Start-up Brinkman electrophoresis of a dielectricsphere for Happel and Kuwabara models

E.I. Saad

Department of Mathematics, Faculty of Science, Damanhour U niversity, Damanhour, Egypt

Abstract

The starting electrophoretic in a charged porous continuumresponse of a homogeneous sus-

pension of spherical particles to the sudden application of an external electric eld is analyzed

semianalytically through the use of a unit cell model. The el ectric double layer in the Brinkman

medium surrounding each sphere is assumed to be thin but nit e, and the eect of dynamic

electroosmosis within it is included. The eld equation for the uid outside the double layers

is solved based on the unit cell model under the start-up Brin kman model. Expressions of the

time-dependent electrophoretic and settling velocities o f the particle in the Laplace transform

as functions of the permeability, the relative mass density , the electrokinetic radius, and the

volume fraction of the particles are performed for two diere nt boundary conditions at the c-

titious surface of the cell. Numerical results indicate tha t the time scale for the development of

electrophoresis and sedimentation is signicant and small for a high permeability, an assemblage

of porous spheres with a higher particle volume fraction, an d a smaller particle to porous uid

density ratio; the electrophoretic mobility is a monotonic ally increasing function of the electroki-

netic particle radius at any instant. The start-up of electr ophoretic mobility decreases with an

increase in the particle volume fraction at a small value of t he particle-to-medium density ratio,

but it may increase as the particle volume fraction increase s at a large value of this density.

The particle interaction eect in an assemblage on the time-d ependent electrophoresis is much

smaller than that on the starting sedimentation of the parti cles. The comparisons of the results

of the cell model with dierent boundary conditions at the vir tual surface of the cell are provided

graphically.

Key words: Starting Brinkman electrophoresis; Unit cell model; Elect rophoretic mobility.

1. Introduction

The electrophoresis phenomena is the process of moving charged m olecules in a so-

lution by applying an imposed electric eld across the mixture. During e lectrophoresis,

mobility is dependent on the charge, shape, size and concentration of the molecules in

Email address:[email protected] (E.I. Saad).

Preprint submitted to Transp Porous Med 5 April 2018

solution. The electrophoresis of macromolecules is normally carried out by applying a

sample in a narrow zone to a solution stabilized by a porous matrix. The matrix can

be composed of a number of dierent materials, cellulose acetate, in cluding paper, or

gels made of polyacrylamide. Most practical applications of electrop horesis are biological

and biochemical research, pharmacology, forensic medicine, clinica l investigations, pro-

tein chemistry, and veterinary science and food control as well as molecular biology 1,2.

The basic equations governing the electric conductivity of a dilute as semblage of charged

colloidal particles also describe the electrophoretic phenomena. O’B rien 3 obtained an

analytical expression for the electric conductivity of a dilute swarm of dielectric spheres

with thin but polarized double layers in a general electrolyte solution. Some approximate

analytical formulas for the electrophoretic mobility and electric con ductivity were derived

for dilute suspensions of colloidal spheres in symmetric electrolytes . 4.

The permeable continuum through which electrokinetic phenomena t ake place is often

modelled as an eective Brinkman continuum. To model the uid ow in a homogeneous

suspension of particles, Brinkman 5 introduced the so-called the B rinkman model, which

is essentially a modied Navier–Stokes equation including an extra ter m representing the

corresponding hydraulic drag force induced by the presence of th e solid part within the

porous media. The hydrodynamic permeability is characterized by th e Darcy permeability

(square of the Brinkman screening distance). This idea was also rep orted independently

by Debye and Bueche 6 about the same time when they discussed th e uid ow in a

dilute polymer molecule solution. Hermans and Fujita 7 considered th e electrophoresis of

an isolated charged porous spherical particle and derived a formula for the electrophoretic

mobility of a porous sphere by introducing Brinkman’s equation for th e internal ow eld

of the particle and assuming that the double layer still remains spher ically symmetric

in the presence of the applied electric eld. Although the Brinkman eq uation is semi-

empirical in nature, it ts well with the experimental data in general 8 and is regarded

as the standard approach describing the uid ow through a porou s media. It was further

extended to cover the corresponding electrokinetic transport p henomena inside the porous

media, with an extra consideration of the electrical driving force 9– 13. Feng et al. 14

studied the motion of a chargeless sphere near planar conning bou ndaries in a ber

matrix modelled as Brinkman’s medium. Tsai et al. 15 investigated the oretically the

electrophoretic motion of a charged spherical particle in a porous c ontinuum by focusing

on the boundary eect of a solid plane toward which the particle move s perpendicularly.

