ISSN 1061-933X, Colloid Journal, 2014, Vol. 76, No. 6, pp. 725–738. © Pleiades Publishing, Ltd., 2014. 725 1 INTRODUCTION The behavior of systems involving the motion of aggregates of small particles relative to fluids in which they are immersed covers a wide range of phenomenon of interest in recent times. There are two techniques for handling boundary value problems involving flow through a swarm of particles, namely, the method of reflection and the cell model technique. In the method of reflection, the boundary conditions are satisfied successively on each of the separate bounding surface involved, including the container walls confining the suspension when the fluid particle system is bounded in extent. The cell method, as discussed by Happel and Bren- ner [1], is used for description of the creeping flow in a porous medium. In cell model, the flow takes place through sedimenting swarm of spherical particles and each such spherical particle is considered as surround- ed by a fluid envelope. Uchida [2] proposed a cubic outer fluid envelope. Happel [3, 4] have proposed cell models in which the particle and outer envelope both are spherical/cylindrical in shape. He solved the prob- lem considering the inner sphere/cylinder as solid with respective boundary conditions on the cell sur- 1 The article is published in the original. face. The Happel model, which assumes the velocity to be uniform and no tangential stress at the cell sur- face, leads to an axially symmetric flow that has a sim- ple analytical solution in closed form and thus can be used for heat and mass transfer calculations. Kuwa- bara [5] proposed a cell model to investigate the flow through a swarm of spherical/cylindrical particles tak- ing nil vorticity condition on the cell surface. Howev- er, the Kuwabara formulation requires a small ex- change of mechanical energy with the environment. The mechanical power given by the sphere to the fluid is not all consumed by viscous dissipation in the fluid layer. Apart from above Happel and Kuwabara formu- lations, Kvashnin [6] and Mehta and Morse [7] have given their respective boundary conditions for the out- er cell surface. Kvashnin [6] proposed the condition that the tangential component of velocity reaches a minimum at the cell surface with respect to radial dis- tance, signifying the symmetry on the cell. However, Mehta and Morse [7] used Cunningham’s [8] ap- proach by assuming the tangential velocity as a com- ponent of the fluid velocity, signifying the homogene- ity of the flow on the cell boundary. Analytical solu- tions of particle-in-cell models discussed earlier are always practically useful to many industrial problems. A calculation of the viscous force exerted by a flow- ing fluid on a dense swarm of particles is given by Hydrodynamic Permeability of a Membrane Composed of Porous Spherical Particles in the Presence of Uniform Magnetic Field 1 Bal Govind Srivastava a , Pramod Kumar Yadav b , Satya Deo a , Pramod Kumar Singh a , and Anatoly Filippov c a Department of Mathematics, University of Allahabad, Allahabad-211002 (U.P.), India b Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004 (U.P.), India c Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas, 119991 Moscow, Leninskii pr. 65-1, Russia e-mail: balgovind01@gmail.com, sd_mathau@yahoo.co.in, pramod547@gmail.com, anatoly.filippov@gmail.com Received March 18, 2014 Abstract—This work concerns the flow of an incompressible viscous fluid past a porous sphere in presence of transverse applied uniform magnetic field, using particle-in-cell method. The Brinkman equations are used in porous region and the Stokes equations for non-porous region. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity compo- nents are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta–Morse’s condition). The hydrodynamic drag force experienced by a porous spherical particle in a cell and hydrodynamic permeability of membrane built up by porous spherical particles are evaluated. The patterns of streamlines are also obtained and discussed. The effect of stress jump coefficient, Hartmann number, dimensionless specific permeability of the porous particle and particle volume fraction on the hydrodynamic permeability and streamlines are discussed. Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. DOI: 10.1134/S1061933X14060167