13 4 Vol.13, No. 4

2011 12 ACTA ANALYSIS FUNCTIONALIS APPLICATA Dec., 2011

DOI : 10.3724/SP.J.1160.2011.00357: 1009-1327(2011)04-0357-09

Study on Fluid Flow Passing Through a Porous Media

YUAN Rong1, Albert S Kim2

1. College of Mathematics, Qingdao University, Qingdao 266071, China

2. Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Hawaii, 96822, USA

Abstract: This is a study on fluid flow passing through a swarm of overlapped ideal aggregates with

quadratically varying permeability. The fluid flow is governed by the Brinkman’s extension of Darcy’s

law and continuity equation. These equations with appropriate boundary conditions are analytically

solved by introducing stream function. This work is compared with some previous works for which

the analytical solutions had been derived by other researchers.

Keywords: porous media; overlapped aggregate; drag force; stream function

CLC number: O177 Document code: A

1 Introduction

Porous media can be found in many instances in natural and engineered systems such as soil and

rocks in nature; animal and plant tissues in biology[2,4]; and aggregates formed during sedimentation

and granular filtration in water and wastewater treatment. When fluid flows pass through these

porous medium, the characteristics of flows are closely related to the hydrodynamic properties of the

porous medium. For example, in water and wastewater treatment using membrane filtration, the

product flow and membrane life-span are affected by permeability and selectivity. Therefore, study

for hydrodynamics of fluid flow passing through porous media is very important in real applications.

The study for hydrodynamics of fluid flow relative to an impermeable sphere dates back to

1851[12]. Fluid flow was governed by the Stokes equation and continuity equation. In 1949, Brinkman[1]

studied fluid flow passing through an isolated sphere of uniform permeability. In 1987[8] , Masliyah,

et al. combined Stokes’ and Brinkman’s models together and studied the creeping flow past a solid

sphere with a uniformly porous shell. In 1958[3], Happel put Stokes’ sphere into a spherical cell to

develop an cell model to study fluid flow passing through a swarm of equal-sized particles. In 1973[10],

Neale et al. followed Happle’s approach to research fluid flow passing through a swarm of permeable

spheres by adding Brinkman’s uniformly porous sphere in Happel’s cell.

In 2005[6], Kim and Yuan included Masliyah’s sphere in a cell to study the creeping flow over

a swarm of the equal-sized composite spheres and provided the most general solution that included

the five previous studies. In real aggregates, the porosity varies radially from a dense core to sparse

edges. Highly porous outer edges allow more fluid flow to penetrate into the inner regions of the

aggregate in comparison to the those of a uniformly porous sphere. In 2005[5], Kim and Yuan studied

hydrodynamics of an ideal aggregate with quadratically increasing permeability. As an extension

of this work, they[7] developed a cell model for a swarm of the ideal aggregates and studied the

Received date: 2011-06-29

Foundation item: This work was supported by the US National Science Foundation (CBET-0444931(CAREER) and

CBET-0552413)

358 13

specific cake resistance of a cake layer composed of deposited aggregates formed in the diffusion-

limited-cluster-aggregation (DLCA) regime[9,11,13]. This work successfully described hydrodynamics

of a concentrated suspension of the DLCA aggregates, but was limited to the permeability expression

for a swarm of non-touching aggregates.

If aggregates overlap each other in the bulk phase, sharing their edges boundary regions, bigger

aggregates through aggregate clustering are formed, and at the same time the number of aggregates

decreases. In this case, the effective permeability of groups of aggregates is less than that of individ-

ual aggregate by interfacial overlapping. Due to the physical characteristic of aggregates, i.e., a dense

core of low permeability and sparse edges of high porosity, overlapping between adjacent aggregates

can occur in the bulk phase as well as more frequently on the membrane surface. For process opti-

mization and maintenance-cost reduction, therefore, theoretical development in this research area is

in a high demand. In this study, we will develop a hydrodynamic theory for a swarm of overlapped

(DLCA) aggregates, modeled as a collection of porous spheres with a varying permeability, located in

a concentric tangential-stress-free cell.

