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Time-dependent lift and drag on a rigid body in a viscous steady linear flow

Published online by Cambridge University Press:  11 February 2019

Fabien Candelier ,
Bernhard Mehlig Open the ORCID record for Bernhard Mehlig [Opens in a new window]  and
Jacques Magnaudet Open the ORCID record for Jacques Magnaudet [Opens in a new window]
Fabien Candelier*
Affiliation:
Aix-Marseille Univ., CNRS, IUSTI (Institut Universitaire des Systèmes Thermiques et Industriels) F-13013 Marseille, France
Bernhard Mehlig
Affiliation:
Department of Physics, Gothenburg University, SE-41296 Gothenburg, Sweden
Jacques Magnaudet*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
*
†Email addresses for correspondence: fabien.candelier@univ-amu.fr, jmagnaud@imft.fr
†Email addresses for correspondence: fabien.candelier@univ-amu.fr, jmagnaud@imft.fr
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Abstract

We compute the leading-order inertial corrections to the instantaneous force acting on a rigid body moving with a time-dependent slip velocity in a linear flow field, assuming that the square root of the Reynolds number based on the fluid-velocity gradient is much larger than the Reynolds number based on the slip velocity between the body and the fluid. As a first step towards applications to dilute sheared suspensions and turbulent particle-laden flows, we seek a formulation allowing this force to be determined for an arbitrarily shaped body moving in a general linear flow. We express the equations governing the flow disturbance in a non-orthogonal coordinate system moving with the undisturbed flow and solve the problem using matched asymptotic expansions. The use of the co-moving coordinates enables the leading-order inertial corrections to the force to be obtained at any time in an arbitrary linear flow field. We then specialize this approach to compute the time-dependent force components for a sphere moving in three canonical flows: solid-body rotation, planar elongation, and uniform shear. We discuss the behaviour and physical origin of the different force components in the short-time and quasi-steady limits. Last, we illustrate the influence of time-dependent and quasi-steady inertial effects by examining the sedimentation of prolate and oblate spheroids in a pure shear flow.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 554 - 595
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Copyright
© 2019 Cambridge University Press 

