Journal of Fluids and Structures 20 (2005) 735–747
Creeping motions of a porous spherical shell in a
concentric spherical cavity
H.J. Keh
, Y.S. Lu
Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC Received 16 September 2004; accepted 19 March 2005
Abstract
The quasisteadytranslation and steadyrotation of a sphericallysymmetric porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid are investigated analytically. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted bythe fluid on the porous shell which are proportional to the translational and angular velocities, respectively, are obtained in closed form. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the porous shell decrease monotonicallywith a decrease in its permeability. The boundary effects of the cavitywall on the creeping motions of a porous shell can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a porous spherical shell in the cavity reduce to those for a solid sphere and for a porous sphere. The hydrodynamic behavior for a porous spherical shell maybe approximated bythat for a permeable sphere when the porous shell is sufficientlythick, depending on its permeability.
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Keywords: Porous spherical shell; Multiphase flow; Creeping flow; Boundaryeffects
1. Introduction
The motion of colloidal particles in a continuous medium at low Reynolds numbers has long been an important subject in the fields of chemical, biomedical, and environmental engineering and science. The majorityof these transport phenomena are fundamental in nature, but permit one to develop rational understanding of manypractical systems and industrial processes such as sedimentation, flotation, electrophoresis, agglomeration, and motion of blood cells in an arteryor vein. The theoretical studyof this subject has grown out of the classic work ofStokes’ (1851)for the creeping translational motion of a rigid sphere in an unbounded incompressible Newtonian fluid.
In most practical applications, particles are not isolated. So, it is important to determine if the presence of neighboring particles and/or boundaries significantly affects the movement of particles. Problems of the hydrodynamic interactions between two or more solid or fluid particles and between these particles and various boundaries have been
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treated extensivelyin the past. Summaries for the useful knowledge in this area and some informative references can be found inHappel and Brenner (1983)andKim and Karrila (1991).
The motion of porous particles relative to a fluid has been studied for a long time, since theycan be good models for polymer coils in a solvent and for flocs of fine particles in a colloidal suspension. An approach which includes a second-order viscous term to Darcy’s equation for the flow in porous media was developed byBrinkman (1947)andDebye and Bueche (1948)independently. Sutherland and Tan (1970)used Darcy’s law for the inside flow field and the Stokes equations for the outside field of a sedimenting porous sphere and concluded that it is reasonable for an isolated sphere on the assumption of immobilized fluid within the porous structure. Their result was proven incorrect byOoms et al. (1970)andNeale et al. (1973), who used the Brinkman equation for the flow within the porous sphere and more general boundaryconditions at the surface of the particle. Experimental studies on the sedimentation of porous particles have been reported by Matsumoto and Suganuma (1977) and Masliyah and Polikar (1980), whose results are in good agreement with the theoretical prediction ofNeale et al. (1973).
The creeping flow of a fluid past a composite sphere having a central solid core and an outer porous shell was solved byMasliyah et al. (1987)using the Brinkman equation for the flow field inside the fluid-permeable surface layer. An analytical formula for the drag force experienced by the particle was derived as a function of the radius of the solid core, the thickness of the porous shell, and the permeabilityof the shell. Theyalso measured the settling velocityof a solid sphere with attached threads and found that theoretical predictions for the composite sphere are in excellent agreement with the experimental results. Later, the boundaryeffect on the motion of a composite sphere perpendicular to one or two plane walls was later examined byChen and Ye (2000). A boundarycollocation method was used to studythe general case where the porous shell thickness and separation distance between the particle and the wall can be arbitrary,
Nomenclature
a internal radius of the porous spherical shell, m
b external radius of the porous spherical shell, m
c radius of the spherical cavity, m
A, B, C1, D1 coefficients in Eq. (12a) for the external flow field, given byEqs. (A.11)–(A.14) A0
, B0
1 coefficients in Eq. (27a) for the external flow
field, given byEqs. (A.28) and (A.29) B0
3 coefficient in Eq. (27c) for the internal flow
field, given byEqs. (A.32)
C3, D3 coefficients in Eq. (12c) for the internal flow field, given byEqs. (A.19) and (A.20) C0, D0 coefficients in Eq. (27b) for the flow field
inside the porous shell, given byEqs. (A.30) and (A.31)
E, F, G, H coefficients in Eq. (12b) for the flow field inside the porous shell, given byEqs. (A.15)–(A.18)
Fd drag force acting on the porous shell, N Fð0Þd drag force acting on the porous shell in the
absence of the cavity, N
k permeabilityin the porous shell, m2 M normalized translational mobilityof the
porous shell
N normalized rotational mobilityof the por-ous shell
p dynamic pressure distribution, N m2 r radial spherical coordinate, m
Td hydrodynamic torque exerted on the porous shell, N m
Tð0Þd hydrodynamic torque exerted on the porous shell in the absence of the cavity, N m U translational velocityof the porous shell,
m s1
v fluid velocityfield, m s1
vr, vy, vfcomponents of fluid velocityin spherical coordinates, m s1 Greek letters a a=k1=2 b b=k1=2 g c=k1=2 z b/c
Z viscosityof the fluid, kg m1s1 y; f angular spherical coordinates
x r=k1=2
trr; try; trf fluid stresses relevant to the surfaces of the
porous shell, N m2
C Stokes stream function of the fluid flow, m3s1
O angular velocityof the porous shell, s1 Subscripts
1 fluid external to the porous shell 2 fluid inside the porous shell 3 fluid internal to the porous shell
and a lubrication theory was used to analyze the special case of a particle coated with a thin permeable layer in near contact with a single plane. Recently, the translational and rotational motions of a composite sphere in a concentric spherical cavityhave been analyzed byKeh and Chou (2004).