2 Modeling Approach

We consider steady, axi-symmetric, creeping flow of a Newtonian fluid of viscosity µ, relative to

the composite sphere. For mathematical convenience, we consider the sphere to be stationary having

its center at the origin of spherical coordinates (r, θ, φ), with the fluid approaching in the +Z direction

at velocity V , as illustrated in Fig. 1. The composite sphere represents an overlapped aggregate. On

the average, the overlapped region can be modeled as a uniformly porous outer later of thickness s−a.The inner region (not overlapped) maintains the structure of quadratically increasing permeability.

Fig.1 Coordinate system for axi-symmetric fluid flow relative to a composite sphere consisting of a porous

sphere with quadratically increasing permeability surrounded by a uniformly porous shell

3 Governing Equations and Boundary Conditions

Within the outer region (a < r < s) and inner region (0 < r < a) of the composite sphere, the

4 YUAN Rong, et al: Study on Fluid Flow Passing Through a Porous Media 359

Brinkman’s extension of Darcy’s law (known as Brinkman’s equation) with continuity equations are

used to describe the prevailing flow field, respectively. The governing equations in both regions are

expressed as follows, respectively.

µ∇2u∗ − µ∗

Ku∗ = ∇p∗, a < r < s (1a)

µ†∇2u† − µ†

k2r2u† = ∇p†, 0 < r < a (1b)

with continuity equations as follows:

∇ · u∗ = 0, a < r < s (2a)

∇ · u† = 0, 0 < r < a (2b)

where u∗ =(

u∗r , u∗θ, u

∗φ

)

and u† =(

u†r, u†θ, u

†φ

)

denote the fluid velocity vector of flow in the outer

region (a < r < s) and inner region (0 < r < a), respectively. p∗ and p† are the fluid pressures

and µ∗ and µ† the effective viscosities in the outer and inner regions, respectively. For mathematical

simplicity without loss of physical meaning, it is usually assumed that

µ∗ = µ† (3)

and

p∗ (a, θ) = p† (a, θ) (4a)

p∗ (s, θ) = p† (s, θ) (4b)

The boundary conditions which describe the present problem are: 1) finite velocity field at the center

of the overlapped DLCA aggregate; 2) continuity of flow field and stress tensor components at the

inter surface of two porous regions; 3) the vanishing tangential stress at the surface of the aggregate

undergoing the ambient fluid velocity V . The boundary conditions can be described as follows:

limr→0

u†r is finite (5a)

limr→0

u†θ is finite (5b)

u†r (a, θ) = u∗r (a, θ) (5c)

u†θ (a, θ) = u∗θ (a, θ) (5d)

τ †rr (a, θ) = τ∗rr (a, θ) (5e)

τ †rθ (a, θ) = τ∗rθ (a, θ) (5f)

u∗r (s, θ) = −V cos θ (5g)

τ∗rθ (s, θ) = 0 (5h)

where (τ∗rr, τ∗rθ) and (τ †rr, τ

†rθ) denote the normal and tangential components of the stress tensor in the

outer and inner regions and are defined by

τ∗rr = −p∗ + 2µ∗ ∂u∗r

∂r(6a)

τ∗rθ = µ∗

(

1

r

∂u∗r∂θ

+∂u∗θ∂r

− u∗θr

)

(6b)

and

τ †rr = −p† + 2µ† ∂u†r

∂r(7a)

τ †rθ = µ†

(

1

r

∂u†r∂θ

+∂u†θ∂r

− u†θr

)

(7b)

respectively.

