# Stokes Flow past a Swarm of Porous Nanocylindrical Particles Enclosing a Solid Core

Authors: Satya Deo; Pramod Kumar Yadav
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008-02-13)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1, 2] and Kuwabara [3]
proposed a cell model in which two concentric cylinders/spheres serve as the
model for fluid moving through an assemblage of circular cylinders/spheres. The
Kuwabara model assumes uniform velocity condition and vanishing of vorticity at
the cell surface, whereas, Happel model assumes vanishing of shear stress
instead of vanishing of vorticity at the cell surface. An analytical study of
the steady incompressible flow past a circular cylinder embedded in a porous
medium based on the Brinkman model has been reported by Pop and Cheng [4]. The
drag on flow past a cylinder with slip was also evaluated by Datta and Shukla
[5]. The problem of Stokes flow through a swarm of spherical particles moving
in arbitrary direction was studied by Dassios and Vafeas [6] by using 3D Happel
model. Stokes flow past a swarm of porous circular cylinders with Happel and
Kuwabara boundary conditions was discussed by Deo [7]. Recently, a new model
for calculating specific resistance of aggregated colloidal cake layers in
membrane filtration processes was discussed by Kim and Yuan [8].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 [9]. Special known results are then also deduced from the
present analysis.


## 2. Statement and Mathematical Formulation of the Problem

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 bea and let the radius of the concentric porous
cylindrical shell enclosing the solid cylindrical core be b(b>a). Further, we assume that this porous shell is
enveloped by a concentric cylinder of radius c(c>b) named as cell surface, and let the radius of each
nano porous cylindrical particle be ap (Figure 1). Also, we assume that the
fluid is approaching towards the cell surface as well as partially passing through
the composite cylinder perpendicular to the axis of cylinder (z-axis) with
velocity U from left to right. The radius c of hypothetical cell is so chosen that the
solid volume fraction γ of
the swarm is equal to the solid volume fraction of the cell,  that is,
(2.1)γ=πb2πc2.Figure 1
The physical model and the coordinate system.


### 2.1. Governing Equations

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 [10])
(2.2)μ1∇2v(1)=∇p(1).
Also, we assume
that the flow inside the porous cylindrical shell is governed by Brinkman’s [11]
equation
(2.3)∇2v(2)−(σ2b2)v(2)=(1μ2)∇p(2).
Here, σ2=βb2/k, with β=μ1/μ2,μ1 is the viscosity of the fluid,μ2 denotes the effective viscosity of porous
medium, withk being the permeability of porous medium. Since,σ is a dimensionless number related with
the permeability, therefore we called it as dimensionless permeability
parameter. Here, v(i),p(i),i=1,2, are the velocity vector and pressure outside
and inside the porous cylindrical shell, respectively.The equations of
continuity for axisymmetric, incompressible viscous fluid in cylindrical polar
coordinates(r,θ,z) in both regions can be written as
(2.4)∂∂r(rvr(i))+∂∂θ(vθ(i))=0,
where vr(i) and vθ(i),i=1,2, are components of velocities in the direction of r and θ,
respectively. The stream functions ψ(i)(r,θ) in both regions satisfying equations of
continuity (2.4) may be defined as
(2.5)vr(i)=1r∂ψ(i)∂θ,vθ(i)=−∂ψ(i)∂r.Therefore,
on elimination of pressures in both (2.2) and (2.3) and on using (2.4), we get
the following fourth-order partial differential equations, respectively, as
(2.6)∇4ψ(1)=∇2(∇2ψ(1))=0,(2.7)∇4ψ(2)−(σ2b2)∇2ψ(2)=∇2(∇2−(σ2b2))ψ(2)=0,
where the Laplacian operator
(2.8)∇2=∂2∂r2+1r∂∂r+1r2∂2∂θ2,
with the macroscopic assumption of μ1/μ2=1.The
range ofr and θ in the above (2.6) and (2.7) within a cylinder
can be given below as
(2.9)0<r<∞,0<θ≤2π.
Furthermore, the
expressions for tangential and normal stresses Trθ(i),Trr(i),i=1,2, respectively, are given by
(2.10)Trθ(i)=μi[1r2∂2ψ(i)∂θ2+1r∂ψ(i)∂r−∂2ψ(i)∂r2],Trr(i)=−p(i)+2μir[∂2ψ(i)∂r∂θ−1r∂ψ(i)∂θ].
Also, the
pressure may be obtained in both regions (Happel and Brenner [10]) by integrating the
following relations, respectively, as
(2.11)∂p(i)∂r=μi[∇2vr(i)−vr(i)r2−2r2∂vθ(i)∂θ−δ2i(σb)2vr(i)],1r∂p(i)∂θ=μi[∇2vθ(i)−vθ(i)r2+2r2∂vr(i)∂θ−δ2i(σb)2vθ(i)],
where δ21=0 and δ22=1.A suitable
stream function solution of the Stokes equation (2.6) can be expressed as
(2.12)ψ(1)(r,θ)=Ub[A1r'+B1r'3+C1r'+D1r'lnr']sinθ.
A particular
solution of the Brinkman equation (2.7) may be written as
(2.13)ψ(2)(r,θ)=Ub[A2r'+B2r'+C2I1(σr')+D2K1(σr')]sinθ.
Here, I1(σr') and K1(σr') are the modified Bessel functions of
the order one of the first and second kinds (Abramowitz and Stegun [12]),
respectively, and the dimensionless variable r'=r/b.


