Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models
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Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models M KRISHNA PRASAD
∗
and TINA BUCHA
Department of Mathematics, National Institute of Technology, Raipur 492 010, India ∗ Corresponding author. E-mail: [email protected]; [email protected] MS received 15 February 2019; revised 23 September 2019; accepted 10 October 2019 Abstract. The present paper studies the impact of applied uniform transverse magnetic field on the flow of incompressible conducting fluid around a weakly permeable spherical particle bounded by a spherical container. Analytical solution of the problem is obtained using Happel and Kuwabara cell models. The concerned flow is parted in two regions, bounded fluid region and internal porous region, to be governed by Stokes and Darcy’s law respectively. At the interface between the fluid and the permeable region, the boundary conditions used are continuity of normal component of velocity, Saffman’s boundary condition and continuity of pressure. For the cell surface, Happel and Kuwabara models together with continuity in radial component of the velocity has been used. Expressions for drag force, hydrodynamic permeability and Kozeny constant acting on the spherical particle under magnetic effect are presented. Representation of hydrodynamic permeability for varying permeability parameters, particle volume fraction, slip parameter and Hartmann numbers are represented graphically. Also, the magnitude of Kozeny constant for weakly permeable and semipermeable sphere under a magnetic effect has been presented. In limiting cases many important results are obtained. Keywords. Weakly permeable spherical particle; Stokes flow; Darcy’s law; Hartmann number; drag force; cell model. PACS Nos 47.15.G−; 47.56.+r; 47.65.−d
1. Introduction Area of flow through porous medium is of great importance and so has attracted the interest of many researchers over the years. Its application is in diverging fields of science including chemical, biomedical, industrial, meteorology, environmental engineering and many more. Weakly permeable formulations are generally of low permeability. Some examples include flow through porous fluidisation, flow of a fluid mixture, flow in packed beds, sedimentation, fall of snow-flakes etc. The type of low permeability porous membrane which permits only specific types of particles to move through it is known as semipermeable membrane. This type of membrane allows only small particles like ions and molecules of water to pass through it. Osmosis is a phenomenon in which the diffusion of water takes place through a semipermeable membrane from an area of high concentration to an area of low concentration. It is seen in plants and animals. 0123456789().: V,-vol
Study of creeping flow through porous particles of varying geometry is generally modelled using Stokes equation for the flow of fluid and Darcy law [1] or Brinkman’s equation [2] for the flow in porous region. Darcy’s law is applicable for the flow through media havin
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