Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T04:27:51.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamic force on a sphere normal to an obstacle due to a non-uniform flow

Published online by Cambridge University Press:  04 April 2017

Bhargav Rallabandi Open the ORCID record for Bhargav Rallabandi [Opens in a new window] ,
Sascha Hilgenfeldt  and
Howard A. Stone
Bhargav Rallabandi*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Sascha Hilgenfeldt
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: vbr@princeton.edu, hastone@princeton.edu
†Email addresses for correspondence: vbr@princeton.edu, hastone@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

For a small sphere suspended in a background fluid flow near an obstacle, we calculate the hydrodynamic force on the sphere in the direction normal to the boundary of the obstacle. Using the Lorentz reciprocal theorem, we obtain analytical expressions for the normal force in the Stokes flow limit, valid for arbitrary separations of the particle from the obstacle, both for solid obstacles and those with free surfaces. The main effect of the boundary is to produce a normal force proportional to extensional flow gradients in the vicinity of the particle. The strength of this force is greatest when the separation between the surfaces of the particle and the obstacle is small relative to the particle size. While the magnitude of the force weakens for large separations between the sphere and the obstacle (decaying quadratically with separation distance), it can significantly modify Faxén’s law even at modestly large separation distances. In addition, we find a second force contribution due to the curvature of the background flow normal to the obstacle, which is also important when the sphere is close to the obstacle. The results of the theory are of importance to the dynamics of particles in confined geometries, whether bounded by a solid obstacle, the wall of a channel or a gas bubble.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Multiphase and Particle-laden Flows: Particle/fluid flow Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 407 - 434
Check for updates
Copyright
© 2017 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

