Creeping motions ofa composite sphere in a concentric spherical cavity
Huan J. Keh
∗, Jing Chou
Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC Received 24 July 2003; received in revised form 26 September 2003; accepted 14 October 2003
Abstract
An analytical study is presented for the quasisteady translation and steady rotation of a spherically symmetric composite particle composed ofa solid core and a surrounding porous shell located at the center ofa spherical cavity 0lled with an incompressible Newtonian 2uid. In the 2uid-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 2ow 0eld of the system, and the hydrodynamic drag force and torque exerted by the 2uid on the particle which is proportional to the translational and angular velocities, respectively, are obtained in closed forms. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the particle decrease monotonically with a decrease in the permeability ofits porous shell. The boundary e;ects ofthe cavity wall on the creeping motions ofa composite sphere can be quite signi0cant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a composite sphere in the cavity reduce to those for a solid sphere and for a porous sphere.
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Keywords: Fluid mechanics; Composite sphere; Porous sphere; Multiphase 2ow; Creeping 2ow; Boundary e;ects
1. Introduction
The area ofthe moving ofcolloidal particles in a con-tinuous medium at low Reynolds number has continued to receive much attention from investigators in the 0elds of chemical, biomedical, and environmental engineering and science. The majority ofthese moving phenomena are fun-damental in nature, but permit one to develop rational under-standing ofmany practical systems and industrial processes such as sedimentation, 2otation, spray drying, agglomera-tion, and motion ofblood cells in an artery or vein. The theoretical study ofthis subject has grown out ofthe classic work ofStokes’ (1851)for a translating rigid sphere in an unbounded viscous 2uid.
In most practical applications, particles are not isolated. So, it is important to determine ifthe presence ofneigh-boring particles and/or boundaries signi0cantly a;ects the movement ofparticles. Problems ofthe hydrodynamic inter-actions between two or more particles and between particles and boundaries have been treated extensively in the past. Summaries for the useful knowledge in this area and some
∗Corresponding author. Tel.: +886-2-2363-5462;
fax: +886-2-2362-3040.
E-mail address:huan@ntu.edu.tw(H.J. Keh).
informative references can be found inHappel and Brenner (1983)andKim and Karrila (1991).
The surface of a colloidal particle is generally not hard and smooth as assumed in many theoretical models. For instance, surface layers are purposely formed by adsorbing polymers to make the suspended particles stable against 2occulation (Napper, 1983). Even the surfaces of model colloids such as silica and polystyrene latex are “hairy” with a gel-like polymeric layer extending a substantial distance into the sus-pending medium from the bulk material inside the particle (Anderson and Solomentsev, 1996). In particular, the sur-face of a biological cell is not a hard smooth sursur-face, but rather is a permeable rough surface with various appendages ranging from protein molecules on the order of nanometers to cilia on the order ofmicrometers (Wunderlich, 1982). Such particles can be modeled as a composite particle hav-ing a central solid core and an outer porous shell (Sasaki, 1985).
The creeping 2ow ofan incompressible Newtonian 2uid past a spherically symmetric composite particle was solved by Masliyah et al. (1987) using the Brinkman equation (which may be regarded as an extension ofDarcy’s law) for the 2ow 0eld inside the 2uid-permeable surface layer and the Stokes equations for the 2ow 0eld external to the par-ticle. An analytical formula for the drag force experienced
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by the particle was derived as a function ofthe radius ofthe solid core, the thickness ofthe porous shell, and the perme-ability ofthe shell. They also measured the settling velocity ofa solid sphere with attached threads and found that the-oretical predictions for the composite sphere are in excel-lent agreement with the experimental results. Employing a unit-cell model for the creeping 2ow relative to an assem-blage ofidentical composite spheres, Prasad et al. (1990)
andKeh and Kuo (1997)later obtained analytical solutions for the dependence of the average drag force of this assem-blage on the volume fraction of the particles.