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References

Aris, R. 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Asmolov, E. S. 1990 Dynamics of a spherical particle in a laminar boundary layer. Fluid Dyn. 25, 886–890.Google Scholar
Asmolov, E. S. & McLaughlin, J. B. 1999 The inertial lift on an oscillating sphere in a linear shear flow. Intl J. Multiphase Flow 25, 739–751.Google Scholar
Auton, T. R., Prud’homme, M. & Hunt, J. C. R. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.Google Scholar
Basset, A. B. 1888 A Treatise on Hydrodynamics. Deighton Bell.Google Scholar
Batchelor, G. K. 1979 Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution. J. Fluid Mech. 95, 369–400.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7, 83–103.Google Scholar
Bedeaux, D. & Rubi, J. M. 1987 Drag on a sphere moving slowly through a fluid in elongational flow. Physica A 144, 285–298.Google Scholar
Blanes, S., Casas, F., Oteo, J. A. & Ros, J. 2009 The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238.Google Scholar
Bluemink, J. J., Lohse, D., Prosperetti, A. & Van Wijngaarden, L. 2010 Drag and lift forces on particles in a rotating flow. J. Fluid Mech. 643, 1–31.Google Scholar
Boussinesq, J. 1885 Sur la résistance qu’oppose un fluide indéfini au repos sans pesanteur au mouvement varié d’une sphère solide qu’il mouille sur toute sa surface quand les vitesses restent bien continues et assez faibles pour que leurs carrés et produits soient négligeables. C. R. Acad. Sci. Paris 100, 935–937.Google Scholar
Brenner, H. 1963 The Stokes resistance of an arbitrary particle. Chem. Engng Sci. 18, 1–25.Google Scholar
Brenner, H. 1964a The Stokes resistance of an arbitrary particle–II: an extension. Chem. Engng Sci. 19, 599–629.Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle–III: shear fields. Chem. Engng Sci. 19, 631–651.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.Google Scholar
Candelier, F. 2008 Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers. J. Fluid Mech. 608, 319–336.Google Scholar
Candelier, F., Einarsson, J., Lundell, F., Mehlig, B. & Angilella, J. R. 2015 The role of inertia for the rotation of a nearly spherical particle in a general linear flow. Phys. Rev. E 91, 053023.Google Scholar
Candelier, F., Einarsson, J. & Mehlig, B. 2016 Angular dynamics of a small particle in turbulence. Phys. Rev. Lett. 117, 204501.Google Scholar
Candelier, F. & Souhar, M. 2007 Time-dependent lift force acting on a particle moving arbitrarily in a pure shear flow, at small Reynolds number. Phys. Rev. E 76, 067301.Google Scholar
Childress, S. 1964 The slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 20, 305–314.Google Scholar
Dabade, V., Marath, N. K. & Subramanian, G. 2016 The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow. J. Fluid Mech. 791, 631–703.Google Scholar
Daitche, A. 2013 Advection of inertial particles in the presence of the history force: higher order numerical schemes. J. Comput. Phys. 254, 93–106.Google Scholar
Drew, D. 1978 The force on a small sphere in slow viscous flow. J. Fluid Mech. 88, 393–400.Google Scholar
Einarsson, J., Candelier, F., Lundell, F., Angilella, J. R. & Mehlig, B. 2015a Effect of weak fluid inertia upon Jeffery orbits. Phys. Rev. E 91, 041002.Google Scholar
Einarsson, J., Candelier, F., Lundell, F., Angilella, J. R. & Mehlig, B. 2015b Rotation of a spheroid in a simple shear at small Reynolds number. Phys. Fluids 27, 063301.Google Scholar
Eringen, A. C. 1967 Mechanics of Continua. Wiley.Google Scholar
Feng, J. & Joseph, D. D. 1995 The unsteady motion of solid bodies in creeping flows. J. Fluid Mech. 303, 83–102.Google Scholar
Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 1, 143–160.Google Scholar
Gavze, E. 1990 The accelerated motion of a rigid bodies in non-steady Stokes flow. Intl J. Multiphase Flow 16, 153–166.Google Scholar
Gavze, E. & Shapiro, M. 1998 Motion of inertial spheroidal particles in a shear flow near a solid wall with special application to aerosol transport in microgravity. J. Fluid Mech. 371, 59–79.Google Scholar
Gotoh, T. 1990 Brownian motion in a rotating flow. J. Stat. Phys. 59, 371–402.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Harper, E. Y. & Chang, I.-D. 1968 Maximum dissipation resulting from lift in a slow viscous shear flow. J. Fluid Mech. 33, 209–225.Google Scholar
Herron, I. H., Davis, S. H. & Bretherton, F. P. 1975 On the sedimentation of a sphere in a centrifuge. J. Fluid Mech. 68, 209–234.Google Scholar