On the other hand,Jones (1973)analyzed the creeping motion of a spherically symmetric porous shell using Darcy’s law and an empirical boundarycondition. Later on, the same problem was investigated byQin and Kaloni (1993)and
Bhatt and Sacheti (1994)using the Brinkman equation and well-defined velocityand stress boundaries. There were several features in their analytical solutions which cannot be seen from Darcy’s solution.
The object of this work is to obtain insights into the boundaryeffects on the translational and rotational motions of an arbitraryporous spherical shell within a small pore. This type of problem is difficult to solve, due to the structural difference for hydrodynamics inside and outside the porous shell and the complexity of the actual system geometry. In order to avoid the mathematical difficulties encountered in the problem of a sphere in a cylinder (which is a widely used model for particles in pores), we choose to examine the motions of a porous spherical shell situated at the center of a spherical cavity. Although the spherical cavity geometry is an idealized abstraction of any real system, the results obtained in this geometryhave been shown to be in good agreement with available solutions for the boundaryeffects on the partition coefficient (Giddings et al., 1968;Glandt, 1981), settling velocity(Bungayand Brenner, 1973;Happel and Brenner, 1983), and electrophoretic mobility(Zydney, 1995; Keh and Chiou, 1996) of a nonporous particle in a cylindrical pore. The spherical symmetry in this model system allows exact analytical solutions to be obtained, and the results show that the boundaryeffects on the motions of a porous shell can be significant in general situations.
2. Translation of a porous spherical shell in a spherical cavity
In this section we consider the quasisteadytranslational motion of a porous spherical shell of external radius b and internal radius a, in a concentric spherical cavityof radius c, filled with an incompressible Newtonian fluid of viscosity Z, as illustrated inFig. 1. The porous shell is assumed to be nondeformable, and its velocityequals U in the positive z (axial) direction. The spherical coordinate system (r, y, f) is established with its origin at the center of the porous shell or cavity. The Reynolds number is assumed to be sufficiently small so that the inertial terms in the fluid momentum equation can be neglected, in comparison with the viscous terms. Our purpose here is to determine the hydrodynamic drag force exerted on the porous shell in the presence of the cavity.
The external region (b r c), the porous region (a r b), and the internal region (r a) are denoted as regions I, II, and III, respectively. Then, the fluid flow in regions I and III is governed by the Stokes equations,
Zr2v i rpi¼0, (1a) r vi¼0 (1b) c z b a θ φ r
Here, v is the fluid velocityfield for the flow relative to the porous shell, p is the corresponding dynamic pressure distribution, i ¼ 1 or 3, and the subscripts 1 and 3 represent the regions I and III, respectively.
For the fluid flow within the porous shell, the relative velocity v2 and dynamic pressure p2 are governed bythe
Brinkman equation, which is preferred to the Darcyequation to accommodate the boundaryconditions at the particle surfaces ðr ¼ a and bÞ, Zr2v 2 Z kv2 rp2¼0 (2a) and r v2¼0, (2b)
where k is the permeabilityof the porous shell, and the quantities with subscript 2 represent macroscopicallyaveraged quantities pertaining to the porous medium. Here, we have assumed that the fluid has the same viscosityinside and outside the permeable shell which is reasonable according to available evidence (Koplik et al., 1983). For some model porous particles made of steel wool (in glycerin–water solution) (Matsumoto and Suganuma, 1977) and plastic foam slab (in silicon oil) (Masliyah and Polikar, 1980), experimental values of k can be as large as 107m2, while in the poly(N-isopropylacrylamide) hydrogel layers on latex particles in electrolyte solutions, values of k were found to be about 1015–1018m2(Makino et al.,1994).