Anderson and Solomentsev (1996)analyzed motions ofa spherical particle covered by a thin layer ofadsorbed poly-mer near an in0nite plane wall and along the centerline of a long cylindrical tube. Using the concept ofhydrodynamic thickness ofthe polymer layer and a method ofre2ections to-gether with the technique ofmatched asymptotic expansions, they determined the boundary e;ects on the particle move-ment to O(2) in increasing powers of up to O(3), where is the ratio ofthe polymer-layer length scale to the parti-cle radius and is the ratio ofthe partiparti-cle radius to the dis-tance between the particle center and the boundary. On the other hand, the quasisteady motion ofa sphere coated with a thin polymer layer normal to an in0nite plane, which can be either a solid wall or a free surface, has been investigated (Kuo and Keh, 1999). A combined analytical-numerical ex-act solution ofthe hydrodynamic e;ect exerted by the plane on the moving particle accurate to O(2) was also obtained using the method ofmatched asymptotic expansions and a boundary collocation technique. Recently, the boundary e;ect on the motion ofa composite sphere perpendicular to one or two plane walls has been examined by Chen and Ye (2000). The boundary collocation method was used to study the general case where the porous shell thickness and separation distance between the particle and the wall can be arbitrary, and a lubrication theory was used to an-alyze the special case ofa particle with a thin permeable layer in near contact with a single plane.
The purpose ofthis work is to obtain insights into the boundary e;ects on the translational and rotational motions ofan arbitrary composite sphere within a small pore. This type ofproblem is diLcult to solve due to the structural dif-ference for hydrodynamics inside and outside the porous sur-face layer and the complexity of the actual system geometry. In order to avoid the mathematical diLculties encountered in the problem ofa sphere in a cylinder (which is a widely used model for particles in pores), we choose to examine the motions ofa composite sphere situated at the center of a spherical cavity. Although the geometry ofspherical cav-ity is an idealized abstraction ofany real system, the results obtained in this geometry have been shown to be in good agreement with available solutions for the boundary e;ects on the partition coeLcient (Giddings et al., 1968; Glandt, 1981), settling velocity (Bungay and Brenner, 1973;Happel and Brenner, 1983), and electrophoretic mobility (Zydney, 1995;Keh and Chiou, 1996) ofa “bare” particle in a
cylin-drical pore. The spherical symmetry in this model system allows exact analytical solutions to be obtained, and the re-sults show that the boundary e;ects on the motions ofa composite particle can be signi0cant in general situations. 2. Translation of a composite sphere in a spherical cavity
In this section we consider the quasisteady translational motion ofa spherical composite particle ofradius b in a con-centric spherical cavity (or pore) ofradius c 0lled with an incompressible Newtonian 2uid ofviscosity , as illustrated in Fig. 1. The composite sphere has a surface layer of ho-mogeneous porous material ofconstant thickness b − a and permeability k so that the radius ofthe rigid impermeable core is a. The porous shell is assumed to be non-deformable, and the particle velocity equals U in the positive z (axial) direction. The spherical coordinate system (r; ; ) is estab-lished with its origin at the particle center. The Reynolds number is assumed to be suLciently small so that the iner-tial terms in the 2uid momentum equation can be neglected, in comparison with the viscous terms. Our purpose here is to determine the hydrodynamic drag force exerted on the particle in the presence ofthe cavity.
The 2uid 2ow between the particle and the cavity (b 6 r 6 c) is governed by the Stokes equations
∇2v − ∇p = 0; (1a)
∇ · v = 0; (1b)
where v is the 2uid velocity 0eld for the 2ow outside and relative to the particle and p is the corresponding dynamic pressure distribution. For the 2uid 2ow within the porous surface layer (a 6 r 6 b), the relative velocity v∗ and
dy-namic pressure p∗are governed by the Brinkman equation,
c b a θ r φ z
Fig. 1. Geometric sketch for the motion of a composite sphere in a concentric spherical cavity.
which is preferred to the Darcy equation to accommodate the boundary conditions at the particle surface (r = b), ∇2v∗−
kv∗− ∇p∗= 0 (2a)
and
∇ · v∗= 0; (2b)
where the superscript ∗ represents a macroscopically aver-aged quantity pertaining to the porous layer region. Here, we have assumed that the 2uid has the same viscosity in-side and outin-side the permeable layer which is reasonable according to available evidence (Koplik et al., 1983).