Hogg, A. J. 1994 The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows. J. Fluid Mech. 272, 285–318.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Kaplun, S. & Lagerstrom, P. A. 1957 Asymptotic expansions of Navier–Stokes solutions for small Reynolds numbers. J. Math. Mech. 6, 585–593.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Lamb, S. H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon Press.Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435–476.Google 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, 3572–3574.Google Scholar
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1, 31–53.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, 561–605.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, 115–142.Google Scholar
Marath, N. K. & Subramanian, G. 2018 The inertial orientation dynamics of anisotropic particles in planar linear flows. J. Fluid Mech. 844, 357–402.Google Scholar
Maxworthy, T. 1965 An experimental determination of the slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 23, 373–384.Google Scholar
Mazur, P. & Bedeaux, D. 1974 A generalization of Faxén’s theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow. Physica 76, 235–246.Google Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.Google Scholar
Meibohm, J., Candelier, F., Rosen, T., Einarsson, J., Lundell, F. & Mehlig, B. 2016 Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number. Phys. Rev. Fluids 1, 084203.Google Scholar
Miyazaki, K. 1995 Dependence of the friction tensor on the rotation of a frame of reference. Physica A 222, 248–260.Google Scholar
Miyazaki, K., Bedeaux, D. & Avalos, J. B. 1995 Drag on a sphere in slow shear flow. J. Fluid Mech. 296, 373–390.Google Scholar
Nir, A. & Acrivos, A. 1973 On the creeping motion of two arbitrary-sized touching spheres in a linear shear field. J. Fluid Mech. 59, 209–223.Google Scholar
Onuki, A. & Kawasaki, K. 1980 Critical phenomena of classical fluids under flow. I: mean field approximation. Prog. Theor. Phys. 63, 122–139.Google Scholar
Oseen, C. W. 1910 Über die Stoke’sche Formel und über die verwandte Aufgabe in der Hydrodynamik. Ark. Mat. Astron. Fys. 6, 1–20.Google Scholar
Pérez-Madrid, A., Rubi, J. M. & Bedeaux, D. 1990 Motion of a sphere through a fluid in stationary homogeneous flow. Physica A 163, 778–790.Google Scholar
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and circular cylinder. J. Fluid Mech. 2, 237–262.Google Scholar
Rosen, T., Einarsson, J., Nordmark, A., Aidun, C. K., Lundell, F. & Mehlig, B. 2015 Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers. Phys. Rev. E 92, 063022.Google Scholar
Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.Google Scholar
Saffman, P. G. 1968 The lift on a small sphere in a slow shear flow-corrigendum. J. Fluid Mech. 31, 624.Google Scholar
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics, 2nd edn. Springer.Google Scholar
Sano, T. 1981 Unsteady flow past a sphere at low Reynolds number. J. Fluid Mech. 112, 433–441.Google Scholar
Sauma-Pérez, T., Johnson, C. G., Yang, L. & Mullin, T. 2018 An experimental study of the motion of a light sphere in a rotating viscous fluid. J. Fluid Mech. 847, 119–133.Google Scholar
Segré, G. & Silberberg, A. 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, 115–135.Google Scholar
Segré, G. & Silberberg, A. 1962b Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14, 136–157.Google Scholar
Shin, M., Koch, D. L. & Subramanian, G. 2009 Structure and dynamics of dilute suspensions of finite-Reynolds-number settling fibers. Phys. Fluids 21, 123304.Google Scholar
Singh, V., Koch, D. L. & Stroock, A. D. 2013 Rigid ring-shaped particles that align in simple shear flow. J. Fluid Mech. 722, 121–158.Google Scholar
Stone, H. A. 2000 Philip Saffman and viscous flow theory. J. Fluid Mech. 409, 165–183.Google Scholar
Subramanian, G. & Koch, D. L. 2005 Inertial effects on fibre motion in simple shear flow. J. Fluid Mech. 535, 383–414.Google Scholar
Taylor, G. I. 1928 The forces on a body placed in a curved or converging stream of fluid. Proc. R. Soc. Lond. A 120, 260–283.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Truesdell, C. & Noll, W. 1965 The Non-Linear Field Theories of Mechanics. Springer.Google Scholar
Van Dyke, M. 1978 Perturbation Methods in Fluid Mechanics, 2nd edn. Parabolic Press.Google Scholar
Van Nierop, E. A., Luther, S., Bluemink, J. J., Magnaudet, J., Prosperetti, A. & Lohse, D. 2007 Drag and lift forces on bubbles in a rotating flow. J. Fluid Mech. 571, 439–454.Google Scholar
Voth, G. & Soldati, A. 2017 Anisotropic particles in turbulence. Annu. Rev. Fluid Mech. 49, 249–276.Google Scholar
Wilkinson, M., Bezuglyy, V. & Mehlig, B. 2009 Fingerprints of random flows? Phys. Fluids 21, 043304.Google Scholar