Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T06:54:17.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of general creeping motion of a sphere inside a cylinder

Published online by Cambridge University Press:  07 December 2009

SUKALYAN BHATTACHARYA ,
COLUMBIA MISHRA  and
SONAL BHATTACHARYA
SUKALYAN BHATTACHARYA*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
COLUMBIA MISHRA
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
SONAL BHATTACHARYA
Affiliation:
Department of Electrical & Computer Engineering, Texas Tech University, Lubbock, TX 79409, USA
*
†Email address for correspondence: s.bhattacharya@ttu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we develop an efficient procedure to solve for the Stokesian fields around a spherical particle in viscous fluid bounded by a cylindrical confinement. We use our method to comprehensively simulate the general creeping flow involving the particle-conduit system. The calculations are based on the expansion of a vector field in terms of basis functions with separable form. The separable form can be applied to obtain general reflection relations for a vector field at simple surfaces. Such reflection relations enable us to solve the flow equation with specified conditions at different disconnected bodies like the sphere and the cylinder. The main focus of this article is to provide a complete description of the dynamics of a spherical particle in a cylindrical vessel. For this purpose, we consider the motion of a sphere in both quiescent fluid and pressure-driven parabolic flow. Firstly, we determine the force and torque on a translating-rotating particle in quiescent fluid in terms of general friction coefficients. Then we assume an impending parabolic flow, and calculate the force and torque on a fixed sphere as well as the linear and angular velocities of a freely moving particle. The results are presented for different radial positions of the particle and different ratios between the sphere and the cylinder radius. Because of the generality of the procedure, there is no restriction in relative dimensions, particle positions and directions of motion. For the limiting cases of geometric parameters, our results agree with the ones obtained by past researchers using different asymptotic methods.

JFM classification

Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 295 - 328
Copyright
Copyright © Cambridge University Press 2009

