Z. angew. Math. Phys. 51 (2000) 481–490
0044-2275/00/030481-10 $ 1.50+0.20/0
c 2000 Birkhäuser Verlag, Basel
Zeitschrift für angewandte
Mathematik und Physik ZAMP
Stokes flow inside a porous spherical shell
G. P. Raja Sekhar and T. Amaranath
Abstract. Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Darcy’s law for the flow in the porous region is
discussed. The formulae for drag and torque are found by deriving the corresponding Faxen’s
laws. It is found that torque does not depend on the thickness of the spherical shell.
Mathematics Subject Classification (1991). 76D07.
Keywords. Stokes flow, Darcy law, porous spherical shell.
1. Introduction
Interest in flow past spherical boundaries began with the pioneering work of Hasimoto [1] who discussed axisymmetric flow past a rigid sphere. Subsequently several
papers ( Shail [2], Collins [3], Palaniappan [4]) appeared on this topic. Recently
an interesting review of Stokes flows past spherical boundaries is due to Hasimoto
[5].
Palaniappan et al. [4] proposed a solution for Stokes equations. This solution
was shown to be a complete general solution by Raja Sekhar et al. [6]. Here we
use this solution to discuss the problem of a non-axisymmetric Stokes flow of a
viscous, incompressible fluid inside a sphere of radius ‘b’ with internal singularities,
enclosed by a porous spherical shell of thickness (a − b) with an impermeable outer
boundary. We assume that the flow in the porous region obeys Darcy law [7] and
the flow inside the liquid sphere obeys Stokes equations. We assume that the flow
is generated by singularities inside the clear fluid region, as in the absence of body
forces fluid flow can be generated by the presence of such singularities. One has
to impose suitable boundary conditions at the interface of clear fluid and porous
region. Several types of boundary conditions were suggested in literature. Beavers
and Joseph [8] have proposed an empirical slip flow condition for a plane boundary.
In the limit of small permeability, Saffman [9] gave a theoretical justification for
the empirical boundary condition proposed by Beavers and Joseph [8]. In the
limit k → 0, where k represents permeability they showed that one can derive the
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G. P. Raja Sekhar and T. Amaranath
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√
k du
+ O(k)
α dn
at the interface, where u is the tangential velocity and n is measured along the
normal to the interface and this result is consistent with that of Beavers and
Joseph. The theoretical and experimental evidence supported the view that the
Darcy equation offers a good representation of the flow in many low porosity
systems and therefore for relatively small permeabilities. In light of this, Saffman’s
condition has gained considerable importance.
Recently Ochoa - Tapia and Whitaker ([10], [11]) suggested a stress jump
boundary condition at the fluid/porous interface. They have used the Brinkman
- extended Darcy law and shown that the process of matching the Brinkman extended Darcy law to the Stokes equations requires a discontinuity in the stress.
Kuznetsov ([12], [13], [14]) has used the stress jump boundary condition at the
interface between a porous medium and a clear fluid to discuss flow in channels
partially filled with a porous medium.
We shall use the Saffman [9] condition together with continuity of the normal
velocity component and pressure at the permeable boundary, as we are using Darcy
law for the porous region and not the Brinkman extended Darcy law. We also
discuss some physical properties like drag and torque by deriving the corresponding
Faxén’s [15] laws.
condition
u=
2. Mathematical formulation
Let us consider an arbitrary Stokes flow of a viscous, incompressible fluid inside a
sphere of radius ‘b’ with internal singularities, enclosed by a porous spherical shell
of thickness (a − b). The flow inside the region r < b is governed by the Stokes
equations
µ∇2 Vi = ∇pi ,
∇.Vi = 0,
(1)
(2)
where µ is the coefficient of viscosity, Vi the velocity and pi the pressure of the
fluid. The superscript ‘i’ denotes the flow inside the region (r < b). Similarly,
we use the superscript ‘e’ for the flow inside the porous region (b ≤ r ≤ a). The
velocity components of Vi in spherical polar coordinates are qri , qθi and qϕi . The
flow inside the porous region (b ≤ r ≤ a) is governed by the Darcy’s law
k
(3)
Ve = − ∇pe ,
µ
∇.Ve = 0,
(4)
where Ve is the volume rate of flow per unit cross section area, pe the pressure and
k > 0 is the permeability coefficient. The velocity components of Ve in spherical
polar coordinates are qre , qθe and qϕe .
