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The reciprocal theorem in fluid dynamics and transport phenomena

  • Hassan Masoud (a1) and Howard A. Stone (a2)

Abstract

In the study of fluid dynamics and transport phenomena, key quantities of interest are often the force and torque on objects and total rate of heat/mass transfer from them. Conventionally, these integrated quantities are determined by first solving the governing equations for the detailed distribution of the field variables (i.e. velocity, pressure, temperature, concentration, etc.) and then integrating the variables or their derivatives on the surface of the objects. On the other hand, the divergence form of the conservation equations opens the door for establishing integral identities that can be used for directly calculating the integrated quantities without requiring the detailed knowledge of the distribution of the primary variables. This shortcut approach constitutes the idea of the reciprocal theorem, whose closest relative is Green’s second identity, which readers may recall from studies of partial differential equations. Despite its importance and practicality, the theorem may not be so familiar to many in the research community. Ironically, some believe that the extreme simplicity and generality of the theorem are responsible for suppressing its application! In this Perspectives piece, we provide a pedagogical introduction to the concept and application of the reciprocal theorem, with the hope of facilitating its use. Specifically, a brief history on the development of the theorem is given as a background, followed by the discussion of the main ideas in the context of elementary boundary-value problems. After that, we demonstrate how the reciprocal theorem can be utilized to solve fundamental problems in low-Reynolds-number hydrodynamics, aerodynamics, acoustics and heat/mass transfer, including convection. Throughout the article, we strive to make the materials accessible to early career researchers while keeping it interesting for more experienced scientists and engineers.

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Corresponding author

Email addresses for correspondence: hmasoud@mtu.edu, hastone@princeton.edu

References

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Achenbach, J. D. 2002 Use of elastodynamic reciprocity theorems for field calculations. In Integral Methods in Science and Engineering (ed. Schiavone, P., Constanda, C. & Mioduchowski, A.), pp. 114. Birkhäuser.
Achenbach, J. D. 2003 Reciprocity in Elastodynamics. Cambridge University Press.
Achenbach, J. D. 2014 A new use of the elastodynamic reciprocity theorem. Math. Mech. Solids 19 (1), 518.
Acree, W. E. 1984 Empirical expression for predicting surface-tension of liquid-mixtures. J. Colloid Interface Sci. 101, 575576.
Acrivos, A. 2015 Reflections on a rheologist: Howard Brenner (1929–2014). Rheol. Bull. 84 (1), 811.
Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer from single spheres in Stokes flow. Phys. Fluids 5 (4), 387394.
Adamson, A. W. & Gast, A. P. 1997 Physical Chemistry of Surfaces. Wiley.
Ajdari, A. & Stone, H. A. 1999 A note on swimming using internally generated traveling waves. Phys. Fluids 11 (5), 12751277.
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21 (1), 6199.
Barber, J. R. 2002 Elasticity. Springer.
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545570.
Becker, L. E., McKinley, G. H. & Stone, H. A. 1996 Sedimentation of a sphere near a plane wall: weak non-Newtonian and inertial effects. J. Non-Newtonian Fluid Mech. 63 (2), 201233.
Bell, C. G., Byrne, H. M., Whiteley, J. P. & Waters, S. L. 2014 Heat or mass transfer at low Péclet number for Brinkman and Darcy flow round a sphere. Intl J. Heat Mass Transfer 68, 247258.
Betti, E. 1872 Teoria della elasticità. Il Nuovo Cimento 7 (1), 6997.
Brady, J. F. 2011 Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives. J. Fluid Mech. 667, 216259.
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20 (1), 111157.
Brenner, H. 1958 Dissipation of energy due to solid particles suspended in a viscous liquid. Phys. Fluids 1 (4), 338346.
Brenner, H. 1961 The Oseen resistance of a particle of arbitrary shape. J. Fluid Mech. 11 (4), 604610.
Brenner, H. 1962 Effect of finite boundaries on the Stokes resistance of an arbitrary particle. J. Fluid Mech. 12 (1), 3548.
