Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-24T05:21:09.642Z Has data issue: false hasContentIssue false

The reciprocal theorem in fluid dynamics and transport phenomena

Published online by Cambridge University Press:  30 September 2019

Hassan Masoud*
Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, Michigan 49931, USA
Howard A. Stone*
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
Email addresses for correspondence:,
Email addresses for correspondence:,


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.

JFM Perspectives
© 2019 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.)


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.Google Scholar
Achenbach, J. D. 2003 Reciprocity in Elastodynamics. Cambridge University Press.Google Scholar
Achenbach, J. D. 2014 A new use of the elastodynamic reciprocity theorem. Math. Mech. Solids 19 (1), 518.10.1177/1081286513505462Google Scholar
Acree, W. E. 1984 Empirical expression for predicting surface-tension of liquid-mixtures. J. Colloid Interface Sci. 101, 575576.10.1016/0021-9797(84)90069-9Google Scholar
Acrivos, A. 2015 Reflections on a rheologist: Howard Brenner (1929–2014). Rheol. Bull. 84 (1), 811.Google Scholar
Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer from single spheres in Stokes flow. Phys. Fluids 5 (4), 387394.10.1063/1.1706630Google Scholar
Adamson, A. W. & Gast, A. P. 1997 Physical Chemistry of Surfaces. Wiley.Google Scholar
Ajdari, A. & Stone, H. A. 1999 A note on swimming using internally generated traveling waves. Phys. Fluids 11 (5), 12751277.10.1063/1.869991Google Scholar
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21 (1), 6199.10.1146/annurev.fl.21.010189.000425Google Scholar
Barber, J. R. 2002 Elasticity. Springer.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545570.10.1017/S0022112070000745Google Scholar
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), 201233.10.1016/0377-0257(95)01424-1Google Scholar
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, 247258.10.1016/j.ijheatmasstransfer.2013.09.017Google Scholar
Betti, E. 1872 Teoria della elasticità. Il Nuovo Cimento 7 (1), 6997.10.1007/BF02824597Google Scholar
Brady, J. F. 2011 Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives. J. Fluid Mech. 667, 216259.10.1017/S0022112010004404Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20 (1), 111157.10.1146/annurev.fl.20.010188.000551Google Scholar
Brenner, H. 1958 Dissipation of energy due to solid particles suspended in a viscous liquid. Phys. Fluids 1 (4), 338346.10.1063/1.1705892Google Scholar
Brenner, H. 1961 The Oseen resistance of a particle of arbitrary shape. J. Fluid Mech. 11 (4), 604610.10.1017/S0022112061000755Google Scholar
Brenner, H. 1962 Effect of finite boundaries on the Stokes resistance of an arbitrary particle. J. Fluid Mech. 12 (1), 3548.10.1017/S0022112062000026Google Scholar
Brenner, H. 1963a Forced convection heat and mass transfer at small Péclet numbers from a particle of arbitrary shape. Chem. Engng Sci. 18 (2), 109122.10.1016/0009-2509(63)80020-2Google Scholar
Brenner, H. 1963b The Stokes resistance of an arbitrary particle. Chem. Engng Sci. 18 (1), 125.10.1016/0009-2509(63)80001-9Google Scholar
Brenner, H. 1964a The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19 (8), 519539.10.1016/0009-2509(64)85045-4Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle. IV: Arbitrary fields of flow. Chem. Engng Sci. 19 (10), 703727.10.1016/0009-2509(64)85084-3Google Scholar
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.Google Scholar
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.Google Scholar
Brenner, H. 1970b Pressure drop due to the motion of neutrally buoyant particles in duct flows. J. Fluid Mech. 43 (4), 641660.10.1017/S0022112070002641Google Scholar
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), 537543.10.1021/i160040a001Google Scholar
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), 561595.10.1017/S002211206300152XGoogle Scholar
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), 283295.10.1080/00986448408940506Google Scholar
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.10.1007/978-94-009-0225-1_10Google Scholar
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, 2734.10.1007/BF02120313Google Scholar
