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MOTION OF A SLIP SPHERE IN A NONCONCENTRIC FICTITIOUS SPHERICAL ENVELOPE OF MICROPOLAR FLUID

Part of: Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  10 September 2014

E. I. SAAD
E. I. SAAD*
Affiliation:
Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt email elsayedsaad74@yahoo.com
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Abstract

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Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.

Keywords

micropolar fluidslip conditioncell modelsnormalized drag force

MSC classification

Secondary: 76A05: Non-Newtonian fluids
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 4 , April 2014 , pp. 383 - 401
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Aero, E. L., Bulygin, A. N. and Kuvshinskii, E. V., “Asymmetric hydromechanics”, J. Appl. Math. Mech. 29 (1965) 333–346; doi:10.1016/0021-8928(65)90035-3.CrossRefGoogle Scholar
Alloui, Z., Beji, H. and Vasseur, P., “Double-diffusive and Soret-induced convection of a micropolar fluid in a vertical channel”, Comput. Math. Appl. 62 (2011) 725–736; doi:10.1016/j.camwa.2011.05.053.Google Scholar
Basset, B. B., A treatise on hydrodynamics 2 (Dover, New York, 1961).Google Scholar
Bocquet, L. and Charlaix, E., “Nanofluidics, from bulk to interfaces”, Chem. Soc. Rev. 39 (2010) 1073–1095; doi:10.1039/B909366B.CrossRefGoogle ScholarPubMed
Chhabra, R. P., Comiti, J. and Machac, I., “Flow of non-Newtonian fluids in fixed and fluidised beds”, Chem. Engrg. Sci. 56 (2001) 1–27; doi:10.1016/S0009-2509(00)00207-4.Google Scholar
Dassios, G., Hajinicolaou, M., Coutelieris, F. A. and Payatakes, A., “Stokes flow in spheroidal particle-in-cell models with Happel and Kuwabara boundary conditions”, Internat. J. Engrg. Sci. 33 (1995) 1465–1490; doi:10.1007/s00707-008-0048-0.Google Scholar
Eringen, A. C., Microcontinuum field theories II: fluent media (Springer, New York, 2001).Google Scholar
Faltas, M. S. and Saad, E. I., “Stokes flow past an assemblage of slip eccentric spherical particle-in-cell models”, Math. Methods Appl. Sci. 34 (2011) 1594–1605; doi:10.1002/mma.1465.Google Scholar
Faltas, M. S. and Saad, E. I., “Slow motion of porous eccentric spherical particle-in-cell models”, Transp. Porous Media 95 (2012) 133–150; doi:10.1007/s11242-012-0036-7.Google Scholar
Gayen, B. and Alam, M., “Algebraic and exponential instabilities in a sheared micropolar granular fluid”, J. Fluid Mech. 567 (2006) 195–233; doi:10.1017/S002211200600214X.Google Scholar
Happel, J., “Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles”, AIChE J. 4 (1958) 197–201.Google Scholar
Happel, J. and Brenner, H., Low Reynolds number hydrodynamics (Martinus Nijoff, The Hague, Netherlands, 1983).CrossRefGoogle Scholar
Hayakawa, H., “Slow viscous flows in micropolar fluids”, Phys. Rev. E 61 (2000) 5477–5492; doi:10.1103/PhysRevE.61.5477.Google Scholar
Keh, H. J. and Lee, T. C., “Axisymmetric creeping motion of a slip spherical particle in a nonconcentric spherical cavity”, Theoret. Comput. Fluid Dyn. 24 (2010) 497–510; doi:10.1016/j.ijengsci.2013.03.010.Google Scholar
Kirby, B. J., Micro- and nanoscale fluid mechanics: transport in microfluidic devices (Cambridge University Press, Cambridge, 2010).Google Scholar