Since the 2ow 0eld is axially symmetric, it is convenient to introduce the Stokes stream function (r; ) which sat-is0es Eq. (1b) and is related to the velocity components in the spherical coordinate system by
vr= −r2sin 1 @@ ; (3a)
v =r sin 1 @@r: (3b)
Taking the curl ofEq. (1a) and applying Eq. (3) gives a fourth-order linear partial di;erential equation for , E4
s = Es2(Es2) = 0 if b 6 r 6 c; (4)
where the axisymmetric Stokes operator E2
s is given by E2 s = @ 2 @r2 + sin r2 @ @ 1 sin @ @ : (5)
Accordingly, Eq. (2) can be expressed in terms ofthe stream function ∗(r; ), which is related to the velocity compo-nents v∗
r and v∗ within the porous layer by Eq. (3), as E4
s∗−1kEs2∗= 0 if a 6 r 6 b: (6)
Due to the continuity ofvelocity and stress components at the outer surface of the porous shell, which is physically realistic and mathematically consistent for the present prob-lem (Neale et al., 1973;Koplik et al., 1983;Chen and Ye, 2000), and the no-slip requirement at the solid surfaces, the boundary conditions for the 2ow 0eld are
r = a : v∗ r = v∗ = 0; (7) r = b : v∗ r = vr; v∗ = v ; (8a,b) ∗ rr= rr; ∗r = r ; (9a,b)
r = c : vr= −U cos ; v = U sin : (10a,b)
Here rr and r are the normal and shear stresses for the
2uid 2ow relevant to the particle surface. Eqs. (7)–(10) take a reference frame that the composite sphere is at rest and the velocity ofthe 2uid at the cavity wall is the par-ticle velocity in the opposite direction. Since we take the
same 2uid viscosity inside and outside the porous shell, use the 2uid velocity continuity given by Eq. (8), and neglect the possible osmotic e;ects in the porous shell, Eq. (9a) is equivalent to the continuity ofpressure (p∗= p at r = b).
A solution to Eqs. (4) and (6) suitable for satisfying boundary conditions on the spherical surfaces is
=1
2kU(A−1+ B + C2+ D4)sin2
if 6 6 ; (11a)
∗=1
2kU[E−1+ F2+ G(−1cosh − sinh )
+ H(−1sinh − cosh )] sin2
if 6 6 ; (11b)
where the dimensionless variables = r=k1=2; =
a=k1=2; = b=k1=2 and = c=k1=2. The dimensionless
con-stants A; B; C; D; E; F; G and H are found from Eqs. (7)– (10) using Eq. (3). The procedure is straightforward but tedious, and the result is given in AppendixA.
The drag force (in the z direction) exerted by the external 2uid on the composite sphere with the spherical boundary r = b can be determined from (Happel and Brenner, 1983) Fd= # # 0 r 3sin3 @ @r E2 s r2sin2 r d : (12)
Substitution ofEq. (11a) into the above integral results in the simple relation
Fd= 4#UBk1=2; (13)
where B is given by Eq. (A.10) in AppendixA.
In the limiting case of = = b=c = 0, the above equation becomes
Fd(0)= −6#bU SR ; (14)
where
R = W cosh − 3 2(V + sinh ) + cosh( − )
×[W (V − cosh ) + 3 2 sinh ]
+ sinh( − )[W cosh + 3 2(V − sinh )]; (15a)
S = ( sinh − cosh )[(W + 3) cosh( − )
+ 3( 2− 1) sinh( − ) − 6 ] (15b)
with
V = sinh − cosh ; (16a)
W = 23+ 3+ 3 : (16b)
The formula given by Eq. (14) is the reduced result for the translation ofan isolated composite sphere in an unbounded 2uid obtained byMasliyah et al. (1987).