Brenner, H. 1963a Forced convection heat and mass transfer at small Péclet numbers from a particle of arbitrary shape. Chem. Engng Sci. 18 (2), 109122.
Brenner, H. 1963b The Stokes resistance of an arbitrary particle. Chem. Engng Sci. 18 (1), 125.
Brenner, H. 1964a The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19 (8), 519539.
Brenner, H. 1964b The Stokes resistance of an arbitrary particle. IV: Arbitrary fields of flow. Chem. Engng Sci. 19 (10), 703727.
Brenner, H. 1967 On the invariance of the heat-transfer coefficient to flow reversal in Stokes and potential streaming flows past particles of arbitrary shape. J. Math. Phys. Sci. 1, 173179.
Brenner, H. 1970a Invariance of the overall mass transfer coefficient to flow reversal during Stokes flow past one or more particles of arbitrary shape. Chem. Engng Prog. Symp. Ser. 66, 123126.
Brenner, H. 1970b Pressure drop due to the motion of neutrally buoyant particles in duct flows. J. Fluid Mech. 43 (4), 641660.
Brenner, H. 1971 Pressure drop due to the motion of neutrally buoyant particles in duct flows. II. Spherical droplets and bubbles. Ind. Engng Chem. Fundam. 10 (4), 537543.
Brenner, H. & Cox, R. G. 1963 The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17 (4), 561595.
Brenner, H. & Haber, S. 1984 Symbolic operator solutions of Laplace’s and Stokes’ equations Part 1. Laplace’s equation. Chem. Engng Commun. 27 (5–6), 283295.
Brenner, H. & Nadim, A. 1996 The Lorentz reciprocal theorem for micropolar fluids. In The Centenary of a Paper on Slow Viscous Flow by the Physicist H. A. Lorentz, pp. 169176. Springer.
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 2734.
Brinkman, H. C. 1948 On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 8186.
Brunet, E. & Ajdari, A. 2004 Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries. Phys. Rev. E 69 (1), 016306.
Brunn, P. 1976a The behavior of a sphere in non-homogeneous flows of a viscoelastic fluid. Rheol. Acta 15 (11-12), 589611.
Brunn, P. 1976b The slow motion of a sphere in a second-order fluid. Rheol. Acta 15 (3–4), 163171.
Brunn, P. 1980 The motion of rigid particles in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 7 (4), 271288.
Bungay, P. M. & Brenner, H. 1973 Pressure drop due to the motion of a sphere near the wall bounding a Poiseuille flow. J. Fluid Mech. 60 (1), 8196.
Candelier, F., Einarsson, J. & Mehlig, B. 2016 Angular dynamics of a small particle in turbulence. Phys. Rev. Lett. 117 (20), 204501.
Carrier, G. F.1953 On slow viscous flow. Tech. Rep. Final Report, Office of Naval Research Contract Nonr-653 (00).
Caswell, B. 1972 The stability of particle motion near a wall in Newtonian and non-Newtonian fluids. Chem. Engng Sci. 27 (2), 373389.
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92 (1), 131170.
Charlton, T. M. 1960 A historical note on the reciprocal theorem and theory of statically indeterminate frameworks. Nature 187 (4733), 231.
Clebsch, R. F. A. 1862 Theorie der Elasticität fester Körper. B. G. Teubner.
Cox, R. G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow – I theory. Chem. Engng Sci. 23 (2), 147173.
Crowdy, D. G. 2013 Wall effects on self-diffusiophoretic Janus particles: a theoretical study. J. Fluid Mech. 735, 473498.
Davis, A. M. J. 1990 Stokes drag on a disk sedimenting toward a plane or with other disks; additional effects of a side wall or free-surface. Phys. Fluids 2, 301312.
Day, R. F. & Stone, H. A. 2000 Lubrication analysis and boundary integral simulations of a viscous micropump. J. Fluid Mech. 416, 197216.
De Hoop, A. T. 1995 Handbook of Radiation and Scattering of Waves. Academic Press.