360 13

4 Solution to the Overlapped DLCA Model

Due to axi-symmetry, we introduce stream functions ψ∗(r, θ) and ψ†(r, θ)

u∗r = − 1

r2 sin θ

∂ψ∗

∂θ(8a)

u∗θ =1

r sin θ

∂ψ∗

∂r(8b)

and

u†r = − 1

r2 sin θ

∂ψ†

∂θ(9a)

u†θ =1

r sin θ

∂ψ†

∂r(9b)

related to the velocities field by Eq.(8) and Eq.(9) in outer and inner regions, respectively. Taking the

curl of each equation in Eq.(1) and considering the continuity equations in Eq.(2), we get

E4ψ∗ − 1

KE2ψ∗ = 0, a < r < s (10a)

E4ψ† − 1

k2r2E2ψ† +

1

(k2r2)2

∂(

k2r2)

∂r

∂ψ†

∂r= 0, 0 < r < a (10b)

where E2 is defined as:

E2 =∂2

∂r2+

sin θ

r2∂

∂θ

(

1

sin θ

∂

∂θ

)

The general solutions of Eqs.(10) are

ψ∗ = −V s2

2

[

A

ξ+Bξ2 + C

(

cosh ξ

ξ− sin ξ

)

+D

(

sinh ξ

ξ− cosh ξ

)]

sin2(θ), α < ξ < β (11a)

ψ† = −V s2

2(Eξn1 + Fξn2 +Gξn3 +Hξn4) sin2(θ), 0 < ξ < α (11b)

where ξ = r/√K, α = a/

√K, β = s/

√K and n1, n2, n3 and n4 are defined by

n1 =3

2− 1

2

√

k2

(

13 k2 + 2√

1 + 4 k2 (−1 + 9 k2) + 2)

k2(12a)

n2 =3

2− 1

2

√

k2

(

13 k2 + 2 − 2√

1 + 4 k2 (−1 + 9 k2))

k2(12b)

n3 =3

2+

1

2

√

k2

(

13 k2 + 2 − 2√

1 + 4 k2 (−1 + 9 k2))

k2(12c)

n4 =3

2+

1

2

√

k2

(

13 k2 + 2√

1 + 4 k2 (−1 + 9 k2) + 2)

k2(12d)

In order to satisfy the required set of boundary conditions stipulated in Eq.(5), the constants A, B,

C, D, E, F , G and H appearing above assume the following specific values with letting γ = β/α:

A =

(

β sinhβ +1

6β3 sinhβ − coshβ − 1

2β2 coshβ

)

C

+

(

− sinhβ + β coshβ − 1

2β2 sinhβ +

1

6β3 coshβ

)

D (13)

B = −1

6

(

−3β coshβ + β2 sinhβ)

C

β2− 1

6

(

−3β sinhβ + β2 coshβ)

D

β2− 1

β2(14)

C =f1(

n4, eα, eβ

)

αn4H + f1(

n3, eα, eβ

)

αn3G+ g1(

eα, eβ)

J1 (eα, eβ)(15)

D =f2(

n4, eα, eβ

)

αn4H + f2(

n3, eα, eβ

)

αn3G+ g2(

eα, eβ)

J1 (eα, eβ)(16)

E = 0 (17)

4 YUAN Rong, et al: Study on Fluid Flow Passing Through a Porous Media 361

F = 0 (18)

G =h(

n4, eα, eβ

)

αn3J2 (n4, n3, eα, eβ)(19)

H = − h(

n3, eα, eβ

)

αn4J2 (n4, n3, eα, eβ)(20)

where the expressions of functions f1, f2, g1, g2, h, J1 and J2 are shown in Appendix A.

The drag force, F, experienced by the composite sphere is calculated by integrating the normal

and tangential stress distributions over the surface:

F = 2πs2∫ π

0

(τ∗rr cos θ − τ∗rθ sin θ)|r=s sin θdθ (21)

The result obtained is

F = 6πµV sΩKY q−olp (22)

where ΩKY q−olp can be calculated as:

ΩKY q−olp = −1

9β[

A− 2β3B + 2 (β sinhβ − coshβ)C + 2 (β coshβ − sinhβ)D]

(23)

5 Comparison to Previous Work

The algebra leading to the expressions Eqs.(13) − (20) and Eq.(23) is exceedingly tedious. How-

ever, their correctness may be confirmed by testing a number of limiting cases of Ω†KY q−olp defined

by the following expression for which analytical solutions are already known.