Vol. 51 (2000)
Stokes flow inside a porous spherical shell
483
3. Boundary conditions
The boundary conditions are assumed to be as follows:
a) the pressure is continuous on r = b
pe (b, θ, ϕ) = pi (b, θ, ϕ),
(5)
b) the normal component of the velocity is continuous on the boundary of the shell
r=b
qre (b, θ, ϕ) = qri (b, θ, ϕ),
(6)
c) Saffman’s conditions for the tangential components of velocity on the shell are
satisfied on r = b
√ ∂q i
k θ (b, θ, ϕ),
∂r
i
√
∂q
ϕ
qϕi (b, θ, ϕ) = k
(b, θ, ϕ),
∂r
qθi (b, θ, ϕ) =
(7)
(8)
d) the normal component of velocity vanishes on r = a
qre (a, θ, ϕ) = 0,
(9)
4. Method of solution
We assume the representation proposed by Palaniappan et al. [4] for the velocity
and pressure in the region r < b, given by
Vi = CurlCurl(rAi ) + Curl(rB i ),
∂
pi = p0 + µ (r∇2 Ai ),
∂r
(10)
(11)
where ∇4 Ai = 0, ∇2 B i = 0 and p0 is a constant. Suppose the basic flow is given
by
A0 (r, θ, ϕ) =
B0 (r, θ, ϕ) =
∞
X
n=1
∞
X
n=1
αn
rn+1
χn
rn+1
0
αn
+ n−
r 1
Tn (θ, ϕ),
!
Sn (θ, ϕ),
(12)
(13)
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G. P. Raja Sekhar and T. Amaranath
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where
Sn (θ, ϕ) =
Tn (θ, ϕ) =
n
X
m=0
n
X
Pnm (ζ)(Anm cos mϕ + Bnm sin mϕ),
(14)
Pnm (ζ)(Cnm cos mϕ + Dnm sin mϕ),
(15)
m=0
0
and ζ = cos θ, Pnm is the associated Legendre polynomial, αn , αn , χn , Anm , Bnm ,
Cnm and Dnm are known constants. The expansions given in (12) and (13) are
valid in the region 0 < c < r < b and we shall explain in specific examples how these
expansions occur. These expansions are used after adding the necessary perturbed
terms to satisfy the boundary conditions on r = b. Also A0 and B0 given in the
equations (12) and (13) satisfy the expressions ∇4 A0 = 0 and ∇2 B0 = 0. Now,
in presence of the rigid boundary r = a and the permeable boundary r = b, the
region of thickness (a − b) being porous, the modified flow is assumed in the region
r < b as follows
!