## 2.1. Governing Equations

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 [10])
(2.2)μ1∇2v(1)=∇p(1).
Also, we assume
that the flow inside the porous cylindrical shell is governed by Brinkman’s [11]
equation
(2.3)∇2v(2)−(σ2b2)v(2)=(1μ2)∇p(2).
Here, σ2=βb2/k, with β=μ1/μ2,μ1 is the viscosity of the fluid,μ2 denotes the effective viscosity of porous
medium, withk being the permeability of porous medium. Since,σ is a dimensionless number related with
the permeability, therefore we called it as dimensionless permeability
parameter. Here, v(i),p(i),i=1,2, are the velocity vector and pressure outside
and inside the porous cylindrical shell, respectively.The equations of
continuity for axisymmetric, incompressible viscous fluid in cylindrical polar
coordinates(r,θ,z) in both regions can be written as
(2.4)∂∂r(rvr(i))+∂∂θ(vθ(i))=0,
where vr(i) and vθ(i),i=1,2, are components of velocities in the direction of r and θ,
respectively. The stream functions ψ(i)(r,θ) in both regions satisfying equations of
continuity (2.4) may be defined as
(2.5)vr(i)=1r∂ψ(i)∂θ,vθ(i)=−∂ψ(i)∂r.Therefore,
on elimination of pressures in both (2.2) and (2.3) and on using (2.4), we get
the following fourth-order partial differential equations, respectively, as
(2.6)∇4ψ(1)=∇2(∇2ψ(1))=0,(2.7)∇4ψ(2)−(σ2b2)∇2ψ(2)=∇2(∇2−(σ2b2))ψ(2)=0,
where the Laplacian operator
(2.8)∇2=∂2∂r2+1r∂∂r+1r2∂2∂θ2,
with the macroscopic assumption of μ1/μ2=1.The
range ofr and θ in the above (2.6) and (2.7) within a cylinder
can be given below as
(2.9)0<r<∞,0<θ≤2π.
Furthermore, the
expressions for tangential and normal stresses Trθ(i),Trr(i),i=1,2, respectively, are given by
(2.10)Trθ(i)=μi[1r2∂2ψ(i)∂θ2+1r∂ψ(i)∂r−∂2ψ(i)∂r2],Trr(i)=−p(i)+2μir[∂2ψ(i)∂r∂θ−1r∂ψ(i)∂θ].
Also, the
pressure may be obtained in both regions (Happel and Brenner [10]) by integrating the
following relations, respectively, as
(2.11)∂p(i)∂r=μi[∇2vr(i)−vr(i)r2−2r2∂vθ(i)∂θ−δ2i(σb)2vr(i)],1r∂p(i)∂θ=μi[∇2vθ(i)−vθ(i)r2+2r2∂vr(i)∂θ−δ2i(σb)2vθ(i)],
where δ21=0 and δ22=1.A suitable
stream function solution of the Stokes equation (2.6) can be expressed as
(2.12)ψ(1)(r,θ)=Ub[A1r'+B1r'3+C1r'+D1r'lnr']sinθ.
A particular
solution of the Brinkman equation (2.7) may be written as
(2.13)ψ(2)(r,θ)=Ub[A2r'+B2r'+C2I1(σr')+D2K1(σr')]sinθ.
Here, I1(σr') and K1(σr') are the modified Bessel functions of
the order one of the first and second kinds (Abramowitz and Stegun [12]),
respectively, and the dimensionless variable r'=r/b.