F† = 2πa2

∫ π

0

(

τ †rr cos θ − τ †rθ sin θ)∣

∣

∣

r=asin θdθ = 6πµV aΩ†

KY q−olp (24)

where Ω†KY q−olp is calculated by

Ω†KY q−olp = − γ2

9

[(

8 + 2n3 −n3

k2− 5n3

2 + n33

)

Gαn3

+

(

2n4 + 8 − 5n42 + n4

3 − n4

k2

)

Hαn4

]

(25)

· Limiting case I: as s→ ∞, k2 → 0 and K → ∞, Stokes Law is retrieved[12].

lims→∞k2→0K→∞

Ω†KY q−olp = ΩS = 1 (26)

· Limiting case II: as k2 → 0, K → ∞ and s = b, This result is identical to Happel’s work[3]

limk2→0K→∞s=b

Ω†KY q−olp = ΩH (27)

where η = 1/γ.

· Limiting case III: as s→ ∞, K → ∞ and a = R, Kim and Yuan’s result [5] is obtained

lims→∞K→∞a=R

Ω†KY q−olp = ΩKY q =

(n4 + 1) (n4 − 2) (n3 + 1) (n3 − 2) k2 + n3n4 + 2

(n4 − 1) (n4 + 1) (n3 − 1) (n3 + 1) k2 + n3n4 + 1(28)

· Limiting case IV: as K → ∞, this result is identical to Kim and Yuan’s[7]

limK→∞a=Rs=b

Ω†KY q−olp = ΩKY qc

= − 2

3J3 [− (n4 + 1) (−2 + n4) (n3 + 1) (n3 − 2) k2 − 2 − n4n3] γ

6

+ 2 [− (n3 − 2) (n3 − 4) (−2 + n4) (n4 − 4) k2 + 8 − n4n3] γ (29)

where

J =2 [(n3 − 1) (n3 + 1) (n4 − 1) (n4 + 1) k2 + 1 + n4n3] γ6

362 13

+ 3 [− (n3 + 1) (n3 − 2) (n4 + 1) (n4 − 2) k2 − n4n3 − 2] γ5

+ 16 + 3 [(n4 − 1) (n4 − 4) (n3 − 1) (n3 − 4) k2 + n4n3 − 4] γ

− 2 (n3 − 2) (n3 − 4) (n4 − 2) (n4 − 4) k2 − 2n4n3 (30)

The above comparison is summarized in Fig. 2

Fig.2 Comparison of the overlapped DLCA model with some previous models

6 Theoretical Results and Discussion

As discussed in Kim and Yuan[5], the prefactor k2 = 0.2 of the ideal aggregate with quadrati-

cally varying permeability adequately describes the hydrodynamics of aggregate formed in the DLCA

regime. We will assume this critical value of 0.2 for the prefactor k2 for the rest of discussion.

Streamlines of the fluid flow through the overlapped dlca aggregates with continuous

permeability profiles We choose K = 1 without loss of generality, and the inner radius of the

aggregate a =√

5 to keep the continuous permeability profile as shown in Fig.3a. In this case, the

streamlines are continuously distributed. The streamline of fluid flow is as shown in Fig. 3b.

Fig.3 Streamlines of the fluid flow through the overlapped DLCA aggregates with continuous permeability

profiles

From the Fig.3, we can see that streamlines in the shell are more curved than those in the

overlapped region. This implies that the outer layer is more permeable than the inner sphere, which

is consistent with the permeability profile, see Fig. 3a.