0
∞
X
0
α
α
n
n
Ai =
+ n−
+ βn rn + βn rn+2 Sn (θ, ϕ),
(16)
rn+1
r 1
Bi =
n=1
∞ X
n=1
χn
rn+1
+ σn rn Tn (θ, ϕ)
(17)
and the pressure in the porous region (b < r < a) is assumed as
pe = p0 +
∞ X
r n n +
n=1
0
0
Sn (θ, ϕ),
n+1 n
1
r
(18)
0
where βn , βn , σn , n and n are unknown constants to be determined from the
boundary conditions. The modified flow given in (16) - (17) is assumed by adding
the perturbed terms to the basic flow (12) - (15). The form of Ai and B i ensures
that they satisfy ∇4 Ai = 0 and ∇2 B i = 0 respectively and that the perturbed
terms vanish as r → 0. Using the scalar functions Ai and B i given in equations
(16) - (17), the velocity components and pressure given in (18) for the modified
flow inside the region (r < b) become
qri =
∞
X
h
0
n(n + 1) αn r−(n+2) + αn r−n
n=1
i
0
+ rn−1 βn + rn+1 βn Sn (θ, ϕ),
qθi =
∞ h
X
n=1
(19)
0
−nr−(n+2) αn − (n − 2)r−n αn + (n + 1)rn−1 βn
Vol. 51 (2000)
Stokes flow inside a porous spherical shell
0
+ (n + 3)rn+1 βn
∞ h
X
qϕi =
i ∂S
h
i ∂T
n
+ csc θ χn r−(n+1) + σn rn
,
∂θ
∂ϕ
n
485
(20)
0
−nr−(n+2) αn − (n − 2)r−n αn + (n + 1)rn−1 βn
n=1
i
i ∂T
0
∂Sn h
n
− χn r−(n+1) + σn rn
,
+ (n + 3)rn+1 βn csc θ
∂ϕ
∂θ
∞ h
X
0
pi = p0 + 2µ
n(2n − 1)r−(n+1) αn
(21)
n=1
i
0
+ (n + 1)(2n + 3)rn βn Sn (θ, ϕ).
(22)
For the porous region (b < r < a) from the equations (3) and (18) the components
of volume rate of flow and the pressure are given respectively by
qre = −
qθe = −
∞
i
0
k X h n−1
nr
n − (n + 1)r−(n+2) n Sn (θ, ϕ),
µ
k
µ
n=1
∞ h
X
0
rn−1 n + r−(n+2) n
n=1
i ∂S
n
∂θ
,
i ∂S
Xh
0
k
n
= − csc θ
rn−1 n + r−(n+2) n
,
µ
∂ϕ
(23)
(24)
∞
qϕe
(25)
n=1
pe = p0 +
∞ X
0
rn n + r−(n+1) n Sn (θ, ϕ).
(26)
n=1
The expressions given in equations (19) - (26) satisfy the boundary conditions
given in equations (5) - (9) if
0
0
(An αn + An αn )
,
Ln
0
0
(Bn αn + Bn αn )
=
,
Ln
0
0
(Cn αn + Cn αn )
=
,
Ln
0
0
(Cn αn + Cn αn )
n
=
a2n+1
,
n+1
Ln
√
[b + (n + 1) k]χn
√
=−
,
[b − n k]b2n+1
βn =
0
βn
n
0
n
σn
where
h
√ An = −(2n + 3) b2 (b − k) na2n+1 + (n + 1)b2n+1
(27)
(28)
(29)
(30)
(31)
486
G. P. Raja Sekhar and T. Amaranath
i
√ + 2n(n + 1)k(b + (n + 2) k) b2n+1 − a2n+1 ,
h
√ 0
An = −(2n + 1) b4 (b − 3 k) na2n+1 + (n + 1)b2n+1
i
√ 2n+1
+ 2kb2 2(n2 + n − 3)b − 3n(n + 1) k
b
− a2n+1 ,
√ Bn = (2n + 1)(b + k) na2n+1 + (n + 1)b2n+1 ,
h
√ 0
Bn = (2n − 1) b2 (b − k) na2n+1 + (n + 1)b2n+1
i
√ + 2n(n + 1)k(b − (n − 1) k) b2n+1 − a2n+1 ,
√
Cn = 2(n + 1)2 (2n + 1)(2n + 3)µ(b + k)b2n+1 ,
h
√ i
0
Cn = 2(n + 1)(2n − 1)(2n + 1)µ b(n + 3) − 3(n + 1) k b2n+3 ,
h
√ Ln = 2b2n+1 b2 (b − 2(n + 1) k) na2n+1 + (n + 1)b2n+1
i
√
− (n + 1)2 (2n + 3)k(b − (n − 1) k)(b2n+1 − a2n+1 ) .