Since the flow field is axiallysymmetric, it is convenient to introduce the Stokes stream functions Ciðr; yÞwhich satisfy
Eqs. (1b) and (2b) and are related to the velocitycomponents in the spherical coordinate system by vri¼ 1 r2sin y @Ci @y , (3a) vyi¼ 1 r sin y qCi qr . (3b)
Taking the curl of Eq. (1a) and applying Eq. (3) gives a fourth-order linear partial differential equation for Ci,
E4sCi¼Es2ðE2sCiÞ ¼0, (4)
where i ¼ 1 or 3, and the axisymmetric Stokes operator E2
s is given by E2 s¼ q2 qr2þ sin y r2 q qy 1 sin y q qy . (5)
Accordingly, Eq. (2a) can be expressed in terms of the stream function C2ðr; yÞas
E4 sC2 1 kE 2 sC2¼0. (6)
Due to the continuityof velocityand stress components at the surfaces of the porous shell, which is physicallyrealistic and mathematicallyconsistent for the present problem (Neale et al., 1973;Koplik et al., 1983;Chen and Ye, 2000), the boundaryconditions for the flow field are
r ¼ a : vr3¼vr2; vy3¼vy2, (7a,b)
trr3¼trr2; try3¼try2, (8a,b)
r ¼ b : vr2¼vr1; vy2¼vy1, (9a,b)
trr2¼trr1; try2¼try1, (10a,b)
r ¼ c : vr1¼ U cos y; vy1¼U sin y. (11a,b)
Here, trrand tryare the normal and shear stresses for the fluid flow relevant to the particle surfaces. Eqs. (7)–(11) take a
reference frame that the porous spherical shell is at rest and the velocityof the fluid at the cavitywall is the particle velocityin the opposite direction. Since we take the same fluid viscosityinside and outside the porous shell, use the fluid velocitycontinuitygiven byEqs. (7) and (9), and neglect the possible osmotic effects in the porous shell, Eqs. (8a) and (10a) are equivalent to the continuityof pressure (p3¼p2at r ¼ a and p2¼p1at r ¼ b).
A solution to Eqs. (4) and (6) suitable for satisfying boundary conditions on the spherical surfaces is (Masliyah et al., 1987;Keh and Chou, 2004)
C1¼ 1 2kU Ax 1þBx þ C 1x2þD1x4 sin2y if b x g, (12a) C2¼ 1 2kU½Ex
1þF x2þGðx1cosh x sinh xÞ þ Hðx1sinh x cosh xÞ sin2y if a x b, (12b)
C3¼12kU ðC3x2þD3x4Þsin2y if x a , (12c)
where the dimensionless variables x ¼ r=k1=2, a ¼ a=k1=2, b ¼ b=k1=2, and g ¼ c=k1=2. The dimensionless constants A, B, C1, D1, C3, D3, E, F, G and H are found from Eqs. (7)–(11) using Eq. (3). The procedure is straightforward but tedious, and the result is given in the appendix.
The drag force (in the z direction) exerted bythe external fluid on the porous shell with the spherical boundaryr ¼ b can be determined from (Happel and Brenner, 1983)
Fd¼pZ Z p 0 r3sin3y@ @rð E2sC1 r2sin2yÞrdy (13)
Substitution of Eq. (12a) into the above integral results in the simple relation
Fd¼4pZUBk1=2, (14)
where B is given byEq. (A.12) in the appendix.
In the limiting case of b=g ¼ b=c ¼ 0, the above equation becomes Fð0Þd ¼ 6pZbU R
bS, (15)
where
R ¼ 2ð3bw2aw1Þcoshðb aÞ þ 2ð3w2abw1Þsinhðb aÞ, (16a)
S ¼60a33½ða2þ15Þð4a3þa w3Þ a2w310að5a23Þ coshðb aÞ ½ða2þ15Þð2a4þ9 aw3Þ 30aw3þ9a2
sinhðb aÞ ð16bÞ
with
w1¼b3ða2þ45Þ a3ða2þ15Þ, (17a)
w2¼b3ð2a2þ15Þ a3ð2a2þ5Þ, (17b)
w3¼bð2b2þ3Þ. (17c)
The formula given byEq. (15) is the reduced result for the translation of an isolated porous spherical shell in an unbounded fluid obtained byBhatt and Sacheti (1994).
