A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxen's laws for drag and torque for the flow past a porous sphere are also given.

Stokes flow past a porous sphere using Brinkman's model

An exact solution for the creeping flow past a porous spherical shell has been obtained using the Stokes & Brinkman equations. Expressions for the velocities and pressure fields both inside and outside of porous spherical shell are obtained and the force acting on the shell is also determined. It is found that the drag on a porous spherical shell is smaller as compared to the shell with solid core. Several limiting cases of the solution are also discussed. In particular it is shown that the solution obtained using Darcy's law can be derived easily with the appropriate choice of boundary conditions.



Eine exakte Losung der schleichenden Stromung hinter einer porosen Kugelschale wird unter Benutzung der Stokes- und Brinkmangleichungen abgeleitet. Ausdrucke fur die Geschwindigkeiten und das Druckfeld innerhalb und auserhalb der porosen Kugelschale werden erhalten, und die auf die Schale wirkende Kraft wird ebenfalls bestimmt. Es zeigt sich, das der Stromungswiderstand auf eine porose Kugelschale, verglichen mit einer Schale mit festem Kern, kleiner ist. Verschiedene Grenzfalle der Losung werden diskutiert. Insbesondere wird gezeigt, das die Losung, die aus dem Darcyschen Gesetz abgeleitet wird, leicht durch passende Wahl der Randbedingungen erhalten werden kann.

Creeping Flow Past a Porous Spherical Shell

low of an incompressible viscous fluid past a porous sphere has been discussed. The flow has been divided in three regions. The Region-I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear fluid. In Regions II and III clear fluid flows and Stokes and Oseen solutions are respectively valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Stream function of Region II is matched with that of Region I at the surface of the sphere by the conditions suggested by Ochao-Tapia and Whitaker and it is matched with that of Oseen’s solutions far away from the sphere. It is found that the drag on the sphere reduces significantly when it is porous and it decreases with the increase of permeability of the medium.

Flow past a porous sphere at small Reynolds number

This paper concerns the flow of an incompressible, viscous fluid past a porous spherical particle enclosing a solid core, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. 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. No-slip and impenetrability boundary conditions on the solid spherical core have been used. The hydrodynamic drag force experienced by a porous spherical particle enclosing a solid core and permeability of membrane built up by solid particles with a porous shell are evaluated. It is found that the hydrodynamic drag force and dimensionless hydrodynamic permeability depends not only on the porous shell thickness, particle volume fraction γ and viscosities of porous and fluid medium, but also on the stress jump coefficient. 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). Some previous results for the hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.

Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition

Creeping flow of micropolar fluid past a porous sphere

A method of solving the Brinkman equation, using Cartesian tensors, is developed. General expressions for the velocity vector and the pressure, which can directly be used in situations where boundary conditions are expressible in Cartesian-tensor form, are obtained. It is shown how the drag on a porous sphere can be directly and easily calculated using this method.

Qin Yu

Papers

A cartesian-tensor solution of the Brinkman equation