Brinkman, H. C. 1948 On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 8186.10.1007/BF02120318Google Scholar
Brunet, E. & Ajdari, A. 2004 Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries. Phys. Rev. E 69 (1), 016306.Google Scholar
Brunn, P. 1976a The behavior of a sphere in non-homogeneous flows of a viscoelastic fluid. Rheol. Acta 15 (11-12), 589611.10.1007/BF01524746Google Scholar
Brunn, P. 1976b The slow motion of a sphere in a second-order fluid. Rheol. Acta 15 (3–4), 163171.10.1007/BF01526063Google Scholar
Brunn, P. 1980 The motion of rigid particles in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 7 (4), 271288.10.1016/0377-0257(82)80019-0Google Scholar
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), 8196.10.1017/S0022112073000054Google Scholar
Candelier, F., Einarsson, J. & Mehlig, B. 2016 Angular dynamics of a small particle in turbulence. Phys. Rev. Lett. 117 (20), 204501.10.1103/PhysRevLett.117.204501Google Scholar
Carrier, G. F.1953 On slow viscous flow. Tech. Rep. Final Report, Office of Naval Research Contract Nonr-653 (00).10.21236/AD0016588Google Scholar
Caswell, B. 1972 The stability of particle motion near a wall in Newtonian and non-Newtonian fluids. Chem. Engng Sci. 27 (2), 373389.10.1016/0009-2509(72)85074-7Google Scholar
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92 (1), 131170.10.1017/S0022112079000562Google Scholar
Charlton, T. M. 1960 A historical note on the reciprocal theorem and theory of statically indeterminate frameworks. Nature 187 (4733), 231.10.1038/187231a0Google Scholar
Clebsch, R. F. A. 1862 Theorie der Elasticität fester Körper. B. G. Teubner.Google Scholar
Cox, R. G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow – I theory. Chem. Engng Sci. 23 (2), 147173.10.1016/0009-2509(68)87059-9Google Scholar
Crowdy, D. G. 2013 Wall effects on self-diffusiophoretic Janus particles: a theoretical study. J. Fluid Mech. 735, 473498.10.1017/jfm.2013.510Google Scholar
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, 301312.10.1063/1.857780Google Scholar
Day, R. F. & Stone, H. A. 2000 Lubrication analysis and boundary integral simulations of a viscous micropump. J. Fluid Mech. 416, 197216.10.1017/S002211200000879XGoogle Scholar
De Hoop, A. T. 1995 Handbook of Radiation and Scattering of Waves. Academic Press.Google Scholar
Debye, P. & Bueche, A. M. 1948 Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J. Chem. Phys. 16, 573579.10.1063/1.1746948Google Scholar
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, 607618.10.1017/jfm.2016.41Google Scholar
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 33293341.10.1063/1.866465Google Scholar
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.10.1007/978-1-4939-2065-5_8Google Scholar
Elfring, G. J. 2015 A note on the reciprocal theorem for the swimming of simple bodies. Phys. Fluids 27 (2), 023101.10.1063/1.4906993Google Scholar
Elfring, G. J. 2017 Force moments of an active particle in a complex fluid. J. Fluid Mech. 829, R3.10.1017/jfm.2017.632Google Scholar
Elfring, G. J. & Goyal, G. 2016 The effect of gait on swimming in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 234, 814.10.1016/j.jnnfm.2016.04.005Google Scholar
Elfring, G. J., Leal, L. G. & Squires, T. M. 2016 Surface viscosity and Marangoni stresses at surfactant laden interfaces. J. Fluid Mech. 792, 712739.10.1017/jfm.2016.96Google Scholar
Eversman, W. 2001 A reverse flow theorem and acoustic reciprocity in compressible potential flows in ducts. J. Sound Vib. 246 (1), 7195.Google Scholar
Fair, M. C. & Anderson, J. L. 1989 Electrophoresis of nonuniformly charged ellipsoidal particles. J. Colloid Interface Sci. 127 (2), 388400.10.1016/0021-9797(89)90045-3Google Scholar
Felderhof, B. U. 1983 Reciprocity in electrohydrodynamics. Physica A 122 (3), 383396.10.1016/0378-4371(83)90038-9Google Scholar
Felderhof, B. U. & Jones, R. B. 1994a Inertial effects in small-amplitude swimming of a finite body. Physica A 202 (1), 94118.10.1016/0378-4371(94)90169-4Google Scholar
Felderhof, B. U. & Jones, R. B. 1994b Small-amplitude swimming of a sphere. Physica A 202 (1), 119144.10.1016/0378-4371(94)90170-8Google Scholar