Through the use of Eqs. (14) and (15), the normalized translational mobilityof a porous spherical shell in a concentric spherical cavitycan be expressed as
M ¼F ð0Þ d Fd ¼ 3R 2BS. (18)
Note that M ¼ 1 as b=g ¼ 0 and 0 Mo1 as 0ob=g 1. The presence of the cavitywall always enhances the hydrodynamic drag on the porous shell since the fluid flow vanishes at the wall as required by Eq. (11).
When k ¼ 0, we have Fð0Þd ¼ 6pZbU (Stokes’ law), and Eq. (18) reduces to M ¼ ð1 9 4z þ 5 2z 39 4z 5þz6Þð1 z5Þ1, (19)
where z ¼ b=c. This is the result for the translation of a solid sphere of radius b in a cavityof radius c (Happel and Brenner, 1983).
When a ¼ 0, Eqs. (15) and (18) become Fð0Þd ¼ 6pZbU 2b
2ðb tanh bÞ
2b3þ3ðb tanh bÞ, (20)
M ¼ ðb cosh b sinh bÞðs21b cosh b 3s23sinh bÞ 2g½bð2b2þ3Þ cosh b 3 sinh b ðs8sinh b s5b cosh bÞ
, (21)
where the dimensionless parameters s5, s8, s21, and s23are defined by Eq. (A.22) in the appendix. The hydrodynamic drag and normalized mobilitygiven byEqs. (20) and (21) describe the translation of a porous (permeable) sphere of radius b in an unbounded fluid (Neale et al., 1973) and in a cavityof radius c, respectively. In the limiting case of b ! 1 (or k ¼ 0), Eq. (20) reduces to the Stokes law Fð0Þd ¼ 6pZbU, while in the limit of the singular case b ¼ 0 (or k ! 1), it becomes Fð0Þd ¼0.
The variation of the normalized mobility M given byEq. (21) for the translation of a porous sphere (with a ¼ 0 or a=b ¼ 0) at the center of a spherical cavitywith the separation parameter b=g for various values of b from zero to infinityis presented inFig. 2. The separation parameter b=g ð¼ b=cÞ, reflecting the extent of closeness between the particle and the cavitywall, ranges from 0 (far apart) to 1 (in contact). The curve with b ¼ 0 (or k ! 1) represents the result for the singular case of a porous sphere with no resistance to the fluid flow, while the curve with b ! 1 (or k ¼ 0) denotes the result for a solid particle. As expected, the normalized mobility M equals unityas b ¼ 0 for anyvalue of b=g (with Fd¼Fð0Þd ¼0 for a given particle velocity U) and is a monotonic decreasing function of b=g for anygiven
value of b40. Obviously, the boundary effect on the particle mobility (or drag force) is stronger when the permeability k of the particle is smaller (or b is greater). For bo1, the particle mobilityvaries slowlywith the separation parameter b=g, compared with the case of lower permeability(or greater b). This weak interaction can be explained bythe fact that, instead of bypassing, the fluid can easily flow through a porous particle with a high permeability, leading to a great reduction in the resistant force. For b410, the value of the particle mobilityis quite close to that of a solid particle [with b ! 1 and M given byEq. (19)] when b=g is small, but the difference becomes more significant as the particle gets closer to the wall. This implies that, far from the wall, a porous particle with a low permeabilitybehaves like a solid one with most fluid flowing over it. When the porous particle and cavitywall become sufficientlyclose together, a large pressure gradient is developed in between to drive more fluid to permeate through the porous medium (Chen and Ye, 2000). Interestingly, for situations with a finite value of b, the particle mobilitydoes not vanish even for the singular case as the particle touches the cavitywall (i.e. as b=g ¼ 1).
After understanding the boundaryeffect on the translation of a porous sphere, we now examine the general case of a translating porous spherical shell in a concentric spherical cavity.Figs. 3(a) and (b)show the normalized mobility M as a function of b=g over the entire ranges of the separation and the parameter a=b for the cases of b ¼ 5 and 1, respectively. Again, M decreases monotonicallywith an increase in b=g for fixed values of a=b and b and with an
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 8 10 5 2 1 β = 0 M β/γ
Fig. 2. Plots of the normalized translational mobility M for a porous sphere ða ¼ 0Þ in a concentric spherical cavityversus the separation parameter b=g for various values of b.