Through the use ofEqs. (13) and (14), the normalized translational mobility ofa composite sphere in a concentric spherical cavity can be expressed as
M =Fd(0)
Fd = −
3R
2BS: (17)
Note that M = 1 as = = 0 and 0 6 M ¡ 1 as 0 ¡ = 6 1. The presence ofthe cavity wall always enhances the hydro-dynamic drag on the particle since the 2uid 2ow vanishes at the wall as required by Eq. (10).
When = or k → ∞; Fd(0)= −6#aU (Stokes’ law) and Eq. (17) reduces to
M = 1 − 94 +523−9 45+ 6 (1 − 5)−1; (18)
where = a=c. This is the result for the translation of a solid sphere ofradius a in a cavity ofradius c.
When = or k = 0; Fd(0)= −6#bU and Eq. (17) still reduces to Eq. (18), but with =b=c. This result corresponds to the translation ofa solid sphere ofradius b in a cavity of radius c.
When a = 0, Eqs. (14) and (17) become Fd(0)= −6#bU 22( − tanh )
23+ 3( − tanh ); (19)
M =
( cosh −sinh )(s21cosh −3s23sinh )
2 [(22+3) cosh −3 sinh ](s8sinh −s5 cosh );
(20) where the dimensionless parameters s5; s8; s21 and s23 are
de0ned by Eq. (A.18) in AppendixA. The hydrodynamic drag and normalized mobility given by Eqs. (19) and (20) describe the translation ofa porous (permeable) sphere of radius b in an unbounded 2uid (Neale et al., 1973) and in a cavity ofradius c, respectively. In the limiting case of →
∞ (or k = 0), Eq. (19) reduces to Fd(0)= −6#bU, while in the limit of = 0 (or k → ∞), it becomes Fd(0)= 0.
The variation ofthe normalized mobility M given by Eq. (20) for the translation of a porous sphere (with a = 0 or ==0) at the center ofa spherical cavity with the separation parameter = for various values of from zero to in0nity is presented in Fig.2. The separation parameter = (=b=c), re2ecting the extent ofcloseness between the particle and the cavity wall, ranges from 0 (far apart) to 1 (in contact). The curve with = 0 (or k → ∞) represents the result for a porous sphere with no resistance to the 2uid 2ow, while the curve with → ∞ (or k = 0) denotes the result for a solid particle. As expected, the normalized mobility M equals unity as =0 for any value of = and is a monotonic decreasing function of = for any given value of ¿ 0. Obviously, the boundary e;ect on the particle mobility (or drag force) is stronger when the permeability k ofthe particle is smaller (or is greater). For ¡ 1, the particle mobility varies slowly with the separation parameter = , compared with the case oflower permeability (or greater ). This
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 ofthe normalized translational mobility M for a porous sphere (a = 0) in a concentric spherical cavity versus the separation parameter = for various values of .
weak interaction can be explained by the fact that, instead ofbypassing, the 2uid can easily 2ow through a porous particle with a high permeability, leading to a great reduction in the resistant force. For ¿ 10, the value ofthe particle mobility is quite close to that ofa solid particle (with →
∞ and M given by Eq. (18) with =b=c) when = is small, but the di;erence becomes more signi0cant as the particle gets closer to the wall. This implies that, far from the wall, a porous particle with a low permeability behaves like a solid one with most 2uid 2owing over it. When the porous particle and cavity wall become suLciently close together, a large pressure gradient is developed in between to drive more 2uid to permeate through the porous medium (Chen and Ye, 2000). Interestingly, for cases with a 0nite value of , the particle mobility does not vanish even as the particle touches the cavity wall (i.e., as = = 1).