## 3. Solution of the Problem with Kuwabara Boundary Condition

The boundary conditions, those are physically realistic
and mathematically consistent for the problem, can be taken as given below. On the solid
cylindrical core,(3.1)vr(2)(a,θ)=0,vθ(2)(a,θ)=0.
On the porous
surface,
(3.2)vr(2)(b,θ)=vr(1)(b,θ),vθ(2)(b,θ)=vθ(1)(b,θ),Trr(2)(b,θ)=Trr(1)(b,θ),Trθ(2)(b,θ)=Trθ(1)(b,θ).
On the
hypothetical cell surface,
(3.3)vr(1)(c,θ)=Ucosθ.
The vanishing of
vorticity on the cell surface, that is, Kuwabara condition, implies that
(3.4)∇2ψ(1)(c,θ)=0.


### 3.1. Determination of Arbitrary Constants

Applying the boundary conditions given by (3.1)–(3.4) and solving
the resulting equations, we get the values of all the arbitrary constants A1,B1,C1,D1,A2,B2,C2, and D2 appearing in (2.12) and (2.13).


## 3.1. Determination of Arbitrary Constants

Applying the boundary conditions given by (3.1)–(3.4) and solving
the resulting equations, we get the values of all the arbitrary constants A1,B1,C1,D1,A2,B2,C2, and D2 appearing in (2.12) and (2.13).


## 4. Evaluation of Drag

Integrating the
normal and tangential stresses over the porous cylindrical shell of radiusb in a cell yields
the experienced drag force F per unit length as
(4.1)F=∫02π(Trr(1)cosθ−Trθ(1)sinθ)r=brdθ=4πμ1UD1,
where
(4.2)D1=−4m4σ2[2I1(σℓ)K1(σ)+ℓ(1+ℓ2)σI2(σℓ)K1(σ)+I1(σ){−2K1(σℓ)+ℓ(1+ℓ2)σK2(σℓ)}]Δ,Δ=−16m4ℓσI2(σℓ)K1(σ)+8ℓ3σI2(σℓ)K1(σ)+16m2ℓ3σI2(σℓ)K1(σ)−8m4ℓ3σI2(σℓ)K1(σ)+ℓσ3I2(σℓ)K1(σ)−4m2ℓσ3I2(σℓ)K1(σ)+3m4ℓσ3I2(σℓ)K1(σ)+ℓ3σ3I2(σℓ)K1(σ)−4m2ℓ3σ3I2(σℓ)K1(σ)+3m4ℓ3σ3I2(σℓ)K1(σ)+32m4I1(σ)K1(σℓ)−2σ2I1(σ)K1(σℓ)+8m2σ2I1(σ)K1(σℓ)−6m4σ2I1(σ)K1(σℓ)−8ℓ2σI2(σℓ)K1(σℓ)+8m4ℓ2σI2(σℓ)K1(σℓ)−8ℓσI1(σ)K2(σ)+8m4ℓσI1(σ)K2(σ)+4ℓσ2I2(σℓ)K2(σ)−8m2ℓσ2I2(σℓ)K2(σ)+4m4ℓσ2I2(σℓ)K2(σ)+4ℓ3σ2I2(σℓ)K2(σ)−8m2ℓ3σ2I2(σℓ)K2(σ)+4m4ℓ3σ2I2(σℓ)K2(σ)−16m4ℓσI1(σ)K2(σℓ)+8ℓ3σI1(σ)K2(σℓ)+16m2ℓ3σI1(σ)K2(σℓ)−8m4ℓ3σI1(σ)K2(σℓ)+ℓσ3I1(σ)K2(σℓ)−4m2ℓσ3I1(σ)K2(σℓ)+3m4ℓσ3I1(σ)K2(σℓ)+ℓ3σ3I1(σ)K2(σℓ)−4m2ℓ3σ3I1(σ)K2(σℓ)+3m4ℓ3σ3I1(σ)K2(σℓ)+4(1−m2)σI2(σ)[−2(1+m2)ℓK1(σ)−(−1+m2){2K1(σℓ)−ℓ(1+ℓ2)σK2(σℓ)}]−4m4ℓσ3I2(σℓ)K1(σ)lnm−4m4ℓ3σ3I2(σℓ)K1(σ)lnm+8m4σ2I1(σ)K1(σℓ)lnm−4m4ℓσ3I1(σ)K2(σℓ)lnm−4m4ℓ3σ3I1(σ)K2(σℓ)lnm+2I1(σℓ)[4(−1+m2)σ{(−1+m2)K2(σ)+(1+m2)ℓ2K2(σℓ)}+K1(σ){σ2−4m2σ2+m4(−16+3σ2)−4m4σ2lnm}].
Also,  the drag coefficient CD can be defined as
(4.3)CD=F(1/2)ρU22b=8πD1Re,
where Re=2bU/ν1 is the Reynolds number, and ν1=μ1/ρ being the kinematic viscosity of fluid.