Debye, P. & Bueche, A. M. 1948 Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J. Chem. Phys. 16, 573579.
Dörr, A., Hardt, S., Masoud, H. & Stone, H. A. 2016 Drag and diffusion coefficients of a spherical particle attached to a fluid–fluid interface. J. Fluid Mech. 790, 607618.
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 33293341.
Elfring, G. J. & Lauga, E. 2015 Theory of locomotion through complex fluids. In Complex Fluids in Biological Systems (ed. Spagnolie, S. E.), chap. 8, pp. 283317. Springer.
Elfring, G. J. 2015 A note on the reciprocal theorem for the swimming of simple bodies. Phys. Fluids 27 (2), 023101.
Elfring, G. J. 2017 Force moments of an active particle in a complex fluid. J. Fluid Mech. 829, R3.
Elfring, G. J. & Goyal, G. 2016 The effect of gait on swimming in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 234, 814.
Elfring, G. J., Leal, L. G. & Squires, T. M. 2016 Surface viscosity and Marangoni stresses at surfactant laden interfaces. J. Fluid Mech. 792, 712739.
Eversman, W. 2001 A reverse flow theorem and acoustic reciprocity in compressible potential flows in ducts. J. Sound Vib. 246 (1), 7195.
Fair, M. C. & Anderson, J. L. 1989 Electrophoresis of nonuniformly charged ellipsoidal particles. J. Colloid Interface Sci. 127 (2), 388400.
Felderhof, B. U. 1983 Reciprocity in electrohydrodynamics. Physica A 122 (3), 383396.
Felderhof, B. U. & Jones, R. B. 1994a Inertial effects in small-amplitude swimming of a finite body. Physica A 202 (1), 94118.
Felderhof, B. U. & Jones, R. B. 1994b Small-amplitude swimming of a sphere. Physica A 202 (1), 119144.
Flax, A. H. 1953 Reverse flow and variational theorems for lifting surfaces in non-stationary compressible flow. J. Aero. Sci. 20 (2), 120126.
Fleury, R., Sounas, D., Haberman, M. R. & Alù, A. 2015 Nonreciprocal acoustics. Acoust. Today 11 (3), 1421.
Ford, M. L. & Nadim, A. 1994 Thermocapillary migration of an attached drop on a solid surface. Phys. Fluids 6 (9), 31833185.
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1980a A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion. J. Fluid Mech. 99 (4), 755783.
Ganatos, P., Weinbaum, S. & Pfeffer, R. 1980b A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J. Fluid Mech. 99 (4), 739753.
Godin, O. A. 1997a Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25 (2), 143167.
Godin, O. A. 1997b Reciprocity relations and energy conservation for waves in the system: inhomogeneous fluid flow–anisotropic solid body. Acoust. Phys. 43, 688693.
Goldstein, R. E. 2011 Evolution of biological complexity. In Biological Physics, pp. 123139. Springer.
Golestanian, R., Liverpool, T. B. & Ajdari, A. 2007 Designing phoretic micro-and nano-swimmers. New J. Phys. 9 (5), 126.
Gonzalez-Rodriguez, D. & Lauga, E. 2009 Reciprocal locomotion of dense swimmers in Stokes flow. J. Phys.: Condens. Matter 21 (20), 204103.
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics, vol. 45. Cambridge University Press.
Haj-Hariri, H., Nadim, A. & Borhan, A. 1990 Effect of inertia on the thermocapillary velocity of a drop. J. Colloid Interface Sci. 140 (1), 277286.
Haj-Hariri, H., Nadim, A. & Borhan, A. 1993 Reciprocal theorem for concentric compound drops in arbitrary Stokes flows. J. Fluid Mech. 252, 265277.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media. Martinus Nijhoff.
Hauge, E. H. & Martin-Löf, A. 1973 Fluctuating hydrodynamics and brownian motion. J. Stat. Phys. 7, 259281.