After understanding the boundary e;ect on the transla-tion ofa porous sphere, we now examine the general case ofa translating composite sphere in a concentric spherical cavity. Figs.3a and b show the normalized mobility M as a function of = over the entire ranges ofthe separation and the parameter = for the cases of = 5 and 1, respec-tively. Again, M decreases monotonically with an increase in = for 0xed values of = and and with an increase in for constant values of = and = . The curve with = =1 represents the result for a solid sphere (given by Eq. (18)) and the curve with = = 0 denotes that for a porous sphere (given by Eq. (20)). All the other curves for a composite sphere lie between these lower and upper bounds and M is a monotonic decreasing function of = for given values of and = . Namely, the hydrodynamic drag acting on the parti-cle is reduced as the porous layer becomes thick for a given particle size, permeability, and separation distance. It can be seen that, for the case of = 5, the behavior ofa composite sphere with = = 0:6 can be roughly approximated by that
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 0.80.6 α/β = 0 M β / γ 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 0.8 0.6 0.4 α/β =0 M β /γ (a) (b)
Fig. 3. Plots ofthe normalized translational mobility M for a composite sphere in a concentric spherical cavity versus the separation parameter = for various values of =: (a) = 5, (b) = 1.
ofa porous one ofequal size and permeability when = ¡ 0:5. This is because when a porous layer has a low-to-moderate permeability, it is diLcult for the 2uid to penetrate deep to reach the core surface as long as the layer is suLciently thick and the cavity wall is not too close. Thus, the solid core hardly feels a relative 2uid motion, merely exerting a negligibly small resistant force on the 2uid (Masliyah et al., 1987;Chen and Ye, 2000). However, this approximation is no longer valid for a porous layer with a large permeability, as for the case of = 1 shown in Fig.3b. Again, for cases with = ¡ 1, the particle mobility is not necessarily equal to zero as = = 1.
3. Rotation of a composite sphere in a spherical cavity We now consider the steady rotational motion ofa spher-ical composite particle ofradius b located at the center of a spherical cavity ofradius c. Again, the composite sphere has a porous surface layer of uniform thickness b − a and permeability k. The angular velocity ofthe particle is , 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 particle in the pres-ence ofthe cavity.
The 2uid 2ow 0elds outside and inside the porous shell ofthe composite sphere are still governed by Eqs. (1) and (2), respectively, and they must be solved subject to the following boundary conditions resulting from the continuity ofvelocity and stress components:
r = a : v∗ = 0; (21) r = b : v∗ = v; (22) ∗ r= r; (23) r = c : v= −,c sin ; (24)
where vis the -component ofthe 2uid velocity 0eld and
r is the shear stress for the rotational 2uid 2ow on the
particle surface. Obviously, the r and components ofthe 2uid velocity disappear and the 2uid dynamic pressure is constant everywhere. Eqs. (21)–(24) take a reference frame that the particle is at rest and the angular velocity ofthe cavity wall is that ofthe particle in the opposite direction.
A solution to Eqs. (1) and (2) suitable for satisfying the boundary conditions (21)–(24) is
v= k1=2,(A−2+ B) sin if 6 6 ; (25a)
v∗
= k1=2,[C(−2cosh − −1sinh )
+ D(−2sinh − −1cosh )] sin
if 6 6 ; (25b)
where the dimensionless variables ; ; and were de0ned right after Eq. (11). The constants A; B; Cand Dcan be determined from Eqs. (21)–(24), and the result is given in AppendixA.
After the 2uid velocity 0eld is solved, the torque (in the z direction) exerted on the rotating composite sphere about its center by the external 2uid can be obtained as
Td= −8#,Ak3=2; (26)
where Ais given by Eq. (A.23). In the limit of = =b=c=0, Eq. (26) becomes
Td(0)= −8#b3, R
2S; (27)
the reduced result for the rotation of a composite sphere in an unbounded 2uid, where
R= ( 2− 3 + 3 ) cosh( − )
+ (2− 3 + 3) sinh( − ); (28a)
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 = 0 N β / γ
Fig. 4. Plots ofthe normalized rotational mobility N for a porous sphere (a = 0) in a concentric spherical cavity versus the separation parameter = for various values of .