increase in b for constant values of a=b and b=g. As expected, the curves for the singular case with a=b ¼ 1 lead to M ¼ 1 (with Fd¼Fð0Þd ¼0 for a given particle velocity U) and the curves with a=b ¼ 0 represent the
result for a porous sphere [given byEq. (21)]. All the other curves for a porous spherical shell lie between these upper and lower bounds and M is a monotonic increasing function of a=b for given values of b and b=g. Namely, the hydrodynamic drag acting on the porous shell is reduced as its thickness becomes smaller for a given shell size, permeability, and separation distance. It can be seen that, for the case of b ¼ 5, the behavior of a porous spherical shell with a=b ¼ 0:6 can be roughlyapproximated bythat of a porous sphere of equal size (or b=g) and permeabilitywhen b=go0:5. This is because, when a porous shell has a low to moderate permeability, it is difficult for the fluid to penetrate deep to reach its inner surface as long as the shell is sufficientlythick and the cavitywall is not too close. Thus, the internal fluid (in region III) remains oblivious of the external motion. However, this approximation is no longer valid for a porous shell with a large permeability, as for the case of b ¼ 1 shown in
Fig. 3(b). Again, for situations with a finite value of b, the particle mobilityis not necessarilyequal to zero for the singular case as b=g ¼ 1.
3. Rotation of a porous spherical shell in a spherical cavity
We now consider the steadyrotational motion of a porous spherical shell of external radius b and internal radius a located at the center of a spherical cavityof radius c. The angular velocityof the porous shell is O in the positive z (axial) direction and the Reynolds number is vanishingly small. The objective in this section is to obtain the hydrodynamic torque acting on the porous shell in the presence of the cavity.
The fluid flow fields outside and inside the porous shell are still governed byEqs. (1) and (2) respectively, and they must be solved subject to the following boundaryconditions resulting from the continuityof velocityand stress components:
r ¼ a : vf3¼vf2; trf3¼trf2, (22,23)
r ¼ b : vf2¼vf1; trf2¼trf1, (24,25)
r ¼ c : vf1¼ Oc sin y; (26)
where vfis the f-component of the fluid velocityfield and trfis the shear stress for the rotational fluid flow relevant to
the surfaces of the porous shell. Obviously, the r and y components of the fluid velocitydisappear and the fluid dynamic
0.9 0 0.6 0.8 α/β = 1 α/β = 1 0.0 0.2 0.4 0.6 0.8 1.0 β/γ 0.0 0.2 0.4 β/γ 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 M 0.7 0.8 0.9 1.0 1.1 0.9 0 0.4 0.6 0.8 M (a) (b)
Fig. 3. Plots of the normalized translational mobility M for a porous spherical shell in a concentric spherical cavityversus the separation parameter b=g for various values of a=b: (a)b ¼ 5; (b) b ¼ 1.
pressure is constant everywhere. Eqs. (22)–(26) take a reference frame that the porous shell is at rest and the angular velocityof the cavitywall is that of the shell in the opposite direction.
A solution to Eqs. (1) and (2) suitable for satisfying the boundary conditions (22)–(26) is (Keh and Chou, 2004)
vf1¼k1=2OðA0x2þB01xÞ sin y if b x g , (27a)
vf2¼k1=2O½C0ðx2cosh x x1sinh xÞ þ D0ðx2sinh x x1cosh xÞ sin y if a x b , (27b)
vf3¼k1=2OB03x sin y if x a , (27c)
where the dimensionless variables x, a, b, and g were defined right after Eq. (12). The constants A0, B0 1, B
0 3, C
0, and D0
can be determined from Eqs. (22)–(26), and the result is given in the appendix.
After the fluid velocityfield is solved, the torque (in the z direction) exerted on the rotating porous spherical shell about its center bythe external fluid can be obtained as
Td¼ 8pZOA01k3=2, (28)
where A0
1is given byEq. (A.28). In the limit of b=g ¼ b=c ¼ 0, Eq. (28) becomes
Tð0Þd ¼ 8pZb3O R
0
b2S0, (29)
the reduced result for the rotation of a porous spherical shell in an unbounded fluid, where
R0¼3½aðb2þ3Þ bða2þ3Þ coshðb aÞ þ ½ða2þ3Þðb2þ3Þ 9ab sinhðb aÞ, (30a)
S0¼3a coshðb aÞ þ ða2þ3Þ sinhðb aÞ. (30b)
The normalized rotational mobilityof a porous spherical shell in a concentric spherical cavitycan be expressed as N ¼T ð0Þ d Td ¼bR 0 A0S0 (31)
The presence of the no-slip cavity wall always enhances the hydrodynamic torque on the rotating porous shell. Thus, 0 N 1 for the entire range of b=g (N ¼ 1 as b=g ¼ 0).