Heaslet, M. A. & Spreiter, J. R.1953 Reciprocity relations in aerodynamics. NACA Report 1119, 253–268.
von Helmholtz, H. 1856 Handbuch der Physiologischen Optik. Leopold Voss.
von Helmholtz, H. 1887 Uber die physikalische bedeutung des prinzips der kleinsten wirkung. J. Reine Angew. Math. 100, 137166.
Higdon, J. J. L. & Kojima, M. 1981 On the calculation of Stokes flow past porous particles. Intl J. Multiphase Flow 7 (6), 719727.
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54 (3), 423425.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365400.
Ho, B. P. & Leal, L. G. 1976 Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid. J. Fluid Mech. 76 (4), 783799.
Howell, L. L. 2001 Compliant Mechanisms. John Wiley & Sons.
Hu, H. H. & Joseph, D. D. 1999 Lift on a sphere near a plane wall in a second-order fluid. J. Non-Newtonian Fluid Mech. 88 (1-2), 173184.
Jafari Kang, S., Dehdashti, E., Vandadi, V. & Masoud, H. 2019 Optimal viscous damping of vibrating porous cylinders. J. Fluid. Mech. 874, 339358.
Joseph, D. D. 1973 Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rat. Mech. Anal. 51, 295303.
Kamrin, K. & Stone, H. A. 2011 The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces. Phys. Fluids 23 (3), 031701.
Kaplun, S. 1957 Low Reynolds number flow past a circular cylinder. J. Math. Mech. 595603.
Kaplun, S. & Lagerstrom, P. A. 1957 Asymptotic expansions of Navier–Stokes solutions for small Reynolds numbers. J. Math. Mech. 6 (5), 585593.
Karrila, S. J. & Kim, S. 1989 Integral equations of the second kind for Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations. Chem. Engng Commun. 82 (1), 123161.
Khair, A. S. & Chisholm, N. G. 2014 Expansions at small Reynolds numbers for the locomotion of a spherical squirmer. Phys. Fluids 26 (1), 011902.
Khair, A. S. & Squires, T. M. 2010 Active microrheology: a proposed technique to measure normal stress coefficients of complex fluids. Phys. Rev. Lett. 105 (15), 156001.
Kim, S. 1986 The motion of ellipsoids in a second order fluid. J. Non-Newtonian Fluid Mech. 21 (2), 255269.
Kim, S. 2015 Ellipsoidal microhydrodynamics without elliptic integrals and how to get there using linear operator theory. Ind. Engng Chem. Res. 54 (42), 1049710501.
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.
Koch, D. L. & Subramanian, G. 2006 The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field. J. Non-Newtonian Fluid Mech. 138 (2-3), 8797.
Kumar, A. & Graham, M. D. 2012 Accelerated boundary integral method for multiphase flow in non-periodic geometries. J. Comput. Phys. 231 (20), 66826713.
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
Lagerstrom, P. A. & Cole, J. D. 1955 Examples illustrating expansion procedures for the Navier–Stokes equations. J. Ration. Mech. Anal. 4, 817882.
Lamb, H. 1887 On reciprocal theorems in dynamics. Proc. Lond. Math. Soc. 1 (1), 144151.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lammert, P. E., Crespi, V. H. & Nourhani, A. 2016 Bypassing slip velocity: rotational and translational velocities of autophoretic colloids in terms of surface flux. J. Fluid Mech. 802, 294304.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon Press.
Lauga, E. & Davis, A. M. J. 2012 Viscous Marangoni propulsion. J. Fluid Mech. 705, 120133.
Lauga, E. & Michelin, S. 2016 Stresslets induced by active swimmers. Phys. Rev. Lett. 117 (14), 148001.
Leal, L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69 (2), 305337.
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435476.
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.
Lee, S. H., Chadwick, R. S. & Leal, L. G. 1979 Motion of a sphere in the presence of a plane interface. Part 1. An approximate solution by generalization of the method of Lorentz. J. Fluid Mech. 93 (4), 705726.
Legendre, D. & Magnaudet, J. 1997 A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow. Phys. Fluids 9, 35723574.
Leshansky, A. M. & Brady, J. F. 2004 Force on a sphere via the generalized reciprocal theorem. Phys. Fluids 16 (3), 843844.