4 YUAN Rong, et al: Study on Fluid Flow Passing Through a Porous Media 363

7 Conclusions

For the overlapped DLCA aggregates, the fluid flow passing through the overlapped DLCA ag-

gregate, i.e., a porous sphere with quadratically vary permeability surrounded by a uniformly porous

shell, is governed by Brinkman’s and continuity equations. The analytical solution to the governing

equations is derived and compared with some limiting cases for which the analytical solutions had

been derived by other researchers. In a special case with continuous permeability profile, the stream-

lines within the overlapped aggregate show the consistency with this permeability. The comparison

of the drag force exerted on the overlapped aggregate with those on the ideal aggregate and DLCA

aggregate shows the drag force increases with occupancy fraction. The dimensionless drag force Ω

with respect to the occupancy fraction λ reflects the structural transition of the aggregate cake layer

from the initial stage of aggregate deposition to the significant overlap between aggregates, possibly

followed by total breakage of deposited DLCA aggregates.

Appendix A Calculation to the Overlapped Model

f1 (x, y, z) = − yα3(

1 + z2)

(x+ 1) γ5 + 3 yα2 (z − 1) (1 + z) (x+ 1) γ4

− 6 yα(

1 + z2)

(x+ 1) γ3 +[

y(

1 + z2)

(x− 2)α3 − 6 z (y − 1) (y + 1)α2

+6 z(

y2 + 1)

(x+ 1)α+ 6 (yz + 1) (−y + z) (x+ 1)]

γ2

− 3 yα2 (z − 1) (1 + z) (x− 2) γ (31)

g1 (x, y) = − 3xα3(

y2 + 1)

γ3 + 9xα2 (y − 1) (y + 1) γ2 − 18xα(

y2 + 1)

γ

− 6 y (x− 1) (x+ 1)α2 + 18 y(

x2 + 1)

α+ 18 (xy + 1) (−x+ y) (32)

f2 (x, y, z) =yα3 (z − 1) (1 + z) (x+ 1) γ5 − 3 yα2(

1 + z2)

(x+ 1) γ4

+ 6 yα (z − 1) (1 + z) (x+ 1) γ3 +[

−y (z − 1) (1 + z) (x− 2)α3 + 6 z(

y2 + 1)

α2

−6 z (y − 1) (y + 1) (x+ 1)α− 6 (yz − 1) (−y + z) (x+ 1)] γ2

+ 3 yα2(

1 + z2)

(x− 2) γ (33)

g2 (x, y) =3xα3 (y − 1) (y + 1) γ3 − 9xα2(

y2 + 1)

γ2 + 18xα (y − 1) (y + 1) γ

+ 6 y(

x2 + 1)

α2 − 18 y (x− 1) (x+ 1)α− 18 (xy − 1) (−x+ y) (34)

J1 (x, y) =α4(

x2 + y2)

γ5 − 3α3 (−x+ y) (x+ y) γ4 + 6α2(

x2 + y2)

γ3 +[(

−y2 − x2)

α4

−3α3 (−x+ y) (x+ y) +(

−3x2 − 3 y2 − 24xy)

α2 − 6 (−x+ y) (x+ y)α]

γ2

+[

3α3 (−x+ y) (x+ y) +(

9 y2 + 9x2)

α2 + 9 (−x+ y) (x+ y)α]

γ (35)

h (x, y, z) =[

3 z2x (y − z) (y + z)α7 − 3 z2(

y2 + z2) (

x2 − 2)

α6

+ 3 z2 (y − z) (y + z) (x− 1) (x− 2) (x+ 1)α5

−3 z2(

y2 + z2)

(x− 1) (x− 2) (x+ 1)α4]

k2 − 3 z2x (y − z) (y + z)α5

+3xz2(

y2 + z2)

α4

γ3 +[

9 z2x(

y2 + z2)

α6 − 9 z2 (y − z) (y + z)(

x2 − 2)

α5

+ 9 z2(

y2 + z2)

(x− 1) (x− 2) (x+ 1)α4

−9 z2 (y − z) (y + z) (x− 1) (x− 2) (x+ 1)α3]

k2 − 9xz2(

y2 + z2)

α4

+9 z2x (y − z) (y + z)α3

γ2 +[

18 z2x (y − z) (y + z)α5

364 13

− 18 z2(

y2 + z2) (

x2 − 2)