The normalized rotational mobility ofa composite sphere in a concentric spherical cavity can be expressed as N =Td(0)
Td =
R
AS: (29)
The presence ofthe no-slip cavity wall always enhances the hydrodynamic torque on the rotating particle. Thus, 0 6 N 6 1 for the entire range of = (N = 1 as = = 0).
When = or k → ∞; Td(0)= −8#a3, and Eq. (29)
reduces to
N = 1 − 3; (30)
where = a=c. This is the result for the rotation of a solid sphere ofradius a in a cavity ofradius c.
When = or k = 0; Td(0)= −8#b3, and Eq. (29) still
reduces to Eq. (30) with = b=c. This result corresponds to the rotation ofa solid sphere ofradius b in a cavity of radius c.
When a = 0, Eqs. (27) and (29) become
Td(0)= −8#b3,(1 + 3−2− 3−1coth ); (31)
N = 1 − 33(1 + 3−2− 3−1coth ): (32)
The hydrodynamic torque and normalized mobility pre-dicted by Eqs. (31) and (32) describe the rotation ofa porous sphere ofradius b in an unbounded 2uid and in a cavity ofradius c, respectively. In the limiting case of → ∞ (or k = 0), Eq. (31) reduces to Td(0)= −8#b3,, while in the limit of =0 (or k → ∞), it results in Td(0)=0. Fig.4shows the plot ofthe normalized rotational mobility N for a porous sphere at the center of a spherical cavity versus the separation parameter = for various values of
over the entire ranges. Analogous to the result ofthe trans-lational mobility M ofthe particle, the rotational mobility N equals unity as = 0 for all values of = and decreases monotonically with an increase in the value of = for a speci0ed value of ¿ 0. The boundary e;ect on the rota-tional mobility (or hydrodynamic torque) ofthe permeable particle is stronger when the permeability k is smaller (or is greater). For ¡ 1, the rotational mobility is not a sensi-tive function of = (except as = → 1), compared with the result for a lower permeability (or greater ). For ¿ 10, the value ofthe rotational mobility ofthe porous sphere is close to that ofa solid particle (with k = 0 or → ∞ and N given by Eq. (30) with =b=c) when = is small, while the di;erence is more signi0cant as = → 1. When the parti-cle is in contact with the cavity wall (= = 1), its rotational mobility does not vanish for cases with a 0nite value of .
In Figs. 5a and b, we present the results ofthe normal-ized rotational mobility N for a composite sphere in a con-centric spherical cavity as a function of the parameters = and = over the entire ranges for the cases of = 5 and 1, respectively. Again, N decreases monotonically with an increase in = for given values of and = and with an increase in for 0xed values of = and = . The curves with = = 1 and = = 0 represent the results for the ro-tation ofa solid sphere and a porous sphere, respectively. All the other curves for a composite sphere locate between these lower and upper bounds and N decreases monotoni-cally with an increase in = for speci0ed values of and = . For a particle with = ¿ 1, its rotational mobility does not vanish as = = 1. For the case of = 5, the normalized rotational mobility ofa composite sphere with = =0:6 can be well approximated by that ofa porous one ofequal size, permeability, and separation distance from the cavity wall. As illustrated in Fig.5b, however, this approximation is no longer valid for a layer with a large permeability.
4. Concluding remarks
The quasisteady translation and steady rotation ofa com-posite sphere (which can reduce to a solid sphere and a porous sphere in the limiting cases) in a concentric spherical cavity 0lled with an incompressible Newtonian 2uid have been theoretically investigated in this study. In the creep-ing 2ow regime, the Stokes and Brinkman equations for the 2uid 2ow 0eld applicable to these axisymmetric motions are analytically solved and the hydrodynamic drag force and torque exerted on the particle as functions of the param-eters = (=a=b); = (=b=c), and are obtained in the closed-form expressions (13) and (26). It has been found that, for a speci0ed geometry (0xed values of = and = ), the wall-corrected translational and rotational mobilities of the particle normalized by their corresponding values in the absence ofthe cavity wall are monotonic decreasing func-tions ofthe parameter (or increasing funcfunc-tions of the permeability k ofthe porous surface layer) ofthe particle.
