Hydrodynamic analysis of porous spheres with infiltrated peripheral shells in linear flow fields

Abstract

The velocity fields inside and around porous spheres with infiltrated peripheral shells were solved for three different far-field flows: simple shear, planar elongation and uniaxial extension. The flow was considered to obey Stokes’ law around the porous sphere and Brinkman's extension of Darcy's law within an infiltrated spherical shell beneath the sphere surface. The infiltrated layer, resulting from capillary action, was assumed to have a thickness that remained unaffected when the porous sphere was subjected to external flow fields. The cumulative hydrodynamic force exerted by the fluid upon the solid portion of the porous sphere was calculated for spherical caps with planar fracture surfaces. The magnitude of the tensile component of the hydrodynamic force per unit area of the base of the cap was shown to increase with the size of the cap and with the thickness of the infiltrated layer. On the other hand, the magnitude of the shear component of the hydrodynamic force per unit area of the base of the cap exhibited a maximum for a given cap size. The cap size that maximized the shear force increased with the thickness of the infiltrated layer. However, in contrast to the tensile force, for a constant cap size, the shear force did not necessarily increase with the thickness of the infiltrated layer. The hydrodynamic force exerted by the fluid upon the solid was found to increase with the inverse of the permeability of the porous sphere. The limit for very low permeability was compared to the case of an impermeable sphere. In the case of the tensile force, there was a difference between the low permeability limit for a permeable sphere and the value corresponding to an impermeable sphere. This difference could be attributed to the presence of fluid within the permeable sphere. The fluid contained in the permeable sphere could transmit pressure to the solid, a contribution absent in the case of an impermeable sphere.