КОЛЛОИДНЫЙ ЖУРНАЛ, 2014, том 76, № 6, с. 777-790
HYDRODYNAMIC PERMEABILITY OF A MEMBRANE COMPOSED OF POROUS SPHERICAL PARTICLES IN THE PRESENCE OF UNIFORM MAGNETIC FIELD
© 2014 Bal Govind Srivastava*, Pramod Kumar Yadav**, Satya Deo*, Pramod Kumar Singh*, Anatoly Filippov***
*Department of Mathematics, University of Allahabad, Allahabad-211002 (U.P.), India E-mail: firstname.lastname@example.org; email@example.com **Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004 (U.P.), India E-mail: firstname.lastname@example.org ***Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas, 119991 Moscow, Leninskii pr. 65-1, Russia E-mail: email@example.com Поступила в редакцию 18.03.2014 г.
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 components are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel's, Ku-wabara'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.
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 Brenner , 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 surrounded by a fluid envelope. Uchida  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 problem considering the inner sphere/cylinder as solid with respective boundary conditions on the cell surface. The Happel model, which assumes the velocity to be uniform and no tangential stress at the cell surface, leads to an axially symmetric flow that has a simple analytical solution in closed form and thus can be used for heat and mass transfer calculations. Kuwa-bara  proposed a cell model to investigate the flow through a swarm of spherical/cylindrical particles taking nil vorticity condition on the cell surface. However, the Kuwabara formulation requires a small exchange 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 formulations, Kvashnin  and Mehta and Morse  have given their respective boundary conditions for the outer cell surface. Kvashnin  proposed the condition that the tangential component of velocity reaches a
minimum at the cell surface with respect to radial distance, signifying the symmetry on the cell. However, Mehta and Morse  used Cunningham's  approach by assuming the tangential velocity as a component of the fluid velocity, signifying the homogeneity of the flow on the cell boundary. Analytical solutions of particle-in-cell models discussed earlier are always practically useful to many industrial problems.
A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles is given by Brinkman  in his classical paper. An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium by using the Brinkman model is presented by Grosan et al. . Barman  has reported an analytical study of the incompressible flow past an impervious sphere embedded in a porous medium of constant porosity based on the Brinkman model. He has obtained a closed form exact solution for the governing equation. Srinivasacharya and Prasad  have considered the creeping flow of an incompressible viscous liquid past and through a porous approximate sphere with an impermeable core. They have obtained an exact solution for the problem and they have also evaluated the drag force for the porous approximate spherical particle.
An empirical slip flow condition at the interface for a plane boundary, which is given for a rectilinear flow of a viscous fluid through a two-dimensional parallel channel formed by an impermeable upper wall and a permeable lower wall has been proposed by Beavers and Joseph . Ochoa-Tapia and Whitaker [14, 15] studied the momentum transfer at the boundary between a porous medium and a homogeneous fluid both theoretically and experimentally. They developed the appropriate jump condition for momentum transport which was also named as stress jump boundary condition. Many authors have used the stress jump boundary condition in various flow problems and reported significant changes in results. Kuznetsov [16, 17] used the stress jump boundary condition at the fluid-porous interface to discuss flow in channels partially filled with porous medium. Raja Sekhar and Ama-ranath  solved the problem of an arbitrary Stokes flow inside a porous spherical shell by using Darcy's law for outside porous region and Stokes flow for inside region. The dependence of drag and torque on jump coefficient for an arbitrary viscous flow past a porous sphere with an impermeable core was studied by Bhattacharya and Raja Shekhar . The Stokes flow through a porous sphere using the stress jump condition at the fluid-porous interface was discussed by Srivastava and Srivastava  and have reported that drag on a porous sphere decreases with increasing permeability of the medium.
The hydrodynamic permeability of membrane of porous spherical particles using Mehta-Morse condition on the cell surface was evaluated by Vasin and Fil-
ippov , Filippov et al. . Vasin et al. [23, 24] compared all four cell models to evaluate the permeability of membrane of porous spherical particles with a permeable shell and discussed the effect of different parameters on the hydrodynamic permeability of the membrane for all the four above mentioned boundary conditions. The hydrodynamic permeability of membranes built up by spherical particles covered by porous shells was discussed by Yadav et al. . The problem of the infinite uniform flow of liquid around the spherical drop coated with the porous layer is solved by Vasin and Kharitonova . They have used the boundary condition of the jump of tangential stresses at the liquid-porous medium interface. The case of hydrodynamic permeability of membranes built up by porous cylindrical or spherical particles with impermeable core is investigated by Deo et al. . They have calculated the hydrodynamic permeability of the membranes by using the cell model technique. They also have considered four known boundary conditions, namely, Happel's, Kuwabara's, Kvash-nin's and Cunningham/Mehta-Morse's on the outer surface of the cell. Comparison of the resulting hydro-dynamic permeability is undertaken by them.
In all the above works, the importance of cell model technique in membrane filtration process and effect of various parameters on hydrodynamic permeability of the membrane were discussed. However, Gold  and Globe  in their work discussed the effect of magnetic field on the flow rate. Later on, many authors discussed the effect of magnetic field on the flow rate and found that the flow rate decreases with increase in magnetic field. Some effects of magnetic field on flow of a Newtonian fluid through a circular tube were considered by Mazumdar et al. . Jay-alakshmamma et al.  have discussed the analytical study of creeping flow past a composite sphere with porous shell in presence of magnetic field. Tiwari et al.  have studied the influence of the magnetic field on the permeability of a membrane of solid cylindrical particles covered with porous layer. They have used the cell model technique for this problem. They have discussed the effect of the Hartmann number on the hydrodynamic permeability of the membrane. The motivations of these papers lead us to discuss the present problem in presence of transverse magnetic field which includes some earlier known results.
In this paper we have discussed the effect of magnetic field for an incompressible viscous fluid past a porous sphere, using particle-in-cell method. Suitable boundary conditions are used both for porous and nonporous regions. The dependence of hydrodynamic permeability of membrane built up by porous spherical particle on stress jump coefficient, Hartman number, particle volume fraction and dimensionless permeability of porous medium is discussed graphically for each Happel, Kuwabara, Kvashnin, Mehta-Mor-se/Cunningham boundary conditions.
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