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 0.8 0.6 =0 N 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 0.8 0.6 =0 N β / γ β / γ α/β α/β (a) (b)
Fig. 5. Plots ofthe normalized rotational mobility N for a composite sphere in a concentric spherical cavity versus the separation parameter = for various values of =: (a) = 5, (b) = 1.
For given values of = and , these normalized mobilities decrease monotonically with an increase in the separation parameter = . The analysis assumes that the porous shell of the composite sphere is non-deformable. The results would be very di;erent, particularly in the case where the particle just 0ts within the cavity, ifthe porous shell were able to deform in response to the 2ow (as might be expected for a layer composed ofentangled polymer).
Our results, which provide useful insights into the actual phenomena regarding the creeping motions ofa compos-ite/porous particle in a small pore, show that the bound-ary e;ect ofthe cavity wall on these motions can be signi0cant in appropriate situations. More detailed analyses ofthe 2uid 2ow for a composite/porous particle in open and closed cylindrical pores will clearly be required to quantify the actual behavior in this system and to determine the overall applicability ofthe results obtained in this paper for the spherical cavity to more realistic pore geometries. Notation
a radius ofthe solid sphere, m b radius ofthe composite sphere, m c radius ofthe spherical cavity, m
A; B; C; D coeLcients in Eq. (11a) for the external 2ow 0eld, given by Eqs. (A.9)–(A.12)
A; B; C; D coeLcients in Eq. (25) for the rotational ve-locity 0eld, given by Eqs. (A.23)–(A.26) E; F; G; H coeLcients in Eq. (11b) for the 2ow 0eld
inside the porous shell, given by Eqs. (A.13)– (A.16)
Fd drag force acting on the particle, N
Fd(0) drag force acting on the particle in the ab-sence ofthe cavity, N
k permeability in the porous shell, m2
M normalized translational mobility ofthe
par-ticle
N normalized rotational mobility ofthe particle
p dynamic pressure distribution, N m−2
r radial spherical coordinate, m
Td hydrodynamic torque exerted on the particle, N m
Td(0) hydrodynamic torque exerted on the particle in the absence ofthe cavity, N m
U translational velocity ofthe particle, m s−1
v 2uid velocity 0eld, m s−1
vr; v ; v components of2uid velocity in spherical co-ordinates, m s−1
Greek letters
=a=k1=2
=b=k1=2
=c=k1=2
ratio ofthe particle radius to the distance be-tween the particle center and the wall
viscosity ofthe 2uid, kg m−1s−1
; angular spherical coordinates
=r=k1=2
rr; r ; r 2uid stresses relevant to the particle surface, N m−2
Stokes stream function of the 2uid 2ow,
m3s−1
, angular velocity ofthe particle, s−1 Superscript
Acknowledgements
This research was supported by the National Sci-ence Council ofthe Republic ofChina under Grant NSC91-2815-C-002-008-E.
Appendix A
For conciseness algebraic equations for the determination ofthe coeLcients in Eqs. (11) and (25) as well as their solutions are presented in this appendix.
