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This paper concerns the Stokes flow of an incompressible viscous fluid past a swarm of porous nanocylindrical particles enclosing a solid cylindrical core with Kuwabara boundary condition. An aggregate of porous nanocylindrical particles is considered as a hydro-dynamically equivalent to a solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. Explicit expressions for the stream functions in both regions have been investigated. The drag force acting at each nanoporous cylindrical particle in a cell is evaluated. Also, we solved the same problem by using Happel boundary condition on the hypothetical cell. In certain limiting cases, drag force converges to pre-existing analytical results, such as the drag on a porous circular cylinder and the drag on a solid cylinder in Kuwabara's cell or Happel's cell. Representative results are then discussed and compared for both cases and presented in graphical form by using Mathematica software.

The classical problems of the motion of objects through fluids
continue to be of interest because of their applications in physical sciences
and chemical engineering. A variety of physical situations arises in which the
size of moving objects varies from micro-(10^{−6} meter) to nano-(10^{−9} meter) scales. The computational predictions of the relevant hydrodynamical
parameters of the flow of a viscous incompressible fluid past a swarm of porous
particles at nanoscale are of considerable practical and theoretical interest
of many physical, engineering, and medical problems.

Happel [

In the present work, the problem of
the Stokes flow past a swarm of porous nanocylindrical particles enclosing a
solid cylindrical core with Kuwabara boundary condition is considered. The
Brinkman equation for the flow inside and the Stokes equation outside the
porous cylindrical shell in their stream function formulations is used. The
drag force experienced by each nanoporous circular cylindrical particle in a
cell is evaluated. Representative results are presented in graphical form by
using Mathematica and they are compared in both cases. The Happel formulation
is slightly superior because it leads to particles-in-cell that are self-sufficient
in mechanical energy [

A primary assumption employed in this
study is that a swarm of nanosized porous coaxial (along z-axis) cylindrical
particles surrounding a solid cylindrical core having the same axis is hydro-dynamically
equivalent to a coaxial porous cylindrical shell surrounding the solid core.
Let the radius of the solid cylindrical core be

The physical model and the coordinate system.

The governing equation of incompressible Newtonian creeping flow for clear fluid, that is, outside the porous cylindrical shell, is governed by Stokes equation (Happel and
Brenner [

The equations of
continuity for axisymmetric, incompressible viscous fluid in cylindrical polar
coordinates

Therefore,
on elimination of pressures in both (

The
range of

A suitable
stream function solution of the Stokes equation (

The boundary conditions, those are physically realistic
and mathematically consistent for the problem, can be taken as given below. On the solid
cylindrical core,

Applying the boundary conditions given by (

Integrating the
normal and tangential stresses over the porous cylindrical shell of radius

If

A known result has been reported earlier
by Deo [

When permeability

Happel assumes that on the cell surface shear stress vanishes
instead of vorticity. In this case,we take the seven boundary conditions in
(

In particular, when

Again, if permeability

Figure

Variation of the drag coefficient

Variation of the drag coefficient

The first author is thankful to the Department of Science and Technology, Government of India for providing the financial assistance under its projects Grant no. SR/FTP/MS-07/2004 during this work. Authors acknowledge their sincere thanks to the reviewer for his valuable suggestions which led to much improvement in the presentation of the paper.