Lorentz, H. A. 1895 Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies (in Dutch). E. J. Brill.
Lorentz, H. A. 1896 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it (in Dutch). Versl. Konigl. Akad. Wetensch. Amst. 5, 168175.
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, 561605.
Love, A. E. H. 2013 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.
Magnaudet, J. 2003 Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow. J. Fluid Mech. 485, 115142.
Magnaudet, J. 2011a A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number. J. Fluid Mech. 689, 564604.
Magnaudet, J. 2011b A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number – CORRIGENDUM. J. Fluid Mech. 689, 605606.
Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115157.
Manga, M. & Stone, H. A. 1993 Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 256, 647683.
Masoud, H. & Stone, H. A. 2014 A reciprocal theorem for Marangoni propulsion. J. Fluid Mech. 741, R4.
Maxwell, J. C. 1864 On the calculation of the equilibrium and stiffness of frames. Phil. Mag. 27 (182), 294299.
Maxwell, J. C. 1881 A Treatise on Electricity and Magnetism. Oxford University Press.
Michaelides, E. E. & Feng, Z. 1994 Heat transfer from a rigid sphere in a nonuniform flow and temperature field. Intl J. Heat Mass Transfer 37 (14), 20692076.
Michelin, S. & Lauga, E. 2015 A reciprocal theorem for boundary-driven channel flows. Phys. Fluids 27 (11), 111701.
Morrison, F. A. & Griffiths, S. K. 1981 On the transient convective transport from a body of arbitrary shape. J. Heat Transfer 103 (1), 9295.
Mozaffari, A., Sharifi-Mood, N., Koplik, J. & Maldarelli, C. 2016 Self-diffusiophoretic colloidal propulsion near a solid boundary. Phys. Fluids 28 (5), 053107.
Munk, M. M. 1950 The reversal theorem of linearized supersonic airfoil theory. J. Appl. Phys. 21 (2), 159161.
Nadim, A., Haj-Hariri, H. & Borhan, A. 1990 Thermocapillary migration of slightly deformed droplets. Particul. Sci. Technol. 8 (3-4), 191198.
Navier, C. L. M. H. 1826 Résumé des Leçons données à l’École des Ponts et Chaussées sur l’Application de la Mécanique à l’Établissement des Constructions et des Machines, vol. 1. Didot.
Nazockdast, E., Rahimian, A., Zorin, D. & Shelley, M. 2017 A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics. J. Comput. Phys. 329, 173209.
Nourhani, A., Lammert, P. E., Crespi, V. H. & Borhan, A. 2015 A general flux-based analysis for spherical electrocatalytic nanomotors. Phys. Fluids 27 (1), 012001.
Nunan, K. C. & Keller, J. B. 1984 Effective viscosity of a periodic suspension. J. Fluid Mech. 142, 269287.
Onsager, L. 1931a Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (4), 405.
Onsager, L. 1931b Reciprocal relations in irreversible processes. II. Phys. Rev. 38 (12), 2265.
Oppenheimer, N., Navardi, S. & Stone, H. A. 2016 Motion of a hot particle in viscous fluids. Phys. Rev. Fluids 1 (1), 014001.
Oseen, C. W. 1910 Stokes formula and a related theorem in hydrodynamics. Ark. Mat. Astron. Fys. 6, 20.
Pak, O. S., Feng, J. & Stone, H. A. 2014 Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers. J. Fluid Mech. 753, 535552.
Pak, O. S., Zhu, L., Brandt, L. & Lauga, E. 2012 Micropropulsion and microrheology in complex fluids via symmetry breaking. Phys. Fluids 24 (10), 103102.
Papavassiliou, D. & Alexander, G. P. 2015 The many-body reciprocal theorem and swimmer hydrodynamics. Europhys. Lett. 110 (4), 44001.
Potton, R. J. 2004 Reciprocity in optics. Rep. Prog. Phys. 67 (5), 717.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Pozrikidis, C. 2016 Reciprocal identities and integral formulations for diffusive scalar transport and Stokes flow with position-dependent diffusivity or viscosity. J. Engng Maths 96 (1), 95114.
