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On the motion of a porous sphere in a Stokes flow parallel to a planar confining boundary

Published online by Cambridge University Press:  10 July 2008

B. C. ROY  and
E. R. DAMIANO
B. C. ROY
Affiliation:
Department of Biomedical Engineering, Boston University, Boston, MA 02215, USA
E. R. DAMIANO
Affiliation:
Department of Biomedical Engineering, Boston University, Boston, MA 02215, USA
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Abstract

An analysis is presented of the three-dimensional creeping flow in and around a porous sphere, modelled as a generalized Brinkman medium, near a smooth plane where the sphere (i) translates uniformly without rotating in an otherwise quiescent Newtonian fluid, (ii) rotates uniformly without translating in an otherwise quiescent Newtonian fluid, and (iii) is fixed in a shear field, which is uniform in the far field and has a linearly increasing velocity profile with increasing distance from the plane. The linear superposition of these three flow regimes is also considered for the special case of the free translational and rotational motion of a neutrally buoyant porous sphere in a shear field that is uniform in the far field. Exact series solutions to the momentum equations are derived for the velocity and pressure fields in the Brinkman and Stokes-flow regions. Coefficients in the series solutions for each flow regime are determined using recursion relations derived from the continuity equations in the Brinkman and Stokes-flow regions, from the interfacial boundary conditions on the porous spherical surface, and from the no-slip condition on the plane. Results are presented in terms of the drag force on the porous sphere and torque about the sphere centre as a function of the dimensionless clearance distance between the sphere and the rigid plane for several values of the dimensionless hydraulic permeability of the Brinkman medium. The free motion of the neutrally buoyant sphere is calculated by requiring that the net hydrodynamic drag force and torque acting on the sphere vanish. Results for this case are presented in terms of the dimensionless translational and rotational speeds of the porous sphere relative to the far-field shear rate as a function of the dimensionless clearance distance for several values of the dimensionless hydraulic permeability. The work is motivated by insights it offers into the behaviour of porous agglomerates, and by its potential utility in industrial, biological, biophysical, medicinal and environmental applications wherever gas or liquid suspensions of porous agglomerates might arise.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 606 , 10 July 2008 , pp. 75 - 104
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Baskurt, O. K. & Meiselman, H. J. 2003 Blood rheology and hemodynamics. Seminars in Thrombosis and Hemostasis 29, 435–450.Google ScholarPubMed
Brenner, H. 1964 The Stokes resistance to an arbitrary particle – IV Arbitrary fields of flow. Chem. Engng. Sci. 19, 703.CrossRefGoogle 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, 27–34.CrossRefGoogle Scholar
Damiano, E. R., Duling, B. R., Ley, K. & Skalak, T. C. 1996 Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer. J. Fluid Mech. 314, 163–189.CrossRefGoogle Scholar
Damiano, E. R., Long, D. S., El-Khatib, F. H. & Stace, T. M. 2004 a On the motion of a sphere in a Stokes flow parallel to a Brinkman medium. J. Fluid Mech. 500, 75–101.CrossRefGoogle Scholar
Damiano, E. R., Long, D. S. & Smith, M. L. 2004 b Estimation of viscosity profiles using velocimetry data from parallel flows of linearly viscous fluids: Application to microvascular hemodynamics. J. Fluid Mech. 512, 1–19.CrossRefGoogle Scholar
Damiano, E. R. & Stace, T. M. 2002 A mechano-electrochemical model of radial deformation of the capillary glycocalyx. Biophys. J. 82, 1153–1175.CrossRefGoogle ScholarPubMed
Damiano, E. R. & Stace, T. M. 2005 Flow and deformation of the capillary glycocalyx in the wake of a leukocyte. Phys. Fluids 17, 031509-1–031509-17.CrossRefGoogle Scholar
Davis, A. M. J. 2001 Axisymmetric flow due to a porous sphere sedimenting towards a solid sphere or a solid wall: Application to scavenging of small particles. Phys. Fluids 13, 3126–3133.CrossRefGoogle 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, 13–24.CrossRefGoogle Scholar
Debye, P. & Bueche, A. M. 1948 Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J. Chem. Phys. 16, 573–579.CrossRefGoogle Scholar
Edwards, D. A., Hanes, J., Caponetti, G., Hrkach, J., Ben-Jebria, A., Eskew, M. L., Mintzes, J., Deaver, D., Lotan, N. & Langer, R. 1997 Large porous particles for pulmonary drug delivery. Science 276, 1868–1871.CrossRefGoogle ScholarPubMed
Feng, J., Ganatos, P. & Weinbaum, S. 1998 Motion of a sphere near planar confining boundaries in a Brinkman medium. J. Fluid Mech. 375, 265–296.CrossRefGoogle Scholar