α4 + 18 z2 (y − z) (y + z) (x− 1) (x− 2) (x+ 1)α3

−18 z2(

y2 + z2)

(x− 1) (x− 2) (x+ 1)α2]

k2 − 18 z2x (y − z) (y + z)α3

+18xz2(

y2 + z2)

α2

γ +[

−24 z3yα6

− 6 z2(

16 yz + 2 zx3y − 10 zx2y − 3xy2 + 4 zxy − 3xz2)

α4

− 18 z2 (y − z) (y + z)(

x2 − 2)

α3 + 18 z2(

y2 + z2)

(x− 1) (x− 2) (x+ 1)α2

−18 z2 (y − z) (y + z) (x− 1) (x− 2) (x+ 1)α]

k2 + 12 z3xyα4

− 18xz2(

y2 + z2)

α2 + 18 z2x (y − z) (y + z)α (36)

J2 (w, x, y, z) =[

(y − z) (y + z) (x− w) z2α7 −(

y2 + z2)

(x− w) (w + x) z2α6

+ (y − z) (y + z) (x+ 1) (w + 1) (w − 3 + x) (x− w) z2α5

− (x− 1) (x+ 1)(

y2 + z2)

(−1 + w) (w + 1) (x− w) z2α4]

k2

− (y − z) (y + z) (x− w) z2α5 − (x− w)(

y2 + z2)

(xw + 1) z2α4

γ5

+[

3(

y2 + z2)

(x− w) z2α6 − 3 (y − z) (y + z) (x− w) (w + x) z2α5

+ 3(

y2 + z2)

(x+ 1) (w + 1) (w − 3 + x) (x− w) z2α4

−3 (x− 1) (x+ 1) (y − z) (y + z) (−1 + w) (w + 1) (x− w) z2α3]

k2

−3(

y2 + z2)

(x− w) z2α4 − 3 (x− w) (y − z) (y + z) (xw + 1) z2α3

γ4

+[

6 (y − z) (y + z) (x− w) z2α5 − 6(

y2 + z2)

(x− w) (w + x) z2α4

+ 6 (y − z) (y + z) (x+ 1) (w + 1) (w − 3 + x) (x− w) z2α3

−6 (x− 1) (x+ 1)(

y2 + z2)

(−1 + w) (w + 1) (x− w) z2α2]

k2

−6 (y − z) (y + z) (x− w) z2α3 − 6 (x− w)(

y2 + z2)

(xw + 1) z2α2

γ3

+[

2 (y − z) (y + z) (x− w) z2α7 − 2(

y2 + z2)

(x− w) (w + x) z2α6

− (y − z) (y + z) (x− w)(

xw2 − 2w2 − 7xw + 4w + x2w − 6 − 2x2 + 4x)

z2α5

+ (x− w)(

w2y2x2 + w2x2z2 − 12 zw2y − 4w2y2 − 4w2z2 − 6xwy2 + 6wz2

− 6wxz2 + 6wy2 + 36 zwy− 24wzxy + 4 z2 − 4x2z2 + 4 y2 − 4 y2x2

−12 zx2y − 24 yz + 6xz2 + 6xy2 + 36 zxy)

z2α4 − 3 (y − z) (y + z) (x− w)(

−xw2 − 2w2 + x2w2 + xw + 4w − x2w + 4 − 2x2 + 4x)

z2α3 + 3 (x+ 1) (w + 1)

(x− w)(

wxz2 + 8wzxy + xwy2 − 16 zwy − 16 zxy− 2 z2 − 2 y2 + 32 yz)

z2α2

−6 (x− 1) (x+ 1) (y − z) (y + z) (−1 + w) (w + 1) (x− w) z2α]

k2

− 2 (y − z) (y + z) (x− w) z2α5

+ (x− w)(

wxz2 + 4 z2 + 12 yz + xwy2 + 4 y2)

z2α4

− 3 (x− w) (y − z) (y + z) (xw + 2) z2α3

+ 3 (x− w)(

16 yz + 8wzxy + xwy2 + wxz2)

z2α2

−6 (x− w) (y − z) (y + z) (xw + 1) z2α

γ2 +[

6(

y2 + z2)

