Viscous flow past a porous sphere within a nonconcentric fictitious spherical cell
Tóm tắt
The quasi steady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous sphere situated at an arbitrary position within a virtual spherical cell along the line connecting their centers is analyzed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman model governs the flow inside the porous particle and the flow in the fictitious envelope medium is governed by Stokes equations. A general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the porous particle and virtual spherical cell. Boundary conditions on the particle’s surface and the fictitious spherical envelope for four known cell models are satisfied by a collocation method for truncated series. Numerical solutions for the drag force exerted on the porous sphere in the presence of the cell are obtained for various cases of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere, the viscosity ratio, the stress jump coefficient, and a coefficient that is proportional to the permeability. Streamlines in and around porous sphere are presented for the unit cell models at different values of relevant physical parameters. In the limits of the motions of the porous particle in the concentric position with the cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.
Tài liệu tham khảo
Adler PM (1981) Streamlines in and around porous particles. J Colloid Interface Sci 81:531–535
Beavers GS, Joseph DD (1967) Boundary conditions at naturally permeable wall. J Fluid Mech 30:197–207
Brinkman HC (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
Cunningham E (1910) On the velocity of steady fall of spherical particles through fluid medium. Proc R Soc Lond Ser A 83:357–369
Dassios G, Hajinicolaou M, Coutelieris FA, Payatakes AC (1995) Stokes flow in spheroidal particle-in-cell models with Happel and Kuwabara boundary conditions. J Int J Eng Sci 33:1465–1490
Ehlers W, Bluhm J (2002) Porous media: theory, experiments and numerical applications. Springer, Berlin
Faltas MS, Saad EI (2012) Slow motion of a porous eccentric spherical particle-in-cell models. Transp Porous Med 95:133–150
Filippov A (2014) Mathematical modeling of filtration processes in porous media, structural properties of porous materials and powders used in different fields of science and technology. Springer, London, pp 267–321
Ganatos P, Weinbaum S, Pfeffer R (1980) A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J Fluid Mech 99:739–753
Gluckman MJ, Pfeffer R, Weinbaum S (1971) A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J Fluid Mech 50:705–740
Happel J (1958) Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J 4:197–201
Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijoff, The Hague
Keh HJ, Tu HJ (2000) Osmophoresis in a dilute suspension of spherical vesicles. Int J Multiph Flow 26:125–145
Kuwabara S (1959) The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. J Phys Soc Jpn 14:527–532
Kvashnin AG (1979) Cell model of suspension of spherical particles. Fluid Dyn 14:598–602
Lee TC, Keh HJ (2014) Osmophoresis of a spherical vesicle in a spherical cavity. Eur J Mech B Fluids 46:28–36
Levine S, Neale GH (1974) The prediction of electrokinetic phenomena within multiparticle systems. I. Electrophoresis and electroosmosis. J Colloid Interface Sci 47:520–529
Mehta GD, Morse TF (1975) Flow through charged membranes. J Chem Phys 63:1878–1889
Nield DA, Bejan A (2017) Convection in porous media, 5th edn. Springer, New York
Ochoa-Tapia JA, Whittaker S (1995) Momentum transfer at the boundary between a porous medium and a homogeneous fluid I: Theoretical development, II: Comparison with experiment. Int J Heat Mass Transf 38:2635–2655
Ohshima H (2000) Cell model calculation for electrokinetic phenomena in concentrated suspensions: an Onsager relation between sedimentation potential and electrophoretic mobility. Adv Colloid Interface Sci 88:1–18
Prakash J, Raja Sekhar GP (2013) Dynamic permeability of an assemblage of soft spherical particles. Math Methods Appl Sci 36:2174–2186
Prakash J, Sekhar GPR (2017) Slow motion of a porous spherical particle with a rigid core in a spherical fluid cavity. Meccanica 52:91–105
Prakash J, Raja Sekhar GP, Kohr M (2011) Stokes flow of an assemblage of porous particles: stress jump condition. Z Angew Math Phys 62:1027–1046
Rashidi S, Nouri-Borujerdi A, Valipour MS, Ellahi R, Pop I (2015) Stress-jump and continuity interface conditions for a cylinder embedded in a porous medium. Transp Porous Media 107:171–186
Saad EI (2016) Axisymmetric motion of a porous sphere through a spherical envelope subject to a stress jump condition. Meccanica 51:799–817
Saad EI, Faltas MS (2014) Slow motion of a porous sphere translating along the axis of a circular cylindrical pore subject to a stress jump condition. Transp Porous Med 102:91109
Sangani AS, Behl S (1989) The planar singular solutions of Stokes and Laplace equations and their application to transport processes near porous surfaces. Phys Fluids A 1:21–37
Sherief HH, Faltas MS, Saad EI (2016) Stokes resistance of a porous spherical particle in a spherical cavity. Acta Mech 227:1075–1093
Sherwood JD (2006) Cell models for suspension viscosity. Chem Eng Sci 61:6727–6731
Tan H, Pillai KM (2009) Finite element implementation of stress-jump and stress-continuity conditions at porous-medium, clear-fluid interface. Comput Fluids 38:1118–1131
Tan H, Chen X, Pillai KM, Papathanasiou TD (2008) Evaluation of boundary conditions at the clear-fluid and porous-medium interface using the boundary element method. In: Proceedings, in the 9th international conference on flow processes in composite materials, Montréal (Québec), Canada, 8–10 July (2008)
Valdes-Parada FJ, Goyeau B, Ramirez JA, Ochoa-Tapia JA (2009) Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation. Transp Porous Med 78:439–457
Yadav PK, Deo S, Yadav MK, Filippov A (2013) On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles. Colloid J 75:611–622