L. 1931 a Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (4), 405.CrossRefGoogle Scholar
Onsager, L. 1931 b Reciprocal relations in irreversible processes. II. Phys. Rev. 38 (12), 2265.CrossRefGoogle Scholar
Oseen, C. W. 1927 Neuere Methoden und Ergebnisse in der Hydrodynamik. Akademische Verlagsgesellschaft mbH.Google Scholar
Padmavathi, B. S., Amaranath, T. & Nigam, S. D. 1993 Stokes flow past a porous sphere using Brinkman's model. Z. Angew. Math. Phys. 44 (5), 929–939.CrossRefGoogle Scholar
Palaniappan, D. 1993 Arbitrary Stokes flow past a porous sphere. Mech. Res. Commun. 20 (4), 309–317.CrossRefGoogle Scholar
Rallison, J. M. 1978 Note on the Faxén relations for a particle in Stokes flow. J. Fluid Mech. 88 (3), 529–533.CrossRefGoogle Scholar
Smoluchowski, M. 1903 Contribution to the theory of electro-osmosis and related phenomena. Bull. Intl Acad. Sci. Cracovie 3, 184–199.Google Scholar
Sri Padmavati, B. & Amaranath, T. 2004 Stokes flow past spherical boundaries. Bull. Marathwada Math. Soc. 5 (1), 8–25.Google Scholar
Swan, J. W. & Brady, J. F. 2010 Particle motion between parallel walls: hydrodynamics and simulation. Phys. Fluids 22, 103301.CrossRefGoogle Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501–523.Google Scholar
Wajnryb, E., Mizerski, K. A., Zuk, P. J. & Szymczak, P. 2013 Generalization of the Rotne–Prager–Yamakawa mobility and shear disturbance tensors. J. Fluid Mech. 731.CrossRefGoogle Scholar
Wetzel, E. D. & Tucker, C. L. 1999 Area tensors for modeling microstructure during laminar liquid-liquid mixing. Intl J. Multiphase Flow 25 (1), 35–61.CrossRefGoogle Scholar
Wetzel, E. D. & Tucker, C. L. 2001 Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J. Fluid Mech. 426, 199–228.CrossRefGoogle Scholar
Yan, W. & Brady, J. F. 2016 The behavior of active diffusiophoretic suspensions: an accelerated Laplacian dynamics study. J. Chem. Phys. 145 (13), 134902.CrossRefGoogle ScholarPubMed
Yu, W. & Bousmina, M. 2003 Ellipsoidal model for droplet deformation in emulsions. J. Rheol. 47 (4), 1011–1039.CrossRefGoogle Scholar