Flax, A. H. 1953 Reverse flow and variational theorems for lifting surfaces in non-stationary compressible flow. J. Aero. Sci. 20 (2), 120126.Google Scholar
Fleury, R., Sounas, D., Haberman, M. R. & Alù, A. 2015 Nonreciprocal acoustics. Acoust. Today 11 (3), 1421.Google Scholar
Ford, M. L. & Nadim, A. 1994 Thermocapillary migration of an attached drop on a solid surface. Phys. Fluids 6 (9), 31833185.10.1063/1.868096Google Scholar
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), 755783.10.1017/S0022112080000882Google Scholar
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), 739753.10.1017/S0022112080000870Google Scholar
Godin, O. A. 1997a Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25 (2), 143167.Google Scholar
Godin, O. A. 1997b Reciprocity relations and energy conservation for waves in the system: inhomogeneous fluid flow–anisotropic solid body. Acoust. Phys. 43, 688693.Google Scholar
Goldstein, R. E. 2011 Evolution of biological complexity. In Biological Physics, pp. 123139. Springer.10.1007/978-3-0346-0428-4_6Google Scholar
Golestanian, R., Liverpool, T. B. & Ajdari, A. 2007 Designing phoretic micro-and nano-swimmers. New J. Phys. 9 (5), 126.10.1088/1367-2630/9/5/126Google Scholar
Gonzalez-Rodriguez, D. & Lauga, E. 2009 Reciprocal locomotion of dense swimmers in Stokes flow. J. Phys.: Condens. Matter 21 (20), 204103.Google Scholar
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics, vol. 45. Cambridge University Press.10.1017/CBO9780511894671Google Scholar
Haj-Hariri, H., Nadim, A. & Borhan, A. 1990 Effect of inertia on the thermocapillary velocity of a drop. J. Colloid Interface Sci. 140 (1), 277286.10.1016/0021-9797(90)90342-LGoogle Scholar
Haj-Hariri, H., Nadim, A. & Borhan, A. 1993 Reciprocal theorem for concentric compound drops in arbitrary Stokes flows. J. Fluid Mech. 252, 265277.10.1017/S0022112093003751Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media. Martinus Nijhoff.10.1007/978-94-009-8352-6Google Scholar
Hauge, E. H. & Martin-Löf, A. 1973 Fluctuating hydrodynamics and brownian motion. J. Stat. Phys. 7, 259281.10.1007/BF01030307Google Scholar
Heaslet, M. A. & Spreiter, J. R.1953 Reciprocity relations in aerodynamics. NACA Report 1119, 253–268.Google Scholar
von Helmholtz, H. 1856 Handbuch der Physiologischen Optik. Leopold Voss.Google Scholar
von Helmholtz, H. 1887 Uber die physikalische bedeutung des prinzips der kleinsten wirkung. J. Reine Angew. Math. 100, 137166.10.1515/crll.1887.100.137Google Scholar
Higdon, J. J. L. & Kojima, M. 1981 On the calculation of Stokes flow past porous particles. Intl J. Multiphase Flow 7 (6), 719727.10.1016/0301-9322(81)90041-0Google Scholar
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54 (3), 423425.10.1017/S0022112072000771Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.10.1017/CBO9781139172189Google Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365400.10.1017/S0022112074001431Google Scholar
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), 783799.10.1017/S002211207600089XGoogle Scholar
Howell, L. L. 2001 Compliant Mechanisms. John Wiley & Sons.Google Scholar
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), 173184.10.1016/S0377-0257(99)00013-0Google Scholar
Jafari Kang, S., Dehdashti, E., Vandadi, V. & Masoud, H. 2019 Optimal viscous damping of vibrating porous cylinders. J. Fluid. Mech. 874, 339358.10.1017/jfm.2019.457Google Scholar
Joseph, D. D. 1973 Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rat. Mech. Anal. 51, 295303.10.1007/BF00250536Google Scholar
Kamrin, K. & Stone, H. A. 2011 The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces. Phys. Fluids 23 (3), 031701.10.1063/1.3560320Google Scholar
Kaplun, S. 1957 Low Reynolds number flow past a circular cylinder. J. Math. Mech. 595603.Google Scholar
Kaplun, S. & Lagerstrom, P. A. 1957 Asymptotic expansions of Navier–Stokes solutions for small Reynolds numbers. J. Math. Mech. 6 (5), 585593.Google Scholar
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), 123161.10.1080/00986448908940638Google Scholar
Khair, A. S. & Chisholm, N. G. 2014 Expansions at small Reynolds numbers for the locomotion of a spherical squirmer. Phys. Fluids 26 (1), 011902.10.1063/1.4859375Google Scholar
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), 156001.10.1103/PhysRevLett.105.156001Google Scholar