When k ¼ 0, Tð0Þd ¼ 8pZb3O; and Eq. (31) reduces to
N ¼ 1 z3, (32)
where z ¼ b=c. This is the result for the rotation of a solid sphere of radius b in a cavityof radius c. When a ¼ 0, Eqs. (29) and (31) become
Tð0Þd ¼ 8pZb3Oð1 þ 3b23b1coth bÞ, (33)
N ¼ 1 b
3
g3ð1 þ 3b
23b1coth bÞ. (34)
The hydrodynamic torque and normalized mobility predicted by Eqs. (33) and (34) describe the rotation of a porous sphere of radius b in an unbounded fluid and in a cavityof radius c, respectively. In the limiting case of b ! 1 (or k ¼ 0), Eq. (33) reduces to Tð0Þd ¼ 8pZb3O, while in the limit of the singular case b ¼ 0 (or k ! 1), it results in Tð0Þd ¼0.
Fig. 4shows the plot of the normalized rotational mobility N for a porous sphere at the center of a spherical cavity versus the separation parameter b=g for various values of b over the entire ranges. Analogous to the result of the translational mobility M of the particle, the rotational mobility N equals unityas b ¼ 0 for all values of b=g (with Td¼Tð0Þd ¼0 for a given angular velocity O of the particle) and decreases monotonicallywith an increase in the value
of b=g for a specified value of b40. The boundary effect on the rotational mobility (or hydrodynamic torque) of the permeable particle is stronger when the permeability k is smaller (or b is greater). For bo1, the rotational mobilityis not a sensitive function of b=g (except as b=g ! 1), compared with the result for a lower permeability(or greater b). For b410, the value of the rotational mobilityof the porous sphere is close to that of a solid particle [with k ¼ 0 or b ! 1 and N given byEq. (32)] when b=g is small, while the difference is more significant as b=g ! 1. When a particle with a finite value of b is in contact with the cavitywall (b=g ¼ 1), its rotational mobilitydoes not vanish for this singular case.
In Figs. 5(a) and (b), we present the results of the normalized rotational mobility N for a porous spherical shell in a concentric spherical cavityas a function of the parameters b=g and a=b over the entire ranges for the cases of b ¼ 5 and 1, respectively. Again, N decreases monotonicallywith an increase in b=g for given values of b and a=b and with an increase in b for fixed values of a=b and b=g. As expected, the curves for the singular case with a=b ¼ 1 give N ¼ 1 (with Td¼Tð0Þd ¼0 for a given angular velocity O of the particle), and the curves
with a=b ¼ 0 represent the result for the rotation of a porous sphere. All the other curves for a porous spherical shell locate between these upper and lower bounds and N increases monotonicallywith an increase in a=b for specified values of b and b=g. For a porous shell with a finite value of b, its rotational mobilitydoes not vanish for the singular case as b=g ¼ 1. For the case of b ¼ 5, the normalized rotational mobilityof a porous spherical shell with a=b ¼ 0:8 can be well approximated bythat of a porous sphere of equal size, permeability, and separation distance from the cavitywall. As illustrated inFig. 5(b), however, this approximation is no longer valid for a porous shell with a large permeability.
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.9 1 0.8 = 0 N 0.92 0.96 1.00 0.8 0.9 1 N 0.0 0.2 0.4 0.6 0.8 1.0 β/γ α/β (a) 0.0 0.2 0.4 0.6 0.8 1.0 β/γ (b) = 0 α/β
Fig. 5. Plots of the normalized rotational mobility N for a porous spherical shell in a concentric spherical cavityversus the separation parameter b=g for various values of a=b: (a)b ¼ 5 and (b) b ¼ 1.
8 10 5 2 1 β = 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 N 0.0 0.2 0.4 0.6 0.8 1.0 β/γ
Fig. 4. Plots of the normalized rotational mobility N for a porous sphere ða ¼ 0Þ in a concentric spherical cavityversus the separation parameter b=g for various values of b.
4. Concludingremarks
In this work, the quasisteadytranslation and steadyrotation of a porous spherical shell (which can reduce to a solid sphere and a porous sphere in the limiting cases) in a concentric spherical cavityfilled with an incompressible Newtonian fluid have been investigated theoreticallyin the creeping flow regime. The Stokes and Brinkman equations for the fluid flow field applicable to these axisymmetric motions are analytically solved and the hydrodynamic drag force and torque exerted on the porous shell as functions of the parameters a=bð¼ a=bÞ, b=gð¼ b=cÞ, and ðb ¼ b=k1=2Þ are obtained in the closed-form expressions (14) and (28). It has been found that, for a specified geometry(fixed values of a=b and b=g), the wall-corrected translational and rotational mobilities of the porous shell, normalized bytheir corresponding values in the absence of the cavitywall, are monotonic decreasing functions of the parameter b (or increasing functions of the permeability k) of the porous shell. For given values of a=b and b, these normalized mobilities decrease monotonicallywith an increase in the separation parameter b=g. The analysis assumes that the porous shell is non-deformable. The results would be verydifferent, particularlyin the case where the porous shell just fits within the cavity, if it were able to deform in response to the flow (as might be expected for a shell composed of entangled polymer).
