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This paper presents steady axisymmetric creeping motion of a conducting, incompressible viscous fluid past a weakly permeable slightly deformed sphere in the presence of transverse magnetic field. The Stokes approximation of momentum equation and Darcy’s law together with Lorentz force are used for the flow outside and within the semi-permeable particle. The governing equations are changed into dimensionless form, and resulting equations are solved using separation of variables method. We have determined the resistance force exerted on the oblate spheroid in the presence of magnetic field as a particular case of an approximate sphere. The impact of Hartmann numbers, permeability and deformation parameter on the coefficient of drag and the streamline pattern is exhibited graphically. Some special cases are deduced from the current study and compared with some previous results. The outcome clarifies that the Hartmann numbers enhance the drag on the oblate spheroid in comparison with prolate spheroid.

This paper presents an analytical study of creeping motion of a permeable sphere in a spherical container filled with a micro-polar fluid. The drag experienced by the permeable sphere when it passes through the center of the spherical container is studied. Stream function solutions for the flow fields are obtained in terms of modified Bessel functions and Gegenbauer functions. The pressure fields, the micro-rotation components, the drag experienced by a permeable sphere, the wall correction factor, and the flow rate through the permeable surface are obtained for the frictionless impermeable spherical container and the zero shear stress at the impermeable spherical container. Variations of the drag force and the wall correction factor with respect to different fluid parameters are studied. It is observed that the drag force, the wall correction factor, and the flow rate are greater for the frictionless impermeable spherical container than the zero shear stress at the impermeable spherical container. Several cases of interest are deduced from the present analysis.

This paper deals the MHD slow viscous motion of electrically conducting fluid over a rigid sphere surrounded by a concentric permeable sphere. Darcy’s equation is adopted to describe the motion in semipermeable region and Stokes equation is applied to describe the flow of viscous fluid region. For both flow fields, stream functions are calculated. The resistance force on composite sphere is obtained and variation of drag force with respect to different parameters has been plotted graphically. Some limiting cases are deduced and compared with the solution derived in other research papers. We observed that drag force increases with increasing magnetic field.

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid through a swarm of immiscible Reiner–Rivlin liquid droplets-in-cell using the cell model technique. The stream function solution of Stokes equation is obtained for the flow in the fictitious envelope region, while for the inner flow field within the liquid drop, the solution is obtained by expanding the stream function in a power series of S. The proper boundary conditions are taken on the surface of the liquid sphere, while the appropriate conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. The analytical solution of the problem for four models: Happel’s, Kuwabara’s, Kvashnin’s and Mehta–Morse’s model (usually referred to as Cunningham’s) is derived. The velocity profile and the pressure distribution outside of the droplet are shown in numerous graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force $$W_{C}$$ acting, in each case, on the spherical droplet-in-cell obtained for different values of the parameters characterizing volume fraction $$\gamma ,$$ the relative viscosity $$\lambda$$ , and the cross-viscosity, i.e., S are presented in tabular and graphical forms as well. It is found that normalized hydrodynamic drag force $$W_{C}$$ is a monotonic increasing function of particle volume fraction $$\gamma .$$ It is also observed that solid sphere in-cell experiences greater drag force $$C_{D}$$ , whereas spherical bubble experiences smaller. One of the important findings of the present investigation is that the cross-viscosity $$\mu _{c}$$ of Reiner–Rivlin fluid decreases $$W_{C}$$ on the liquid droplet-in-cell. Further, the drag coefficient $$C_{D}$$ get reduced to analytical results obtained earlier.

This paper presents an analytical study of an infinite expanse of uniform flow of steady axisymmetric Stokes flow of an incompressible Newtonian fluid around the spherical drop of Reiner-Rivlin liquid coated with the permeable layer with the assumption that the liquid located outside the capsule penetrates into the permeable layer, but it is not mingled with the liquid located in the internal concave of capsule. The flow inside the permeable layer is described by the Brinkman equation. The viscosity of the permeable medium is assumed to be same as pure liquid. The stream function solution for the outer flow field is obtained in terms of modified Bessel functions and Gegenbauer functions, and for the inner flow field, the stream function solution is obtained by expanding the stream function in terms of S. The flow fields are determined explicitly by matching the boundary conditions at the pure liquid-porous interface, porous-Reiner-Rivlin liquid interface, and uniform velocity at infinity. The drag force experienced by the capsule is evaluated, and its variation with regard to permeability parameter a, dimensionless parameter S, ratio of viscosities l 2 , and thickness of permeable layer d is studied and graphs plotted against these parameters. Several cases of interest are deduced from the present analysis. It is observed that the cross-viscosity increases the drag force, whereas the thickness d decreases the drag on capsule. It is also observed that the drag force is increasing or decreasing function of permeability parameter for l 2 < 1.

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid past an immiscible Reiner–Rivlin liquid sphere embedded in porous medium using the validity of Brinkman’s model. The stream function solution of Brinkman equation is obtained for the flow in porous region, while for the inner flow field, the solution is obtained by expanding the stream function in a power series of $$S$$ . The flow fields are determined explicitly by matching the boundary conditions at the interface of porous region and the liquid sphere. Relevant quantities such as shearing stresses and velocities on the surface of the liquid sphere are obtained and presented graphically. It is found that dimensionless shearing stress on the surface is of periodic nature and its absolute value decreases with permeability parameter $$\alpha $$ and almost constant for all the representative values of $$S$$ ; on the other hand, the permeability parameter increases the velocity in the vicinity of the liquid sphere. The mathematical expression of separation parameter SEP is also calculated which shows that no flow separation occurs for the considered flow configuration and also validated by its pictorial depiction. The drag coefficient experienced by a liquid sphere embedded in a porous medium is evaluated. The dependence of the drag coefficient on permeability parameter, viscosity ratio and dimensionless parameter $$S$$ is presented graphically and discussed. Some previous well-known results are then also deduced from the present analysis.

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.