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

Bhattacharya, S. 2008 a Cooperative motion of spheres arranged in periodic grids between two parallel walls. J. Chem. Phys. 128, 074709.CrossRefGoogle ScholarPubMed
Bhattacharya, S. 2008 b History force on an asymmetrically rotating body in Poiseuille flow inducing particle-migration across a slit-pore. Phys. Fluids 20, 093301.CrossRefGoogle Scholar
Bhattacharya, S. & Bławzdziewicz, J. 2002 Image system for Stokes-flow singularity between two parallel planar walls. J. Math. Phys. 43, 5720–5731.CrossRefGoogle Scholar
Bhattacharya, S., Bławzdziewicz, J. & Wajnryb, E. 2005 a Hydrodynamic interactions of spherical particles in suspensions confined between two planar walls. J. Fluid Mech. 541, 263–292.CrossRefGoogle Scholar
Bhattacharya, S., Bławzdziewicz, J. & Wajnryb, E. 2005 b Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method. Physica A 356, 294–340.CrossRefGoogle Scholar
Bhattacharya, S., Bławzdziewicz, J. & Wajnryb, E. 2006 a Far-field approximation for hydrodynamic interactions in parallel-wall geometry. J. Comput. Phys. 212, 718–738.CrossRefGoogle Scholar
Bhattacharya, S., Bławzdziewicz, J. & Wajnryb, E. 2006 b Hydrodynamic interactions of spherical particles in Poiseuille flow between two parallel walls. Phys. Fluids 18 (5).CrossRefGoogle Scholar
Brenner, H. 1970 Pressure drop due to the motion of neutrally buoyant particles in duct flows. J. Fluid Mech. 43, 641–660.CrossRefGoogle Scholar
Brenner, H. & Happel, J. 1958 Slow viscous flow past a sphere in a cylindrical tube. J. Fluid Mech. 4, 195–213.CrossRefGoogle Scholar
Bungay, P. M. & Brenner, H. 1973 a Pressure drop due to the motion of a sphere near the wall bounding a Poisseuille flow. J. Fluid Mech. 60, 81–96.CrossRefGoogle Scholar
Bungay, P. M. & Brenner, H. 1973 b The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiph. Flow 1, 25–56.CrossRefGoogle Scholar
Chiu, J. J., Wang, D. L., Chien, S., Skalak, R. & Usami, S. 1998 Effects of disturbed flow on endothelial cells. J. Biomech. Engng – Trans. ASME 120, 2–8.CrossRefGoogle ScholarPubMed
Cichocki, B. & Jones, R. B. 1998 Image representation of a spherical particle near a hard wall. Physica A 258, 273–302.CrossRefGoogle Scholar
Cichocki, B., Jones, R. B., Kutteh, R. & Wajnryb, E. 2000 Friction and mobility for colloidal spheres in Stokes flow near a boundary: the multipole method and applications. J. Chem. Phys. 112, 2548–61.CrossRefGoogle Scholar
Cox, R. G. & Brenner, H. 1967 Effect of finite boundaries on Stokes resistance of an arbitrary particle. 3. Translation and rotation. J. Fluid Mech. 28, 391.CrossRefGoogle Scholar
Cox, R. G. & Mason, S. G. 1971 Suspended particles in fluid flow through tubes. Annu. Rev. Fluid Mech. 3, 291–316.CrossRefGoogle Scholar
Drazer, G., Khusid, B., Koplik, J. & Acrivos, A. 2005 Wetting and particle adsorption in nanoflows. Phys. Fluids 17, ARTN:017102.CrossRefGoogle Scholar
Durlofsky, L. J. & Brady, J. F. 1989 Dynamic simulation of bounded suspensions of hydrodynamically interacting particles. J. Fluid. Mech. 200, 39–67.CrossRefGoogle Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.CrossRefGoogle Scholar
Falade, A. & Brenner, H. 1985 Stokes wall effects for particles moving near cylindrical boundaries. J. Fluid Mech. 154, 145–162.CrossRefGoogle Scholar
Felderhof, B. U. & Jones, R. B. 1989 Displacement theorems for spherical solutions of the linear Navier–Stokes equations. J. Math. Phys. 30, 339–42.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – I. Motion through a quiscent fluid. Chem. Engng Sci. 22, 637–651.CrossRefGoogle Scholar
Greenstein, T. & Happel, J. 1968 Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube. J. Fluid Mech. 34, 705–710.CrossRefGoogle Scholar
Greenstein, T. & Happel, J. 1970 Viscosity of dilute uniform suspensions of sphere spheres. Phys. Fluids 13, 18–21.CrossRefGoogle Scholar
Higdon, J. J. L. & Muldowney, G. P. 1995 Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes. J. Fluid Mech. 298, 193–210.CrossRefGoogle Scholar
Ladd, A. J. C. 1988 Hydrodynamic interactions in suspensions of spherical particles. J. Chem. Phys. 88, 5051.CrossRefGoogle Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Lunsmann, W. J., Genieser, L., Armstrong, R. C. & Brown, R. A. 1993 Finite-element analysis of steady viscoelastic flow around a sphere in a tube-calculations with constant viscosity models. J. Non-Newton. Fluid Mech. 48, 63–99.CrossRefGoogle Scholar
Morris, J. F. & Brady, J. F. 1998 Pressure-driven flow of a suspension: buoyancy effects. Intl J. Multiph. Flow 24, 105–30.CrossRefGoogle Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions – simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle 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, 705–724.CrossRefGoogle Scholar
Pozrikidis, C. 2005 Numerical simulation of cell motion in tube flow. Ann. Biomed. Engng 33, 165–178.CrossRefGoogle ScholarPubMed
Queguiner, C. & BarthesBiesel, D. 1997 Axisymmetric motion of capsules through cylindrical channels. J. Fluid Mech. 348, 349–376.CrossRefGoogle Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–46.CrossRefGoogle Scholar
SugiharaSeki, M. 1996 The motion of an ellipsoid in tube flow at low Reynolds numbers. J. Fluid Mech. 324, 287–308.CrossRefGoogle Scholar
SugiharaSeki, M. & Skalak, R. 1997 Asymmetric flows of spherical particles in a cylindrical tube. Biorheology 34, 155–169.CrossRefGoogle Scholar
Sushko, N. & Cieplak, M. 2001 Motion of grains, droplets, and bubbles in fluid-filled nanopores. Phys. Rev. E 64.CrossRefGoogle ScholarPubMed
Tozeren, H. 1982 Torque on eccentric spheres flowing in tubes. J. Appl. Mech. Trans. ASME 49, 279–283.CrossRefGoogle Scholar
Tozeren, H. 1983 Drag on eccentrically positioned spheres translating and rotating in tubes. J. Fluid Mech. 129, 77–90.CrossRefGoogle Scholar
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75–96.CrossRefGoogle Scholar