Our results, which provide useful insights into the actual phenomena regarding the creeping motions of a porous particle in a small pore, show that the boundaryeffect of the cavitywall on these motions can be significant in appropriate situations. Also, these results will serve as useful limiting solutions against which the results of more sophisticated models (which account for shell deformability, for instance) may be benchmarked. More detailed analyses of the fluid flow for a porous particle in open and closed cylindrical pores will still be required to quantify the actual behavior in this system and to determine the overall applicability of the results obtained in this work for the spherical cavityto more realistic pore geometries.
Acknowledgements
Part of this research was supported bythe National Science Council of the Republic of China.
Appendix
For conciseness, algebraic equations for the determination of the coefficients in Eqs. (12) and (27) as well as their solutions are presented in this appendix.
Applying the boundaryconditions given byEqs. (7)–(11) to the general solution given byEq. (12) for the translation of a porous spherical shell in a concentric spherical cavity, one obtains
C3a3þD3a5¼E þ F a3Gða sinh a cosh aÞ þ Hðsinh a a cosh aÞ, (A.1)
2C3a34D3a5¼E 2F a3G½a sinh a ða2þ1Þ cosh a
H½a cosh a ða2þ1Þ sinh a , ðA:2Þ
6D3a5¼6E þ 3ða2þ2ÞðG cosh a þ H sinh aÞ aða2þ6ÞðG sinh a þ H cosh aÞ, (A.3)
20D3a3¼E 2F a3, (A.4)
A þ Bb2þC1b3þD1b5¼E þ F b3Gðb sinh b cosh bÞ
þHðsinh b b cosh bÞ, ðA:5Þ
A Bb22C
1b34D1b5¼E 2F b3G½b sinh b ðb2þ1Þ cosh b
H½b cosh b ðb2þ1Þ sinh b , ðA:6Þ
6A þ 6D1b5¼6E þ 3ðb2þ2ÞðG cosh b þ H sinh bÞ
2B þ 20D1b3¼E 2F b3, (A.8)
A þ Bg2þC
1g3þD1g5¼g3, (A.9)
A Bg22C
1g34D1g5¼ 2g3. (A.10)
The above simultaneous algebraic equations can be solved to yield the ten unknown constants as A ¼ 2Db2g3f30a3bðg310b3Þ
þ3½a3bð2a2þ5Þs1a4s0s2b2ð2a2þ15Þs3þabða2þ45Þs4 coshðb aÞ
þ ½a4bs0s19a3ð2a2þ5Þs2ab2ða2þ45Þs3þ9bð2a2þ15Þs4 sinhðb aÞg, ðA:11Þ
B ¼ 6Dgf300a3b6
þ ½3b4ð2a2þ15Þs5þa4s0s63a3bð2a2þ5Þs7ab3ða2þ45Þs8 coshðb aÞ
þ ½ab4ða2þ45Þs5þ3a3ð2a2þ5Þs6a4bs0s73b3ð2a2þ15Þs8 sinhðb aÞg, ðA:12Þ
C1¼Dgf30a3ðs1045b5Þ þ3fbð2a2þ15Þ½s9þ4b2g5þ30b3ðg29Þ 2a5s10a3s1145as12 5a4b2s0s14gcoshðb aÞ þ f135b7ð2a2þ15Þ þ abða2þ45Þs 9þ45a3b2g2ð2a2þ5Þ 2g5s27 þ9b5s2815ab4s29 þb3½4ag5ða2þ45Þ 5a4s
0ðg236Þ 90ð2a2þ15Þðg29Þ g sinhðb aÞg, ðA:13Þ