(x− w) z2α6

− 6 (y − z) (y + z) (x− w) (w + x) z2α5

− 3(

y2 + z2)

(x− w)(

xw2 − 2w2 − 7xw + 4w + x2w − 6 − 2x2 + 4x)

z2α4

+ 3 (y − z) (y + z) (x− w)(

x2w2 − 4w2 − 6xw + 6w − 2 − 4x2 + 6x)

z2α3

4 YUAN Rong, et al: Study on Fluid Flow Passing Through a Porous Media 365

− 9 (x+ 1) (x− 2)(

y2 + z2)

(w + 1) (w − 2) (x− w) z2α2

+9 (x+ 1) (x− 2) (y − z) (y + z) (w + 1) (w − 2) (x− w) z2α]

k2

− 6(

y2 + z2)

(x− w) z2α4 + 3 (x− w) (y − z) (y + z) (xw + 4) z2α3

−9 (x− w)(

y2 + z2)

(xw + 2) z2α2 + 9 (x− w) (y − z) (y + z) (xw + 2) z2α

γ

References:

[1] Brinkman H C. On the permeability of media consisting of closely packed porous particles[J]. Appl Sci

Res, 1949, 1(1): 81–86.

[2] Keh-Jim Dunn and David J Bergman. Self diffusion of nuclear spins in a porous medium with a periodic

microstructure[J]. J Chem Phys, 1995, 102(8): 3041–3054.

[3] John Happel. Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical

particles[J]. AIChE J, 1958, 4: 197–201.

[4] Uzi Hizi, David J Bergman. Molecular diffusion in periodic porous media[J]. J Appl Phys, 2000, 87(4):

1704.

[5] Albert S Kim, Yuan R. Hydrodynamics of an ideal aggregate with quadratically increasing permeability[J].

J Colloid Interface Sci, 2005, 285(2): 627–633.

[6] Albert S Kim and Yuan R. A new model for calculating specific resistance of aggregated colloidal cake

layers in membrane filtration processes[J]. J Membrane Sci, 2005, 249(1-2): 89–101.

[7] Albert S Kim and R Yuan. Cake resistance of aggregates formed in the diffusion-limited-cluster-aggregation

(dlca) regime[J]. J Membrane Sci, 2006, 286(1-2): 260–268.

[8] Masliyah J H, Graham H Neale, Malysa K, and Van De Ven T G M. Creeping flow over a composite

sphere: Solid core with porous shell. Chem Eng Sci, 1987, 42(2): 245–253.

[9] Paul Meakin. Diffusion-controlled cluster formation in two, three, and four dimen-sions[J]. Phys Rev A,

1983, 27(1): 604–607.

[10] Graham H Neale, Epstein N, and Nader W. Creeping ow relative to permeable spheres[J]. Chem Eng Sci,

1973, 28(10): 1865–1874.

[11] Risovic D, and Martinis M. Fractal dimensions of suspended particles in seawater[J]. J Colloid Interface

Sci, 1996, 182(1): 199–203.

[12] Stokes G G. On the effect of internal friction of fluids on the motion of pendulums[J]. Trans Camb Phil

Soc, 1851, 9: 1–106.

[13] Weitz D A, and Oliveria M. Fractal structures formed by kinetic aggregation of aqueous gold colloids[J].

Phys Rev Lett, 1984, 52(16): 1433–1436.

!"#$%&('

) *1, Albert S Kim2

1. +,-./0 , +, 266071

2. Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Hawaii, 96822, USA

12: 3456789:;<=>?@A9BCDEFHGI95HI . 5I5KJMLNOP9QRSTUVWXZYZ[Z\Z]Z^Z_

, `Z6ZaZbZ5ZZ. , cZdZeZf [Z\ZgZhZijZklmn 9oZo , pZqZrZsZtZuZvxwzyZ9rs|K~M.

: ; BCFGI ; ; 5.