The creeping flow of an incompressible, bounded micropolar fluid past a porous shell is investigated. The porous shell is modeled using a Darcy equation, sandwiched between a pair of transition Brinkman regions. Analytical expressions for the stream function, pressure, and microrotations are given for each region. Streamline patterns are presented for variations in hydraulic resistivity, micropolar constants, porous layer thickness, and Ochoa-Tapia stress jump coefficient. An expression for the dimensionless drag for the unbounded case of the system is presented, and its variation with hydraulic resistivity and porous shell thickness is presented. The unbounded case represents a theoretical model for oral drug delivery using porous microspheres. It was found that optimal circulation between the porous region and the outer fluid occurred for low values of hydraulic resistivity and for a complete porous sphere.

Controlled release drug delivery employs drug-encapsulating devices from which therapeutic agents may be released at controlled rates for long periods of time, ranging from days to months. Such systems offer numerous advantages over traditional methods of drug delivery, including tailoring of drug release rates, protection of fragile drugs, and increased patient comfort and compliance. Porous microspheres are ideal vehicles for many controlled delivery applications due to their ability to encapsulate a variety of drugs, biocompatibility, high bioavailability, and sustained drug release characteristics. The types of microspheres used in controlled drug delivery include Bioadhesive, Magnetic, Floating, Radioactive, and Polymeric Microspheres.

Considerable work has been carried out on micropolar fluids since the introduction of its theory [

Rao and Rao [

Bhatt [

Srinivasacharya and Rajyalakshmi [

Saad [

Hoffmann

Goharzadeh

Stability analysis has also been performed on a porous medium modeled as a transition Brinkman layer overlying a Darcy layer. Hill and Straughan [

The intent of the problem is to investigate the hydrodynamics associated with a porous polymer cell, geometrically modeled as a porous microsphere, that is orally administered as a drug carrier for a sustained drug delivery system to treat colon cancer. A porous spherical shell is used to generalize the geometry, and micropolar fluid is used to model the complex Non-Newtonian fluid present in the colon. Creeping axisymmetric flow of an incompressible micropolar fluid past a porous shell is considered. The porous region is modeled using Darcy equation sandwiched between two transition Brinkman regions. A stream function formulation is used to solve the system. The accompanying boundary conditions used are continuity of velocity, normal and tangential stresses, and non homogeneous microrotations across the fluid porous region interfaces. At the Brinkman- Darcy interfaces, continuity of velocity, normal stresses, microrotations, and the Beavers and Joseph condition are implemented. Plots of the stream functions as they vary with hydraulic resistivity, micropolar coupling parameter, tangential stress jump parameter, and the thickness of the porous shell are presented and discussed. An analytical expression for the dimensionless drag, in the unbounded fluid case, is derived, and plots of the dimensionless drag as it varies with hydraulic resistivity and porous layer thickness are presented. The results presented by Ramkissoon and Majumdar [

We consider the case of a steady, creeping, axisymmetric, incompressible micropolar fluid flow past a porous shell.

The ambient velocity

The velocity vector

For axisymmetric flow, Stokes Stream functions

Region I consists of micropolar fluid outside the porous sphere, and Region III consists of micropolar fluid inside the inner radius of the shell,

The micropolar constants

Region B1,

Region II,

A dimensionless analysis is performed on the governing equations for all the regions. The dimensionless variables

The dimensionless variables are substituted into the governing equations and for simplicity, the superscript

The governing equations in dimensionless form for the various regions are given by Region I and III (

Region B1 and B2

Region II:

Solutions for Regions I and III, as well as regions B1 and B2, are given by Srinivasacharya and Rajyalakshmi [

Following the analysis given by Happel and Brenner [

where

The following dimensionless boundary conditions are used to obtain the twenty-five (25) total constants present in the general solutions for all the regions:

1. The radial velocity

2. The radial and tangential velocity fields are defined at

3. The microrotations

4. Continuity of both radial

5. Continuity of normal stresses at

6. Ochoa-Tapia jump in tangential Stress Jump condition at

7. The micro-rotations

8. Beavers and Joseph boundary condition at

The solutions for the different regions given by

The streamline patterns are displayed in spherical polar coordinates

Streamline patterns for varying hydralic resistivity

Streamline patterns for varying porous lauer thickness

For unbounded micropolar fluid flow past a porous shell the boundary conditions remain the same as for the bounded system, except that the three (3) boundary conditions at

The stream function

The drag formula for a sphere in a micropolar fluid is given by Ramkissoon and Majumdar [

Substituting

In

There is a decrease in dimensionless drag with increasing hydraulic resistivity

Increasing

The parameters of specific interest are the manufacturable porous parameters, hydraulic resistivity

The circulation regions, on either side of the porous shell, crosses the porous regions and enters the outer micropolar region. This effect is most significant for lower values of

For higher values of

Circulation regions become more pronounced and extend into the outer micropolar fluid region for a porous layer thickness of

To ensure effective delivery of the drugs stored in the porous region, significant mixing between the porous and outer fluid regions must take place. The results show that mixing between the porous and outer fluid region is at its most pronounced for low values of