ZAMP
(32)
(33)
(34)
(35)
(36)
(37)
(38)
5. Faxén’s laws for a porous spherical shell
Faxén’s [15] laws provide expressions for the drag and torque acting on the rigid
sphere of radius ‘a’ in an unbounded, arbitrary Stokes flow. We now give the
corresponding Faxén’s laws for the porous spherical shell. These expressions enable
us to calculate the drag D and the torque T from the basic velocity directly. The
force D exerted on the porous spherical shell by the fluid in the region r < b is
found to be
!
√
(b + k)(a3 + 2b3 )
6πµ
√
[Vo ]0
D=− 3
b
[b2 (b − 4 k)(a3 + 2b3 ) − 20kb(b3 − a3 )]
!
√
πµ
[b2 (b − k)(a3 + 2b3 ) + 4kb(b3 − a3 )]
√
− 3
[∇2 Vo ]0 ,
(39)
5b
[b2 (a3 + 2b3 )(b − 4 k) − 20kb(b3 − a3 )]
and the torque T on the shell is
4πµ
T= 3
b
√ !
3 k
√
1+
[∇ × Vo ]0 ,
b− k
(40)
where Vo is the velocity corresponding to the basic flow and [ ]0 is evaluation at
the origin r = 0.
Vol. 51 (2000)
Stokes flow inside a porous spherical shell
487
6. Examples
(i). Stokeslet
Consider a Stokeslet of strength F1 /8πµ located at (0, 0, c), (c < b < a) whose
axis is along the positive direction of the x-axis. The corresponding expressions
for A0 and B0 due to the Stokeslet are
F1
cos ϕ
(r cos θ − c + R1 )
,
8πµc
r sin θ
F1
sin ϕ
(r cos θ − c + R1 )
,
B0 =
8πµc
r sin θ
A0 =
(41)
(42)
where R12 = r2 + c2 − 2rc cos θ.
The expressions for A0 and B0 given in (41) and (42) are expanded in terms of
spherical harmonics. These expansions are valid for 0 < c < r and are given by
A0 =
∞
F1 X
cn−1
(n + 3)cn+1
[ n−1
−
]P 1 (ζ) cos ϕ,
8πµ
r
n(n + 1) n(n + 1)(2n + 3)rn+1 n
n=1
B0 =
F1
8πµ
∞
X
n=1
(43)
n
c
P 1 (ζ) sin ϕ.
rn+1 n(n + 1) n
(44)
The drag and torque are obtained using the formulae (39) and (40) as follows
!
√
3c2
(b + k)(a3 + 2b3 )
√
D= 3
F1 î
5b
[b2 (b − 4 k)(a3 + 2b3 ) − 20kb(b3 − a3 )]
!
√
1
[b2 (b − k)(a3 + 2b3 ) + 4kb(b3 − a3 )]
√
− 3
F1 î,
(45)
2b
[b2 (b − 4 k)(a3 + 2b3 ) − 20kb(b3 − a3 )]
√ !
c
3 k
√
F1 ĵ.
(46)
T= 3 1+
2b
b− k
Since Darcy’s law is supposed to be valid for small values of permeability, the
variation of drag and torque with the permeability is shown graphically for the
range 0 ≤ k ≤ 0.04 cm2 . The graphs are plotted for the values c = 0.5cm,
a = 2cm, b = 1cm and F1 = 1dyne.