Feng, J. & Weinbaum, S. 2000 Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans. J. Fluid Mech. 422, 281–317.CrossRefGoogle Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1980 A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion. J. Fluid Mech. 99, 755–783.CrossRefGoogle Scholar
Goldman, A. J. 1966 Investigations in low Reynolds number fluid–particle dynamics. PhD Thesis, New York University.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall—II Couette flow. Chem. Engng. Sci. 22, 653–660.CrossRefGoogle 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, 325–338.CrossRefGoogle Scholar
Halvorsen, K. 2007 Probing weak single-molecule interactions: Development and demonstration of a new instrument. PhD Thesis, Boston University.Google Scholar
Heinrich, V., Wong, W. P., Halvorsen, K. & Evans, E. 2008 Imaging biomolecular interactions by fast 3D-tracking of laser-confined carrier particles. Langmuir 24, 1194–1203.CrossRefGoogle Scholar
Hou, J. S., Holmes, M. H., Lai, W.M. & Mow, V. C. 1989 Boundary conditions at the cartilage synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Engng. 111, 78–87.CrossRefGoogle ScholarPubMed
Jeffery, G. B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 2, 327–338.CrossRefGoogle Scholar
Kargol, A., Przestalski, M. & Kargol, M. 2005 A study of porous structure of cellular membranes in human erythrocytes. Cryobiology. 50, 332–337.CrossRefGoogle ScholarPubMed
Kim, A.S. & Stolzenbach, K.D. 2004 Aggregate formation and collision efficiency in differential settling. J. Colloid Interface Sci. 271, 110–119.CrossRefGoogle ScholarPubMed
Long, D. S., Smith, M. L., Pries, A. R., Ley, K. & Damiano, E. R. 2004 Microviscometry reveals reduced blood viscosity and altered shear rate and shear stress profiles in microvessels after hemodilution. Proc. Natl Acad. Sci. USA 101, 10060–10065.CrossRefGoogle ScholarPubMed
Moon, P. & Spencer, D. 1961 Field Theory Handbook. Springer.CrossRefGoogle Scholar
Nir, A. & Pismen, L.M. 1977 Simulations of interparticle forced convection, diffusion and reaction in porous catalyst. Chem. Engng. Sci. 32, 35–41.CrossRefGoogle Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11, 67–74.CrossRefGoogle Scholar
O'Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear ow. Chem. Engng. Sci. 23, 1293–1298.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
Pruppacher, H. R. & Klett, J. D. 1978 Microphysics of Clouds and Precipitation. D. Reidel.CrossRefGoogle Scholar
Saffman, P. G. 1971 On the boundary condition at the surface of a porous medium. Stud. Appl. Maths 1, 93–101.CrossRefGoogle Scholar
Saffman, P. G. & Turner, J. S. 1956 On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30.CrossRefGoogle Scholar
Smith, M. L., Long, D. S., Damiano, E. R. & Ley, K. 2003 Near-wall μ-PIV reveals a hydrodynamically relevant relevant endothelial surface layer in venules in vivo. Biophys. J. 85, 637–645.CrossRefGoogle Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. 3, 110–116.Google Scholar
Truesdell, C. & Toupin, R. 1960 The classical field theories. In Handbuch der Physik (ed. Flügge, S.), pp. 226–793. Springer.Google Scholar
Tsapis, N., Bennett, D., Jackson, B., Weitz, D. A. & Edwards, D. A. 2002 Trojan particles: Large porous carriers of nanoparticles for drug delivery. Proc. Natl Acad. Sci. USA 99, 12001–12005.CrossRefGoogle ScholarPubMed
Vainshtein, P. & Shapiro, M. 2006 Porous agglomerates in the general linear flow field. J. Colloid Interface Sci. 298, 183–191.CrossRefGoogle ScholarPubMed
Vainshtein, P., Shapiro, M. & Gutfinger, C. 2004 Mobility of permeable aggregates: effects of shape and porosity. J. Aerosol Sci. 35, 383–404.CrossRefGoogle Scholar
Vink, H. & Duling, B. R. 1996 Identification of distinct luminal domains for macromolecules, erythrocytes, and leukocytes within mammalian capillaries. Circ. Res. 79, 581–589.CrossRefGoogle ScholarPubMed
Weinbaum, S., Ganatos, P. & Yan, Z. 1990 Numerical multipole and boundary integral equation techniques in Stokes flow. Annu. Rev. Fluid Mech. 22, 275–316.CrossRefGoogle Scholar
Weinbaum, S., Zhang, X., Han, Y., Vink, H. & Cowin, S. C. 2003 Mechanotransduction and flow across the endothelial glycocalyx. Proc. Natl Acad. Sci. USA 100, 7988–7995.CrossRefGoogle ScholarPubMed
Wong, W. P. 2006 Exploring single-molecule interactions through 3D optical trapping and tracking: From thermal noise to protein refolding. PhD Thesis, Harvard University, Cambridge, MA.Google Scholar
Wong, W. P. & Halvorsen, K. 2006 The effect of integration time on fluctuation measurements: Calibrating an optical trap in the presence of motion blur. Opt. Express 14, 12517–12531.CrossRefGoogle ScholarPubMed
Wong, W. P., Heinrich, V. & Evans, E. 2004 Exploring reaction pathways of single-molecule interactions through the manipulation and tracking of a potential-confined microsphere in three dimensions. Mat. Res. Soc. Symp. Proc. 790, P5.1.1–P5.1.12.Google Scholar
Zhao, M. L., Chien, S. & Weinbaum, S. 2001 Dynamic contact forces on leukocyte microvilli and their penetration of the endothelial glycocalyx. Biophys. J. 80 1124–1140.CrossRefGoogle ScholarPubMed