Dierent boundary conditions are suggested at the cell outer (c titious) surface. A

cell may consist of a single impermeable/porous particle surrounded by a uid envelope.

The cell model was originally proposed by Happel 16 and Kuwabara 1 7 to describe the

Stokes ow in an array of solid cylinders and spheres. The two appro aches diered in the

kind of boundary condition adopted on the cell outer surface. The hydrodynamic model of

Happel assumes no-slip boundary condition on the inner represent ative particle and zero

shear stress on the outer envelope, while the Kuwabara model pro posed a nil shear stress

condition, based on the requirement that no mechanical energy sh ould be exchanged

between the cell and the environment 8, Kuwabara’s model relied on the kinematical

argument suggesting a zero vorticity condition on the cell, both mod els usually give similar

results. The Kuwabara and Happel unit cell models are used in sever al applications as

indicated above. Both models considered the eects of adjacent bers. However, many

2

researchers have been approved Kuwabara model because “it is more representative of the

ow around the ber in the case of low Reynolds number” 18. Severa l researchers 19–21

attempted to analyze the transient response of electrolyte solut ion in the porous medium

constructed by a swarm of parallel charged circular cylinders to th e step application of

an electric eld and a pressure gradient in the axial and in the transv erse direction,

respectively, through the use of a unit cell model. Recently, the un it cell model was

employed to investigate the transient electrophoresis of a swarm o f dielectric spheres

with constant zeta potential after the application of a step funct ion electric eld for the

case of thin but nite double layer 21. Saad and Faltas 22 studied se mitheoretically the

time-dependent electrophoresis of a charged spherical particle in an electrolyte solution

saturated in a charged porous medium after the sudden application of an external electric

eld.

In this paper, both the Happel and Kuwabara cell models are adopt ed to describe

the start-up of electrophoresis in a homogeneous assemblage of d ielectric spheres in a

saturated porous medium with constant zeta potential due to a su dden application of a

step function electric eld for the case of thin but nite double layer s in which the start-up

response of the electroosmosis ow at the surface of a sphere-in -cell model is incorporated.

The time-dependent response of the electrophoretic mobility of th e spherical particles

to a step change in the imposed electric eld as a function of the elect rokinetic particle

radius, particle volume fraction, particle-to-medium density ratio, and permeability of the

porous medium is obtained exactly. The time-evolving starting sedime ntation response of

an assemblage of spherical particles to the body-force eld (such as the gravitational

eld) is also derived and compared with the electrophoresis results. In the limiting case

of clear uid, our formulas reduce to the corresponding results ob tained for the startup

electrophoresis in a homogeneous suspension of spherical particle s 21.

2. Time-dependent electrophoresis of a sphere-in-cell mod el with a thin dou-

ble layer

We examine the unsteady electrophoretic in a homogeneous swarm o f dielectric spheres

with thin electric double layers in a saturated charged porous medium of permeabilityK

and density ?.The porous continuum is assumed to be homogeneous and isotropic, and

therefore, permeability and porosity corresponding to the given p orous/Brinkman con-

tinua are constants. As illustrated in Fig. 1, we use a unit cell model in which each

particle of radius ais enclosed by a ctitious spherical envelope of ionic uid having an

outer radius of b,so that the particle/cell volume ratio is equal to the apparent part icle

volume fraction ?throughout the entire swarm, that is ?= ( a/b)3

. The cell as a whole

is electrically neutral. The particle is placed in an electrolyte-saturat ed porous medium,

which can be determined either by theoretical prediction or by expe rimental measure-

ment. Brinkman’s equation has been extensively used in computing th e hydrodynamic

interaction and slow transport of solid sphere in ordered or disorde red brous continuum.