Kim, S. 1986 The motion of ellipsoids in a second order fluid. J. Non-Newtonian Fluid Mech. 21 (2), 255269.10.1016/0377-0257(86)80039-8Google Scholar
Kim, S. 2015 Ellipsoidal microhydrodynamics without elliptic integrals and how to get there using linear operator theory. Ind. Engng Chem. Res. 54 (42), 1049710501.10.1021/acs.iecr.5b01552Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
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), 8797.10.1016/j.jnnfm.2006.03.019Google Scholar
Kumar, A. & Graham, M. D. 2012 Accelerated boundary integral method for multiphase flow in non-periodic geometries. J. Comput. Phys. 231 (20), 66826713.10.1016/ Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.Google Scholar
Lagerstrom, P. A. & Cole, J. D. 1955 Examples illustrating expansion procedures for the Navier–Stokes equations. J. Ration. Mech. Anal. 4, 817882.Google Scholar
Lamb, H. 1887 On reciprocal theorems in dynamics. Proc. Lond. Math. Soc. 1 (1), 144151.10.1112/plms/s1-19.1.144Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
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, 294304.10.1017/jfm.2016.460Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon Press.Google Scholar
Lauga, E. & Davis, A. M. J. 2012 Viscous Marangoni propulsion. J. Fluid Mech. 705, 120133.10.1017/jfm.2011.484Google Scholar
Lauga, E. & Michelin, S. 2016 Stresslets induced by active swimmers. Phys. Rev. Lett. 117 (14), 148001.10.1103/PhysRevLett.117.148001Google Scholar
Leal, L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69 (2), 305337.10.1017/S0022112075001450Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435476.10.1146/annurev.fl.12.010180.002251Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.10.1017/CBO9780511800245Google Scholar
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), 705726.10.1017/S0022112079001981Google Scholar
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, 35723574.10.1063/1.869466Google Scholar
Leshansky, A. M. & Brady, J. F. 2004 Force on a sphere via the generalized reciprocal theorem. Phys. Fluids 16 (3), 843844.Google Scholar
Lorentz, H. A. 1895 Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies (in Dutch). E. J. Brill.Google Scholar
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.Google Scholar
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, 561605.10.1017/S0022112093002885Google Scholar
Love, A. E. H. 2013 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.Google Scholar
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, 115142.10.1017/S0022112003004464Google Scholar
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, 564604.10.1017/jfm.2011.363Google Scholar
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, 605606.10.1017/jfm.2011.475Google Scholar
Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115157.10.1017/S0022112002002902Google Scholar
Manga, M. & Stone, H. A. 1993 Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 256, 647683.10.1017/S0022112093002915Google Scholar
Masoud, H. & Stone, H. A. 2014 A reciprocal theorem for Marangoni propulsion. J. Fluid Mech. 741, R4.10.1017/jfm.2014.8Google Scholar
Maxwell, J. C. 1864 On the calculation of the equilibrium and stiffness of frames. Phil. Mag. 27 (182), 294299.Google Scholar
Maxwell, J. C. 1881 A Treatise on Electricity and Magnetism. Oxford University Press.Google Scholar
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), 20692076.10.1016/0017-9310(94)90308-5Google Scholar
Michelin, S. & Lauga, E. 2015 A reciprocal theorem for boundary-driven channel flows. Phys. Fluids 27 (11), 111701.10.1063/1.4935415Google Scholar
Morrison, F. A. & Griffiths, S. K. 1981 On the transient convective transport from a body of arbitrary shape. J. Heat Transfer 103 (1), 9295.10.1115/1.3244438Google Scholar
Mozaffari, A., Sharifi-Mood, N., Koplik, J. & Maldarelli, C. 2016 Self-diffusiophoretic colloidal propulsion near a solid boundary. Phys. Fluids 28 (5), 053107.10.1063/1.4948398Google Scholar
Munk, M. M. 1950 The reversal theorem of linearized supersonic airfoil theory. J. Appl. Phys. 21 (2), 159161.10.1063/1.1699616Google Scholar
Nadim, A., Haj-Hariri, H. & Borhan, A. 1990 Thermocapillary migration of slightly deformed droplets. Particul. Sci. Technol. 8 (3-4), 191198.10.1080/02726359008906566Google Scholar