Kuwabara, S., “The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers”, J. Phys. Soc. Japan 14 (1959) 527–532.Google Scholar
Lauga, E., Brenner, M. P. and Stone, H. A., Handbook of experimental fluid dynamics (Springer, New York, 2007); doi:10.1007/9783540302995.Google Scholar
Lee, T., Charrault, E. and Neto, C., “Interfacial slip on rough, patterned and soft surfaces: a review of experiments and simulations”, Adv. Colloid Interface Sci. 210 (2014) 21–38 ;doi:10.1016/j.cis.2014.02.015.Google Scholar
Łukaszewicz, G., Micropolar fluids: theory and applications (Birkhäuser, Basel, 1999).Google Scholar
Maxwell, J. C., “On stresses in rarified gases arising from inequalities of temperature”, Philos. Trans. R. Soc. Lond. 170 (1879) 231–256.Google Scholar
Migun, N. P., “On hydrodynamic boundary conditions for microstructural fluids”, Rheol. Acta 23 (1984) 575–581; doi:10.1007/BF01438797.CrossRefGoogle Scholar
Navier, C. L. M. H., “Mémoire sur les lois du mouvement des fluids”, Mem. Acad. R. Sci. Inst. France 6 (1823) 389–416.Google Scholar
Neto, C., Evans, D. R., Bonaccurso, E., Butt, H. J. and Craig, V. S. J., “Boundary slip in Newtonian liquids: a review of experimental studies”, Rep. Progr. Phys. 68 (2005) 2859–2897; doi:10.1088/00344885/68/12/R05.Google Scholar
Prakash, J., Raja Sekhar, G. P. and Kohr, M., “Stokes flow of an assemblage of porous particles: stress jump condition”, Z. Angew. Math. Phys. 62 (2011) 1027–1046; doi:10.1007/s00033-011-0123-6.CrossRefGoogle Scholar
Rosensweig, R. E. and Johnston, R. J., “Aspects of magnetic fluid flow with nonequilibrium magnetization”, in: Continuum mechanics and its applications (eds Graham, G. A. C. and Malik, S. K.), (Hemisphere, New York, 1989), 707–729.Google Scholar
Saad, E. I., “Cell models for micropolar flow past a viscous fluid sphere”, Meccanica 47 (2012) 2055–2068; doi:10.1007/s11012-012-9575-9.CrossRefGoogle Scholar
Saad, E. I., “Motion of a spheroidal particle in a micropolar fluid contained in spherical envelope”, Canad. J. Phys. 86 (2008) 1039–1056; doi:10.1139/P08045.Google Scholar
Sherief, H. H., Faltas, M. S. and Saad, E. I., “Slip at the surface of an oscillating spheroidal particle in a micropolar fluid”, ANZIAM J. 55 (2013) E1–E50.Google Scholar
Sherief, H. H., Faltas, M. S. and Saad, E. I., “Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid”, Z. Angew. Math. Phys. 59 (2008) 293–312; doi:10.1007/s000330076078y.Google Scholar
Sherwood, J. D., “Cell models for suspension viscosity”, Chem. Engrg. Sci. 61 (2006) 6727–6731; doi:10.1016/j.ces.2006.07.016.Google Scholar
Sokovnin, O. M., Zagoskina, N. V. and Zagoskin, S. N., “Choice of boundary conditions for studying the behavior of the swarm of spherical particles traveling through a non-Newtonian liquid”, Theor. Found. Chem. Engrg. 42 (2008) 366–376; doi:10.1134/S0040579508040040.Google Scholar
Whitmer, J. K. and Luijten, E., “Fluid–solid boundary conditions for multiparticle collision dynamics”, J. Phys.: Condens. Matter 22 (2010) 104–106; doi:10.1134/S0040579508040040.Google ScholarPubMed
Willmott, G., “Dynamics of a sphere with inhomogeneous slip boundary conditions in Stokes flow”, Phys. Rev. E 77 (2008) 055302; doi:10.1103/PhysRevE.77.055302.Google Scholar