### 4.1. Deductions of Some Known Results

#### 4.1.1. Drag on a Porous Circular Cylinder in a Cell

Ifa=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. In this case, we get the value of the drag coefficient CD as
(4.4)CD=32πσ2I1(σ)Re[{−4(σ2(1−γ)−4)+σ2(−(2logγ+1)+γ2)}I1(σ)+2σ{σ(1−γ2)I1(σ)+2(1−γ)2I2(σ)}],
where γ=(πb2)/(πc2)=1/m2 being the particle volume fraction.A known result has been reported earlier
by Deo [7] for the drag force experienced by a porous circular cylinder in a
cell.


#### 4.1.2. Drag on a Solid Cylinder in Kuwabara Cell Model (k→0)

When permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.5)CD=32πRe(4γ−γ2−3−2lnγ).
A known result
for the drag has been reported earlier by Kuwabara [3].


#### 4.1.3. Happel Boundary Condition

Happel assumes that on the cell surface shear stress vanishes
instead of vorticity. In this case,we take the seven boundary conditions in
(3.1)–(3.3) to be the
same as in the previous case but in place of eighth condition (3.4), Happel boundary
condition is used. Thus, vanishing of shear stress on the cell surface implies
that
(4.6)Trθ(1)(c,θ)=0.
Applying the
boundary conditions (3.1)–(3.3) with (4.6)
and solving the resulting equations, we get the values of unknown constants
appearing in (2.12) and (2.13). Therefore, we get the explicit expressions of
the stream functions, and, hence, velocity distributions, pressure distributions,
stress, vorticity, and the drag force may be evaluated. Instead, we report the
values of the drag coefficient for the simpler cases as mentioned below.


#### 4.1.4. Drag on a Porous Circular Cylinder in a Cell

In particular, whena=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. Thus, the value of the drag coefficient CD will come out as
(4.7)CD=16πσ2[σI1(σ)(1+γ2)−4γ2I2(σ)]Re[σ2(γ2−1)[σI1(σ)−2I2(σ)]+(8−σ2lnγ)[σ(1+γ2)I1(σ)−4γ2I2(σ)]].


#### 4.1.5. Drag on a Solid Cylinder in Happel Cell Model (k→0)

Again, if permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.8)CD=16π{Re[(lnγ+1)−2γ2/(1+γ2)]},
which agrees with the result reported
earlier by Happel [2] for the drag force experienced by a solid cylinder in a
cell.


## 4.1. Deductions of Some Known Results

### 4.1.1. Drag on a Porous Circular Cylinder in a Cell

Ifa=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. In this case, we get the value of the drag coefficient CD as
(4.4)CD=32πσ2I1(σ)Re[{−4(σ2(1−γ)−4)+σ2(−(2logγ+1)+γ2)}I1(σ)+2σ{σ(1−γ2)I1(σ)+2(1−γ)2I2(σ)}],
where γ=(πb2)/(πc2)=1/m2 being the particle volume fraction.A known result has been reported earlier
by Deo [7] for the drag force experienced by a porous circular cylinder in a
cell.


### 4.1.2. Drag on a Solid Cylinder in Kuwabara Cell Model (k→0)

When permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.5)CD=32πRe(4γ−γ2−3−2lnγ).
A known result
for the drag has been reported earlier by Kuwabara [3].


### 4.1.3. Happel Boundary Condition

Happel assumes that on the cell surface shear stress vanishes
instead of vorticity. In this case,we take the seven boundary conditions in
(3.1)–(3.3) to be the
same as in the previous case but in place of eighth condition (3.4), Happel boundary
condition is used. Thus, vanishing of shear stress on the cell surface implies
that
(4.6)Trθ(1)(c,θ)=0.
Applying the
boundary conditions (3.1)–(3.3) with (4.6)
and solving the resulting equations, we get the values of unknown constants
appearing in (2.12) and (2.13). Therefore, we get the explicit expressions of
the stream functions, and, hence, velocity distributions, pressure distributions,
stress, vorticity, and the drag force may be evaluated. Instead, we report the
values of the drag coefficient for the simpler cases as mentioned below.


