Topic: Tech & EngineeringApplications

Last updated: February 1, 2019

Different boundary conditions are suggested at the cell outer (fictitious) surface. A cell may consist of a single impermeable/porous particle surrounded by a fluid envelope. The cell model was originally proposed by Happel~cite{Happel58} and Kuwabara~cite{Kuwabara59} to describe the Stokes flow in an array of solid cylinders and spheres. The two approaches differed in the kind of boundary condition adopted on the cell outer surface. The hydrodynamic model of Happel assumes no-slip boundary condition on the inner representative particle and zero shear stress on the outer envelope, while the Kuwabara model proposed a nil shear stress condition, based on the requirement that no mechanical energy should be exchanged between the cell and the environment~cite{Happel83}, Kuwabara’s model relied on the kinematical argument suggesting a zero vorticity condition on the cell, both models usually give similar results. The Kuwabara and Happel unit cell models are used in several applications as indicated above.

Both models considered the effects of adjacent fibers. However, many researchers have been approved Kuwabara model because “it is more representative of the flow around the fiber in the case of low Reynolds number”~cite{Hutten16}. Several researchers~cite{Keh14,Chiang15,Keh15a} attempted to analyze the transient response of electrolyte solution in the porous medium constructed by a swarm of parallel charged circular cylinders to the step application of an electric field and a pressure gradient in the axial and in the transverse direction, respectively, through the use of a unit cell model. Recently, the unit cell model was employed to investigate the transient electrophoresis of a swarm of dielectric spheres with constant zeta potential after the application of a step function electric field for the case of thin but finite double layer~cite{Keh15a}. Saad and Faltas~cite{Saad18} studied semitheoretically the time-dependent electrophoresis of a charged spherical particle in an electrolyte solution saturated in a charged porous medium after the sudden application of an external electric field.

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