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2 (3), 237262.
Rallabandi, B., Yang, F. & Stone, H. A.2019 Motion of hydrodynamically interacting active particles. arXiv:1901.04311.
Rallabandi, B., Hilgenfeldt, S. & Stone, H. A. 2017a Hydrodynamic force on a sphere normal to an obstacle due to a non-uniform flow. J. Fluid Mech. 818, 407434.
Rallabandi, B., Saintyves, B., Jules, T., Salez, T., Schönecker, C., Mahadevan, L. & Stone, H. A. 2017b Rotation of an immersed cylinder sliding near a thin elastic coating. Phys. Rev. Fluids 2 (7), 074102.
Rallison, J. M. 1978 Note on the Faxén relations for a particle in Stokes flow. J. Fluid Mech. 88 (3), 529533.
Rallison, J. M. 2012 The stress in a dilute suspension of liquid spheres in a second-order fluid. J. Fluid Mech. 693, 500507.
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89 (1), 191200.
Ramachandran, A. & Khair, A. S. 2009 The dynamics and rheology of a dilute suspension of hydrodynamically Janus spheres in a linear flow. J. Fluid Mech. 633, 233269.
Ranger, K. B. 1978 The circular disk straddling the interface of a two-phase flow. Intl J. Multiphase Flow 4, 263277.
Rayleigh, Lord 1873 Investigation of the character of an incompressible fluid of variable density. Proc. Lond. Math. Soc. 4, 363.
Rayleigh, Lord 1876 On the application of the principle of reciprocity to acoustics. Proc. R. Soc. Lond. 25, 118122.
Rayleigh, Lord 1877 The Theory of Sound, vol. 1. Macmillan.
Relyea, L. M. & Khair, A. S. 2017 Forced convection heat and mass transfer from a slender particle. Chem. Engng Sci. 174, 285289.
Reyes, D. R. 2015 The art in science of MicroTAS: the 2014 issue. Lab on a Chip 15 (9), 19811983.
Roper, M. & Brenner, M. P. 2009 A nonperturbative approximation for the moderate Reynolds number Navier–Stokes equations. Proc. Natl Acad. Sci. USA 106 (9), 29772982.
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385400.
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189 (4760), 209.
Segre, G. & Silberberg, A. 1963 Non-Newtonian behavior of dilute suspensions of macroscopic spheres in a capillary viscometer. J. Colloid Sci. 18 (4), 312317.
Segre, G. & Silberberg, A. J. 1962a Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J. Fluid Mech. 14 (1), 115135.
Segre, G. & Silberberg, A. J. 1962b Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14 (1), 136157.
Sen, A., Ibele, M., Hong, Y. & Velegol, D. 2009 Chemo- and phototactic nano/microbots. Faraday Discuss. 143, 1527.
Sherwood, J. D. 1980 The primary electroviscous effect in a suspension of spheres. J. Fluid Mech. 101 (3), 609629.
Sherwood, J. D. 1982 Electrophoresis of rods. J. Chem. Soc. Faraday Trans. 2 78 (7), 10911100.
Sherwood, J. D. & Stone, H. A. 1995 Electrophoresis of a thin charged disk. Phys. Fluids 7 (4), 697705.
Shoele, K. & Eastham, P. S. 2018 Effects of nonuniform viscosity on ciliary locomotion. Phys. Rev. Fluids 3 (4), 043101.
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115146.
Solomentsev, Y. & Anderson, J. L. 1994 Electrophoresis of slender particles. J. Fluid Mech. 279, 197215.
Squires, T. M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.
Stokes, G. G. 1849 On the Perfect Blackness of the Central Spot in Newton’s Rings, and on the Verification of Fresnel’s Formula for the intensities of Reflected and Reflacted Rays. In Cambridge Library Collection – Mathematics, vol. 2, pp. 89103. Cambridge University Press.