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.Google Scholar
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.Google Scholar
Nourhani, A., Lammert, P. E., Crespi, V. H. & Borhan, A. 2015 A general flux-based analysis for spherical electrocatalytic nanomotors. Phys. Fluids 27 (1), 012001.10.1063/1.4904951Google Scholar
Nunan, K. C. & Keller, J. B. 1984 Effective viscosity of a periodic suspension. J. Fluid Mech. 142, 269287.10.1017/S0022112084001105Google Scholar
Onsager, L. 1931a Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (4), 405.Google Scholar
Onsager, L. 1931b Reciprocal relations in irreversible processes. II. Phys. Rev. 38 (12), 2265.10.1103/PhysRev.38.2265Google Scholar
Oppenheimer, N., Navardi, S. & Stone, H. A. 2016 Motion of a hot particle in viscous fluids. Phys. Rev. Fluids 1 (1), 014001.10.1103/PhysRevFluids.1.014001Google Scholar
Oseen, C. W. 1910 Stokes formula and a related theorem in hydrodynamics. Ark. Mat. Astron. Fys. 6, 20.Google Scholar
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, 535552.10.1017/jfm.2014.380Google Scholar
Pak, O. S., Zhu, L., Brandt, L. & Lauga, E. 2012 Micropropulsion and microrheology in complex fluids via symmetry breaking. Phys. Fluids 24 (10), 103102.10.1063/1.4758811Google Scholar
Papavassiliou, D. & Alexander, G. P. 2015 The many-body reciprocal theorem and swimmer hydrodynamics. Europhys. Lett. 110 (4), 44001.10.1209/0295-5075/110/44001Google Scholar
Potton, R. J. 2004 Reciprocity in optics. Rep. Prog. Phys. 67 (5), 717.10.1088/0034-4885/67/5/R03Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.10.1017/CBO9780511624124Google Scholar
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.Google Scholar
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), 237262.10.1017/S0022112057000105Google Scholar
Rallabandi, B., Yang, F. & Stone, H. A.2019 Motion of hydrodynamically interacting active particles. arXiv:1901.04311.Google Scholar
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.Google Scholar
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.Google Scholar
Rallison, J. M. 1978 Note on the Faxén relations for a particle in Stokes flow. J. Fluid Mech. 88 (3), 529533.Google Scholar
Rallison, J. M. 2012 The stress in a dilute suspension of liquid spheres in a second-order fluid. J. Fluid Mech. 693, 500507.10.1017/jfm.2011.544Google Scholar
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), 191200.10.1017/S0022112078002530Google Scholar
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, 233269.10.1017/S0022112009007472Google Scholar
Ranger, K. B. 1978 The circular disk straddling the interface of a two-phase flow. Intl J. Multiphase Flow 4, 263277.10.1016/0301-9322(78)90002-2Google Scholar
Rayleigh, Lord 1873 Investigation of the character of an incompressible fluid of variable density. Proc. Lond. Math. Soc. 4, 363.Google Scholar
Rayleigh, Lord 1876 On the application of the principle of reciprocity to acoustics. Proc. R. Soc. Lond. 25, 118122.Google Scholar
Rayleigh, Lord 1877 The Theory of Sound, vol. 1. Macmillan.Google Scholar
Relyea, L. M. & Khair, A. S. 2017 Forced convection heat and mass transfer from a slender particle. Chem. Engng Sci. 174, 285289.Google Scholar
Reyes, D. R. 2015 The art in science of MicroTAS: the 2014 issue. Lab on a Chip 15 (9), 19811983.10.1039/C5LC90049BGoogle Scholar
Roper, M. & Brenner, M. P. 2009 A nonperturbative approximation for the moderate Reynolds number Navier–Stokes equations. Proc. Natl Acad. Sci. USA 106 (9), 29772982.10.1073/pnas.0810578106Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385400.10.1017/S0022112065000824Google Scholar
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189 (4760), 209.10.1038/189209a0Google Scholar
Segre, G. & Silberberg, A. 1963 Non-Newtonian behavior of dilute suspensions of macroscopic spheres in a capillary viscometer. J. Colloid Sci. 18 (4), 312317.Google Scholar
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), 115135.10.1017/S002211206200110XGoogle Scholar
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.Google Scholar
Sen, A., Ibele, M., Hong, Y. & Velegol, D. 2009 Chemo- and phototactic nano/microbots. Faraday Discuss. 143, 1527.10.1039/b900971jGoogle Scholar