Applying the boundary conditions given by Eqs. (7)–(10) to the general solution given by Eq. (11) for the translation ofa composite sphere in a concentric spherical cavity, one obtains
E + F 3+ G(cosh − sinh )
+ H(sinh − cosh ) = 0; (A.1)
E − 2F 3− G[ sinh − ( 2+ 1)cosh ]
− H[ cosh − ( 2+ 1) sinh ] = 0; (A.2)
A + B2+ C3+ D5
= E + F3− G( sinh − cosh )
+ H(sinh − cosh ); (A.3)
A − B2− 2C3− 4D5
=E − 2F3− G[ sinh − (2+ 1) cosh ]
− H[ cosh − (2+ 1) sinh ]; (A.4)
6A + 6D5= 6E + 3(2+ 2)(G cosh + H sinh )
− (2+ 6)(G sinh + H cosh ); (A.5)
2B + 20D3= E − 2F3; (A.6)
A + B 2+ C 3+ D 5= 3; (A.7)
A − B 2− 2C 3− 4D 5= −2 3: (A.8)
The above simultaneous algebraic equations can be solved to yield the eight unknown constants as
A = 2/3 3[18 ( 3− 103) + ( s0s1− 9 2s2+ 22s3) cosh( − ) − 3(s0s2− 2s1+ 2s4) sinh( − )]; (A.9) B = 6/ [60 2− (24s 5− 3 2s6+ s0s7) cosh( − ) + (23s 8+ s0s6− 3 2s7) sinh( − )]; (A.10) C = / [455s 10+ (9 22s9− s0s10−23s11) cosh(− ) + 3( 2s 0s9− 9 2s10+ 6s12) sinh( − )]; (A.11) D = 3/ {6 3+ [24(s 13− 6) − 3 2s14+ s0s13] ×cosh( − ) + [23(s 14+ 6) + 3 2s13− s0s14] ×sinh( − )}; (A.12) E = 12/ [3s 15− (3 s16− s0s17) cosh( − ) + (3 s17− s0s16) sinh( − )]; (A.13) F = 6/ [ s18+ 2s16cosh( − ) − 2s17sinh( − )]; (A.14) G = 6/ [3 2s 18cosh + 6 s16cosh − (23s
15+ s0s18) sinh − 6 s17sinh ]; (A.15)
H = 6/ [(23s
15+ s0s18)cosh + 6 s17cosh
− s18sinh − 2s16sinh ]; (A.16)
where / = [12 s22+ (9 2s19− s0s20− 2s21) cosh( − ) + 3(2s23+ s0s19− 2s20) sinh( − )]−1; (A.17) s0= 2+ 3; s1= 3+ 45 − 3; s2= 23+ 15 − 3; s3= 5+ 153− 2 3− 6 3; s4= 25+ 53− 2 3− 2 2; s5= 5+ 153− 5; s6= 65+ 453− 5; s7= 5+ 453− 5; s8= 65+ 153− 5; s9= 2− 152; s10= 4 5− 95+ 53( 2− 36); s11= 4 5− 95+ 53( 2− 24) + 30( 2− 9); s12= 2 5− 157+ 5(5 2− 72) + 103( 2− 9); s13= 2− 2; s14= 32− 2; s15= 35− 53 2+ 2 5; s16= 65− 53( 2− 9) − 5; s17= 215− 53( 2− 9) − 5; s18= 35− 53( 2− 18) + 2 5;
s19= 85− 154 + 603+ 102 3− 3 5; s20= 46− 95 + 1804+ 103 ( 2− 18) − 9 5+ 4 6; s21= 48− 97 + 606+ 25 (5 2− 63) − 33 (3 4− 20 2+ 90) + 42 2+ 6 6; s22= 206− 275 + 53 ( 2− 18) + 2 6; s23= 88− 157 + 206+ 25 (5 2− 36) − 3(3 4− 20 2+ 90) + 2 6: (A.18)
Application ofthe boundary conditions (21)–(24) to the general solution (25) for the rotation of a composite sphere in a concentric spherical cavity yields
C( −2cosh − −1sinh )
+ D( −2sinh − −1cosh ) = 0; (A.19)
A−2+ B = C(−2cosh − −1sinh )
+ D(−2sinh − −1cosh ); (A.20)
3A= [C(2+ 3) − 3D] cosh
− [3C− D(2+ 3)] sinh ; (A.21)
A −2+ B = − : (A.22)
The four unknown constants appearing in the above equa-tions can easily be solved, with the result
A= / 3R; (A.23)
B= / 3S; (A.24)
C= 3/ 3( cosh − sinh ); (A.25)
D= 3/ 3(cosh − sinh ); (A.26)
where
/= { 3[cosh( − ) + sinh( − )] − R}−1 (A.27)
and R and S are de0ned by Eq. (28).
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