Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

Abstract

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.

Sedimentation Velocity and Potential in Dilute Suspensions of Charged Porous Shells

“…The net force acting on the sedimenting porous shell vanishes at the steady state. Applying this constraint to the summation of eqs , , and , we obtain the sedimentation velocity of the spherical porous shell as expressed by eq 11 with where U 0 is the sedimentation velocity of an uncharged porous shell. , Because both the fixed charges of the porous shell and the local induced sedimentation potential gradient are of the order Q , the effect of the fixed charge density on the shell velocity resulting from their interaction starts from the second order Q 2 .…”

“…where U 0 is the sedimentation velocity of an uncharged porous shell. 21,22 Because both the fixed charges of the porous shell and the local induced sedimentation potential gradient are of the order Q, the effect of the fixed charge density on the shell velocity resulting from their interaction starts from the second order Q 2 . It can be found that U 2 depends on the inner-to-outer radius ratio a/b, electrokinetic radius κb (ratio of the outer radius to the Debye length), and shielding parameter λb (ratio of the outer radius to the flow penetration length) of the shell as well as the ionic diffusion coefficients in the electrolyte solution.…”

“…A porous shell is often employed to model a permeable microcapsule or vesicle, − and the sedimentation and other electrokinetic phenomena of a charged porous shell are of interest. , Recently, analytical formulas of the electrophoretic mobility and diffusiophoretic mobility were derived for a charged spherical porous shell. , These formulas reduce to those obtained for an intact porous sphere ,− when the inner radius of the spherical shell equals zero. However, the sedimentation velocity and potential in solutions of charged porous shells have not been analyzed yet.…”