A solution for the problem of Stokes flow past a porous sphere
A general non-axisymmetric Stokes flow of a viscous, incompressible fluid in the presence of a porous sphere is discussed starting from the Stokes equations as the governing differential equations of motion for the flow outside the sphere. The velocity and the pressure then are described using a representation due to \textit{D. Palaniappan, S. D. Nigam, T. Amaranath} and \textit{R. Usha} [Q. J. Mech. Appl. Math. 45, No. 1, 47-56 (1992; Zbl 0754.76018)] in spherical polar coordinates, at which two functions arise, one being harmonic and the other one being biharmonic. The flow inside the porous sphere is governed by Darcy's law. The boundary conditions at the surface of the sphere are set up (no pressure jump, continuous normal velocity component, no slip of the fluid relative to the sphere). The two functions mentioned above are evaluated in series expansions at first for the undisturbed basic flow near the surface of the sphere and then in a modified form for the flow in the presence of the porous sphere. The components of velocity and pressure as well as those of the related volume rate of flow inside the sphere containing a permeability coefficient, are described in detail. Some coefficients of the modified series expansion mentioned above are defined with the help of the boundary conditions. Due to Faxén's law for a porous sphere, drag and torque are found from integrals taken over the surface of the sphere, which contain some terms related to pressure and to velocity components. For a zero permeability coefficient, drag is found to coincide with that for a rigid sphere on the one hand, on the other hand for a harmonic undisturbed velocity drag is less than that obtained in the rigid case, and it is higher for a pure biharmonic undisturbed velocity. Since the torque does not depend on the permeability coefficient, it is the same as in case of a rigid sphere. As an illustrative example, a certain Stokeslet with given strength is discussed in detail for the cases without and with a porous sphere.
scientific article; zbMATH DE number 168462