Sherwood, J. D. 1980 The primary electroviscous effect in a suspension of spheres. J. Fluid Mech. 101 (3), 609629.Google Scholar
Sherwood, J. D. 1982 Electrophoresis of rods. J. Chem. Soc. Faraday Trans. 2 78 (7), 10911100.10.1039/f29827801091Google Scholar
Sherwood, J. D. & Stone, H. A. 1995 Electrophoresis of a thin charged disk. Phys. Fluids 7 (4), 697705.10.1063/1.868595Google Scholar
Shoele, K. & Eastham, P. S. 2018 Effects of nonuniform viscosity on ciliary locomotion. Phys. Rev. Fluids 3 (4), 043101.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115146.Google Scholar
Solomentsev, Y. & Anderson, J. L. 1994 Electrophoresis of slender particles. J. Fluid Mech. 279, 197215.10.1017/S0022112094003885Google Scholar
Squires, T. M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.Google Scholar
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.Google Scholar
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.Google Scholar
Stone, H. A. 1989 Heat/mass transfer from surface films to shear flows at arbitrary Peclet numbers. Phys. Fluids 1 (7), 11121122.Google Scholar
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.Google Scholar
Stone, H. A. & Masoud, H. 2015 Mobility of membrane-trapped particles. J. Fluid Mech. 781, 494505.Google Scholar
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77, 41024104.10.1103/PhysRevLett.77.4102Google Scholar
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, 307358.10.1017/jfm.2010.654Google Scholar
Subramanian, R. S. 1985 The Stokes force on a droplet in an unbounded fluid medium due to capillary effects. J. Fluid Mech. 153, 389400.10.1017/S0022112085001306Google Scholar
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, 225256.10.1017/S002211209400234XGoogle Scholar
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), 333346.10.1080/00986449608936523Google Scholar
Taylor, G. I. 1960 Low Reynolds Number Flow (16 mm film). Educational Services Inc.Google Scholar
Teubner, M. 1982 The motion of charged colloidal particles in electric fields. J. Chem. Phys. 76 (11), 55645573.10.1063/1.442861Google Scholar
Thiébaud, M. & Misbah, C. 2013 Rheology of a vesicle suspension with finite concentration: a numerical study. Phys. Rev. E 88, 062707.Google Scholar
Ursell, F. & Ward, G. N. 1950 On some general theorems in the linearized theory of compressible flow. Q. J. Mech. Appl. Maths 3 (3), 326348.10.1093/qjmam/3.3.326Google Scholar
Van Dyke, M. D. 1964 Perturbation Methods in Fluid Dynamics. Academic Press.Google Scholar
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.Google Scholar
Vandadi, V., Jafari Kang, S. & Masoud, H. 2017 Reverse Marangoni surfing. J. Fluid Mech. 811, 612621.10.1017/jfm.2016.695Google Scholar
Villat, H. 1943 Leçons sur les Fluides Visqueux. Gauthier-Villars.Google Scholar
Wang, S. & Ardekani, A. 2012 Inertial squirmer. Phys. Fluids 24 (10), 101902.10.1063/1.4758304Google Scholar
Whitehead, A. N. 1889 Second approximations to viscous fluid motion. Q. J. Maths 23, 143152.Google Scholar
Würger, A. 2014 Thermally driven Marangoni surfers. J. Fluid Mech. 752, 589601.10.1017/jfm.2014.349Google Scholar
Yano, H., Kieda, A. & Mizuno, I. 1991 The fundamental solution of Brinkman’s equation in two dimensions. Fluid Dyn. Res. 7 (3-4), 109118.10.1016/0169-5983(91)90051-JGoogle Scholar
Yariv, E. & Brenner, H. 2003 Near-contact electrophoretic motion of a sphere parallel to a planar wall. J. Fluid Mech. 484, 85111.Google Scholar
Yariv, E. & Brenner, H. 2004 The electrophoretic mobility of a closely fitting sphere in a cylindrical pore. SIAM J. Appl. Maths 64 (2), 423441.10.1137/S0036139902411119Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69 (2), 377403.10.1017/S0022112075001486Google Scholar
Zhang, W. & Stone, H. A. 1998 Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J. Fluid Mech. 367, 329358.10.1017/S0022112098001670Google Scholar
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), 37263744.10.1016/ Scholar
Zhao, H. & Shaqfeh, E. S. G. 2011 The dynamics of a vesicle in simple shear flow. J. Fluid Mech. 674, 578604.10.1017/S0022112011000115Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2008 Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys. 227 (16), 78417888.10.1016/ Scholar