Stone, H. A. & Duprat, C. 2016 Low-Reynolds-number flows. In Fluid-structure Interactions in Low-Reynolds-Number Flows (ed. Duprat, C. & Stone, H. A.), chap. 2, pp. 2577. Royal Society of Chemistry.
Stone, H. A. 1989 Heat/mass transfer from surface films to shear flows at arbitrary Peclet numbers. Phys. Fluids 1 (7), 11121122.
Stone, H. A., Brady, J. F. & Lovalenti, P. M.2016 Inertial effects on the rheology of suspensions and on the motion of individual particles. Available from the authors.
Stone, H. A. & Masoud, H. 2015 Mobility of membrane-trapped particles. J. Fluid Mech. 781, 494505.
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77, 41024104.
Subramanian, G., Koch, D. L., Zhang, J. & Wang, C. 2011 The influence of the inertially dominated outerregion on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles. J. Fluid Mech. 674, 307358.
Subramanian, R. S. 1985 The Stokes force on a droplet in an unbounded fluid medium due to capillary effects. J. Fluid Mech. 153, 389400.
Tanzosh, J. P. & Stone, H. A. 1994 Motion of a rigid particle in a rotating viscous flow: an integral equation approach. J. Fluid Mech. 275, 225256.
Tanzosh, J. P. & Stone, H. A. 1996 A general approach for analyzing the arbitrary motion of a circular disk in a Stokes flow. Chem. Engng Commun. 148 (1), 333346.
Taylor, G. I. 1960 Low Reynolds Number Flow (16 mm film). Educational Services Inc.
Teubner, M. 1982 The motion of charged colloidal particles in electric fields. J. Chem. Phys. 76 (11), 55645573.
Thiébaud, M. & Misbah, C. 2013 Rheology of a vesicle suspension with finite concentration: a numerical study. Phys. Rev. E 88, 062707.
Ursell, F. & Ward, G. N. 1950 On some general theorems in the linearized theory of compressible flow. Q. J. Mech. Appl. Maths 3 (3), 326348.
Van Dyke, M. D. 1964 Perturbation Methods in Fluid Dynamics. Academic Press.
Vandadi, V., Jafari Kang, S. & Masoud, H. 2016 Reciprocal theorem for convective heat and mass transfer from a particle in Stokes and potential flows. Phys. Rev. Fluids 1 (2), 022001.
Vandadi, V., Jafari Kang, S. & Masoud, H. 2017 Reverse Marangoni surfing. J. Fluid Mech. 811, 612621.
Villat, H. 1943 Leçons sur les Fluides Visqueux. Gauthier-Villars.
Wang, S. & Ardekani, A. 2012 Inertial squirmer. Phys. Fluids 24 (10), 101902.
Whitehead, A. N. 1889 Second approximations to viscous fluid motion. Q. J. Maths 23, 143152.
Würger, A. 2014 Thermally driven Marangoni surfers. J. Fluid Mech. 752, 589601.
Yano, H., Kieda, A. & Mizuno, I. 1991 The fundamental solution of Brinkman’s equation in two dimensions. Fluid Dyn. Res. 7 (3-4), 109118.
Yariv, E. & Brenner, H. 2003 Near-contact electrophoretic motion of a sphere parallel to a planar wall. J. Fluid Mech. 484, 85111.
Yariv, E. & Brenner, H. 2004 The electrophoretic mobility of a closely fitting sphere in a cylindrical pore. SIAM J. Appl. Maths 64 (2), 423441.
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69 (2), 377403.
Zhang, W. & Stone, H. A. 1998 Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J. Fluid Mech. 367, 329358.
Zhao, H., Isfahani, A. H. G., Olson, L. N. & Freund, J. B. 2010 A spectral boundary integral method for flowing blood cells. J. Comp. Phys. 229 (10), 37263744.
Zhao, H. & Shaqfeh, E. S. G. 2011 The dynamics of a vesicle in simple shear flow. J. Fluid Mech. 674, 578604.
Zinchenko, A. Z. & Davis, R. H. 2008 Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys. 227 (16), 78417888.
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