Adamczyk, Z., Adamczyk, M. & Van de Ven, T. G. M. 1983 Resistance coefficient of a solid sphere approaching plane and curved boundaries. J. Colloid Interface Sci. 96 (1), 204–213.Google Scholar
Aponte-Rivera, C. & Zia, R. N. 2016 Simulation of hydrodynamically interacting particles confined by a spherical cavity. Phys. Rev. Fluids 1 (2), 023301.Google Scholar
Asmolov, E. S. 1995 Dusty-gas flow in a laminar boundary layer over a blunt body. J. Fluid Mech. 305, 29–46.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 (4), 739–751.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (03), 545–570.Google Scholar
Batchelor, G. K. & Green, J. T. 1972a The determination of the bulk stress in a suspension of spherical particles to order c 2 . J. Fluid Mech. 56 (03), 401–427.Google Scholar
Batchelor, G. K. & Green, J. T. 1972b The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (02), 375–400.Google 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), 201–233.Google Scholar
Bhattacharya, S., Bławzdziewicz, J. & Wajnryb, E. 2006 Far-field approximation for hydrodynamic interactions in parallel-wall geometry. J. Comput. Phys. 212 (2), 718–738.Google Scholar
Bossis, G., Meunier, A. & Sherwood, J. D. 1991 Stokesian dynamics simulations of particle trajectories near a plane. Phys. Fluids 3 (8), 1853–1858.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3), 242–251.Google Scholar
Cardinaels, R. & Stone, H. A. 2015 Lubrication analysis of interacting rigid cylindrical particles in confined shear flow. Phys. Fluids 27 (7), 072001.Google Scholar
Cox, R. G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface. II. Small gap widths, including inertial effects. Chem. Engng Sci. 22 (12), 1753–1777.Google Scholar
Dance, S. L. & Maxey, M. R. 2003 Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comput. Phys. 189 (1), 212–238.Google Scholar
Dean, W. R. & O’Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika 10 (01), 13–24.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.Google Scholar
Feuillebois, F., Ekiel-Jeżewska, M. L., Wajnryb, E., Sellier, A. & Bławzdziewicz, J. 2015 High-frequency viscosity of a dilute suspension of elongated particles in a linear shear flow between two walls. J. Fluid Mech. 764, 133–147.Google Scholar
Gallier, S., Lemaire, E., Lobry, L. & Peters, F. 2016 Effect of confinement in wall-bounded non-colloidal suspensions. J. Fluid Mech. 799, 100–127.Google Scholar
Giddings, J. C., Yang, F. J. & Myers, M. N. 1976 Flow-field-flow fractionation: a versatile new separation method. Science 193 (4259), 1244–1245.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1966 The slow motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Engng Sci. 21 (12), 1151–1170.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Engng Sci. 22 (4), 637–651.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Engng Sci. 22 (4), 653–660.Google Scholar
Goren, S. L. & O’Neill, M. E. 1971 On the hydrodynamic resistance to a particle of a dilute suspension when in the neighbourhood of a large obstacle. Chem. Engng Sci. 26 (3), 325–338.Google Scholar
Haber, S. & Brenner, H. 1999 Hydrodynamic interactions of spherical particles in quadratic Stokes flows. Intl J. Multiphase Flow 25 (6), 1009–1032.Google Scholar
Haddadi, H. & Morris, J. F. 2015 Topology of pair-sphere trajectories in finite inertia suspension shear flow and its effects on microstructure and rheology. Phys. Fluids 27 (4), 043302.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Application to Particulate Media. Prentice-Hall.Google Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.Google Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (02), 365–400.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
Hood, K., Lee, S. & Roper, M. 2015 Inertial migration of a rigid sphere in three-dimensional Poiseuille flow. J. Fluid Mech. 765, 452–479.Google Scholar
Izard, E., Bonometti, T. & Lacaze, L. 2014 Modelling the dynamics of a sphere approaching and bouncing on a wall in a viscous fluid. J. Fluid Mech. 747, 422–446.Google Scholar
Jeffrey, D. J. 1992 The calculation of the low Reynolds number resistance functions for two unequal spheres. Phys. Fluids 4 (1), 16–29.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1981 The slow motion of a cylinder next to a plane wall. Q. J. Mech. Appl. Maths 34 (2), 129–137.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261–290.Google Scholar
Kempe, T. & Fröhlich, J. 2012a Collision modelling for the interface-resolved simulation of spherical particles in viscous fluids. J. Fluid Mech. 709, 445–489.Google Scholar
Kempe, T. & Fröhlich, J. 2012b An improved immersed boundary method with direct forcing for the simulation of particle laden flows. J. Comput. Phys. 231 (9), 3663–3684.Google Scholar
Ladd, A. J. C. & Verberg, R. 2001 Lattice-Boltzmann simulations of particle-fluid suspensions. J. Stat. Phys. 104 (5–6), 1191–1251.Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12 (1), 435–476.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
Lovalenti, P. M. & Brady, J. F. 1995 The temporal behaviour of the hydrodynamic force on a body in response to an abrupt change in velocity at small but finite Reynolds number. J. Fluid Mech. 293, 35–46.Google Scholar
Maude, A. D. 1961 End effects in a falling-sphere viscometer. Brit. J. Appl. Phys. 12 (6), 293–295.Google Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.Google Scholar
McLaughlin, J. B. 1993 The lift on a small sphere in wall-bounded linear shear flows. J. Fluid Mech. 246, 249–265.Google Scholar
Nadim, A. & Stone, H. A. 1991 The motion of small particles and droplets in quadratic flows. Stud. Appl. Maths 85 (1), 53–73.Google Scholar
Nguyen, N.-Q. & Ladd, A. J. C. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. E 66 (4), 046708.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 (02), 209–223.Google Scholar
O’Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (01), 67–74.Google Scholar
O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (04), 705–724.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (02), 385–400.Google Scholar
Sangani, A. S., Acrivos, A. & Peyla, P. 2011 Roles of particle-wall and particle-particle interactions in highly confined suspensions of spherical particles being sheared at low Reynolds numbers. Phys. Fluids 23 (8), 083302.Google Scholar
Segré, G. & Silberberg, A. 1962 Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14 (1), 136–157.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.Google Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111 (757), 110–116.Google Scholar
Swan, J. W. & Brady, J. F. 2007 Simulation of hydrodynamically interacting particles near a no-slip boundary. Phys. Fluids 19 (11), 113306.Google Scholar
Swan, J. W. & Brady, J. F. 2011 The hydrodynamics of confined dispersions. J. Fluid Mech. 687, 254–299.Google Scholar
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80 (03), 561–591.Google Scholar
Zurita-Gotor, M., Bławzdziewicz, J. & Wajnryb, E. 2007 Swapping trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres. J. Fluid Mech. 592, 447–469.Google Scholar