### 4.1.4. Drag on a Porous Circular Cylinder in a Cell

In particular, whena=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. Thus, the value of the drag coefficient CD will come out as
(4.7)CD=16πσ2[σI1(σ)(1+γ2)−4γ2I2(σ)]Re[σ2(γ2−1)[σI1(σ)−2I2(σ)]+(8−σ2lnγ)[σ(1+γ2)I1(σ)−4γ2I2(σ)]].


### 4.1.5. Drag on a Solid Cylinder in Happel Cell Model (k→0)

Again, if permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.8)CD=16π{Re[(lnγ+1)−2γ2/(1+γ2)]},
which agrees with the result reported
earlier by Happel [2] for the drag force experienced by a solid cylinder in a
cell.


## 4.1.1. Drag on a Porous Circular Cylinder in a Cell

Ifa=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. In this case, we get the value of the drag coefficient CD as
(4.4)CD=32πσ2I1(σ)Re[{−4(σ2(1−γ)−4)+σ2(−(2logγ+1)+γ2)}I1(σ)+2σ{σ(1−γ2)I1(σ)+2(1−γ)2I2(σ)}],
where γ=(πb2)/(πc2)=1/m2 being the particle volume fraction.A known result has been reported earlier
by Deo [7] for the drag force experienced by a porous circular cylinder in a
cell.


## 4.1.2. Drag on a Solid Cylinder in Kuwabara Cell Model (k→0)

When permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.5)CD=32πRe(4γ−γ2−3−2lnγ).
A known result
for the drag has been reported earlier by Kuwabara [3].


## 4.1.3. Happel Boundary Condition

Happel assumes that on the cell surface shear stress vanishes
instead of vorticity. In this case,we take the seven boundary conditions in
(3.1)–(3.3) to be the
same as in the previous case but in place of eighth condition (3.4), Happel boundary
condition is used. Thus, vanishing of shear stress on the cell surface implies
that
(4.6)Trθ(1)(c,θ)=0.
Applying the
boundary conditions (3.1)–(3.3) with (4.6)
and solving the resulting equations, we get the values of unknown constants
appearing in (2.12) and (2.13). Therefore, we get the explicit expressions of
the stream functions, and, hence, velocity distributions, pressure distributions,
stress, vorticity, and the drag force may be evaluated. Instead, we report the
values of the drag coefficient for the simpler cases as mentioned below.


## 4.1.4. Drag on a Porous Circular Cylinder in a Cell

In particular, whena=0,
that is, ℓ=a/b=0, then cylindrical shell will reduce to a porous circular
cylinder of radius b. Thus, the value of the drag coefficient CD will come out as
(4.7)CD=16πσ2[σI1(σ)(1+γ2)−4γ2I2(σ)]Re[σ2(γ2−1)[σI1(σ)−2I2(σ)]+(8−σ2lnγ)[σ(1+γ2)I1(σ)−4γ2I2(σ)]].


## 4.1.5. Drag on a Solid Cylinder in Happel Cell Model (k→0)

Again, if permeabilityk vanishes, that is, permeability parameter σ→∞, then the porous circular cylinder behaves
like a solid cylinder of radius b. In this case, the value of the drag
coefficient CD will become as
(4.8)CD=16π{Re[(lnγ+1)−2γ2/(1+γ2)]},
which agrees with the result reported
earlier by Happel [2] for the drag force experienced by a solid cylinder in a
cell.


## 5. Conclusions

Figure2 shows that the
comparison between Happel and Kuwabara results in the porous circular cylindrical shell for
various values of the particle volume fraction γ,
when permeability parameter σ varies as parameter and ℓ=0.4.
It is seen that the variation of the drag coefficient CD is large in case of Kuwabara boundary condition
in comparison to the case of Happel boundary condition. Figure 3 shows that the
comparison between Happel and Kuwabara results in the porous circular cylinder for various values
of the permeability parameter σ,
when particle volume fraction γ varies and ℓ=0.4.
It is seen that the variation of the drag coefficient CD is large in case of Kuwabara boundary condition
in comparison to the case of Happel boundary condition when particle volume
fraction γ varies as parameter.Figure 2
Variation of the drag coefficientCD versus permeability parameter σ for various values of particle volume fraction γ.Figure 3
Variation of the drag coefficientCD versus particle volume fraction γ for various values of permeability parameter σ.



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*Source: 651910-2008-04-30.xml*