D1¼3Dgf30a3b3 þ ½3b4ð2a2þ15Þð6 s 13Þ þa4s0s1445ab3ðs14þ6Þ þ 6a5bs13 a3bs 25 coshðb aÞ þ ½45ab4ð6 s13Þ 3b3ð2a2þ15Þðs14þ6Þ þ 6a5s14þa4bs0s13
þa3s26 sinhðb aÞg, ðA:14Þ
E ¼ 12Da3gf5b3s15 ½3bð2a2þ5Þs16as0s17 coshðb aÞ
þ ½3ð2a2þ5Þs17abs0s16 sinhðb aÞg, ðA:15Þ
F ¼ 6Dgf5a3s18 ½3bð2a2þ15Þs16aða2þ45Þs17 coshðb aÞ
þ ½3ð2a2þ15Þs17abða2þ45Þs16 sinhðb aÞg, ðA:16Þ
G ¼ 6Dgf½a4s0s18ab3ða2þ45Þs15 cosh a þ 30a3bs16cosh b
þ3½b3ð2a2þ15Þs15a3ð2a2þ5Þs18 sinh a 30a3s17sinh bg, ðA:17Þ
H ¼ 6Dgf3½a3ð2a2þ5Þs18b3ð2a2þ15Þs15 cosh a þ 30a3s17cosh b
þ ½b3ða2þ45Þs
15a3s0s18 sinh a 30a2bs16sinh bg, ðA:18Þ
C3¼3Dgfa5s18b3ða230Þs15
þ6½bð7a2þ15Þs16það2a2þ15Þs17 coshðb aÞ
D3¼3Dgfb3s15a3s18þ6½bs16as17 coshðb aÞ
þ6½abs16s17 sinhðb aÞg, ðA:20Þ
where
D ¼ f60a3s22þ3½a4s0s192a5s20þbð2a2þ15Þs2145as23a3s24 coshðb aÞ
þ ½18a5s19a4s0s20þ45abs219ð2a2þ15Þs23þa3ð45s22bs24Þ sinhðb aÞg1, ðA:21Þ
s0¼a2þ15; s1¼b3þ45b g3; s2¼2b3þ15b g3, s3¼b5þ15b3b2g36g3; s4¼2b5þ5b3b2g32g3, s5¼b5þ15b3g5; s6¼6b5þ45b3g5, s7¼b5þ45b3g5; s8¼6b5þ15b3g5, s9¼6g59b7þb5ð5g2126Þ; s10¼4g59b5þ5b3ðg236Þ, s11¼22g515b7þb5ð5g2117Þ þ 5b3ð7g2198Þ, s12¼2g515b7þb5ð5g272Þ þ 10b3ðg29Þ, s13¼g2b2; s14¼3b2g2, s15¼3b55b3g2þ2g5; s16¼6b55b3ðg29Þ g5, s17¼21b55b3ðg29Þ g5; s18¼3b55b3ðg2þ18Þ þ 2g5, s19¼8b515b4g þ 60b3þ10b2g33g5, s20¼4b69b5g þ 180b4þ10b3gðg218Þ 9bg5þ4g6, s21¼4b89b7g þ 60b6þ2b5gð5g263Þ 3b3gð3g420g2þ90Þ þ 4b2g6þ6g6, s22¼20b627b5g þ 5b3gðg218Þ þ 2g6, s23¼8b815b7g þ 20b6þ2b5gð5g236Þ b3gð3g420g2þ90Þ þ 2g6, s24¼8b815b7g þ 40b6þ900b445bg5þ22g6þb5gð10g2117Þ b3gð3g470g2þ990Þ, s25¼3b415g2b2ðg221Þ, s26¼b6þ45b215g2b4ðg26Þ, s27¼2a6þ30a4þ18a2þ135, s28¼a6þ15a4þ1080 75g22a2ð5g272Þ, s29¼18a4a2ð2g263Þ 90ðg29Þ. ðA:22Þ
Application of the boundaryconditions (22)–(26) to the general solution (27) for the rotation of a porous spherical shell in a concentric spherical cavityyields
B0 3a ¼ C
0ða2cosh a a1sinh aÞ þ D0ða2sinh a a1cosh aÞ, (A.23)
0 ¼ ½C0ða2þ3Þ 3D0a cosh a ½3C0a D0ða2þ3Þ sinh a, (A.24)
A0b2þB01b ¼ C0ðb2cosh b b1sinh bÞ þ D0ðb2sinh b b1cosh bÞ, (A.25)
3A0¼ ½C0ðb2þ3Þ 3D0b cosh b ½3C0b D0ðb2þ3Þ sinh b, (A.26)
A0g2þB0
1g ¼ g. (A.27)
The five unknown constants appearing in the above equations can easilybe solved, with the result A0
1¼ D
B0 1¼3D
0bg3S0, (A.29)
C0¼3D0bg3½3a cosh a ð3 þ a2Þsinh a , (A.30)
D0¼3D0bg3½ð3 þ a2Þcosh a 3a sinh a , (A.31)
B0 3¼ 3D
0bg3, (A.32)
where
D0¼ f3½b2ð3 þ a2Þ að3b þ b3g3Þ coshðb aÞ
þ ½9ab2 ð3 þ a2Þð3b þ b3g3Þ sinhðb aÞg1. ðA:33Þ
and R0and S0are defined byEq. (30).
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