(ii). Rotlet
Consider a rotlet of strength F2 /8πµ at (0, 0, c),(c < b < a) whose axis is along
the positive y-axis direction. The corresponding expressions for A0 and B0 due to
the rotlet are
F2
cos ϕ
(r cos θ − c + R1 )
,
8πµc
r sin θ
F2
sin ϕ
B0 = −
(r cos θ − c + R1 )
,
8πµcR1
r sin θ
A0 = −
(47)
(48)
488
G. P. Raja Sekhar and T. Amaranath
ZAMP
0.45
torque
drag
–0.25
–0.30
–0.35
0.40
0.35
0.30
–0.40
0.25
0.00 0.01 0.02 0.03 0.04
permeability
variation of torque for Stokeslet
0.00 0.01 0.02 0.03 0.04
permeability
variation of drag for Stokeslet
0.56
0.54
0.52
0.50
0.48
0.46
0.44
0.42
0.40
torque
drag
Figure 1.
0.00 0.01 0.02 0.03 0.04
permeability
variation of drag for Rotlet
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
0.00 0.01 0.02 0.03 0.04
permeability
variation of torque for Rotlet
Figure 2.
y
P
(0, 0, c): Location of the
singularity on z-axis
L: Liquid region
P: Porous region
L
x
(0, 0, c)
z
Figure 3.
where R12 = r2 + c2 − 2rc cos θ
Vol. 51 (2000)
Stokes flow inside a porous spherical shell
489
The corresponding expressions in series form for 0 < c < r are
∞
F2 X
cn
A0 = −
P 1 (ζ) cos ϕ,
8πµ
rn+1 n(n + 1) n
B0 = −
F2
8πµ
n=1
∞
X
n=1
cn−1 1
P (ζ) sin ϕ.
rn+1 n n
(49)
(50)
The drag and torque are obtained using the formulae (39) and (40) as follows
3c
D= 3
4b
T=
1
b3
!
√
(b + k)(a3 + 2b3 )
√
F2 î,
b2 (b − 4 k)(a3 + 2b3 ) − 20kb(b3 − a3 )
√ !
3 k
√
F2 ĵ.
1+
b− k
(51)
(52)
For the values c = 0.5cm, a = 2cm, b = 1cm and F2 = 1dyne, the variation of
drag and torque with permeability is shown graphically in Fig.2 (i) and Fig.2(ii)
respectively for the range 0 ≤ k ≤ 0.04cm2 . In Fig.2(i), it is seen that the drag
decreases in magnitude and is in the negative x-direction.
The expressions given for A0 and B0 in equations (43) and (44) for a Stokeslet
at (0, 0, c) and (49) and (50) for a rotlet at (0, 0, c) respectively are what we have
presented in equations (12) and (13) in a general case, which are valid in c < r.
Conclusions
We have discussed a non-axisymmetric Stokes flow inside a sphere of radius b
bounded by a porous spherical shell of thickness (a − b). The flow is generated
by the presence of singularities inside the sphere as, in absence of body forces
fluid flow can be generated by singularities. Expressions for drag and torque
are presented in equations (39) and (40) respectively. It can be observed from
equation (40) that torque is always greater than that corresponding to a rigid
sphere. It does not depend on the thickness of the shell. Torque depends on the
permeability and increases with permeability. This fact can be observed from the
figures (fig. 1(i) , fig. 2(i)) in the case when the flow is generated by a Stokeslet and
a rotlet respectively. However, drag behaves differently. In the case of a Stokeslet
drag first increases and then decreases in magnitude in the -ve x direction with
increasing permeability (fig. 1(ii)). Whereas drag increases first and then decreases
in magnitude in the +ve x direction with increasing permeability (fig. 2(ii)) in the
case of a rotlet.
490
G. P. Raja Sekhar and T. Amaranath
ZAMP
Acknowledgements
The authors thank the referees for their constructive suggestions on their earlier
version which have helped them to improve the presentation and in particular for
bringing to their attention the works of Ochoa-Tapia and Whitaker and Kuznetsov.
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G. P. Raja Sekhar and T. Amaranath
Department of Mathematics & Statistics
University of Hyderabad
Hyderabad - 500 046
India
e-mail: tasm@uohyd.ernet.in
(Received: May 26, 1998; revised: September 28, 1998)