When Brinkman’s equation is scaled with respect to the radius of the s phereaand made

non-dimensional a parameter, ??

= a/K 1

/2

, will show up. This parameter describes the

ratio of the sphere radius to the bre interaction layer thickness, K1

/2

, therefore Kde-

3

z

E 0

r

f

Oa

qd

U

z

b

Fig. 1. Coordinate graph for the electrophoresis of a sphere in the unit cell model.

pends on the bre spacing f

s. A sphere of radius

awill not be capable to translate in a

rigid periodic bre array when the open spacing between bres, f

s is less than the spher-

ical particle diameter. Then for a solid bre array, Brinkman’s param eter??

will have a

maximum value if the spherical particle is not to be trapped by the br es. For a periodic

bre array, we have to satisfy a constraint on Kthat is f

s ;

2a, and not all values of K

are acceptable unless the bres are highly exible. It should be note d here that for Stokes

ows, the drag acting on a sphere is the same for the two cases of a sphere translates near

a conning boundaries with the uid is at rest at innity and when the s pherical particle

held xed with a uniform ow at innity; while in a Brinkman continuum the se two cases

are not commutable because for a uniform ow at innity there is a va lue for the pressure

gradient at innity arises from the Darcy resistance 14. The origin o f the spherical coor-

dinate system ( r, ?, ?) is established at the center of the particle/cell and the axis ?= 0

points toward the positive z-direction, and the problem for each cell is axially symmetric

about the z-axis and independent of the coordinate ?. Further, the particle enveloped by

a nite thin electric double layer (say, ?a?2) in an electrolyte uid–saturated porous

medium within the virtual cell. Initially, t= 0 a constant electric eld E

0 is imposed

suddenly throughout the system in the polar axis ?= 0 and maintained afterwards. As

a result, the particles experience electrophoretic motion with a time -dependent velocity

in the same direction as E

0.

Note that U(t) is unknown and has to be determined with

U (0) = 0 .Because the Reynolds number associated with electrokinetic ow is a ssumed

small, therefore the motion of the electrolyte solution-saturated porous medium outside

the double layer can be modelled by the unsteady Brinkman equation. It is convenient to

use the stream function which is related to the velocity component sq

r(

r, ? ) and q

(

r, ? )

by

qr =

? 1

r

2

sin ??

?? , q

= 1r

sin ??

?r .

(2.1)

Therefore, the stream function satised by the fourth-order d ierential equation 22:

L ?1

L ?1 ?

?2

? 1

???t

= 0 , (2.2)

4

where?= 1 /?K

is the dimensional permeability parameter, ?is the eective kinematic

viscosity, µis the viscosity of the uid, and L

?1 is the axisymmetric Stokesian operator:

L ?1 = ?

2

?r

2+ sin

?r2 ???

1sin ????

. (2.3)

Solving the Laplace equation, ?2

V = 0 governing the electric potential Vin the medium

outside the thin double layer surrounding the dialectic spherical par ticle and applying the

non-conducting (zero-ux) condition at the outer edge of the do uble layer in a virtual cell,

?V /?r = 0 at r= a+

as well as the condition: V=?E

0r

cos ?at r= b.The solution of

this problem is V=?E

0/

(2 + ?)2 r+ a3

/r 2

cos ?.Therefore, the local tangential electric

eld E

in

?–direction at the surface of the dielectric spherical particle cause d by the

imposed electric eld is

E=

?1

r?V??

r = a+ =

? 32 +

?E

0sin

?. (2.4)

This tangential electric eld interacts with the thin double layer to pr oduce a local elec-

troosmotic velocity at r= a+

. Thus, the initial and boundary conditions for the velocity

components q

r and

q

of the uid through the porous medium as well as at the ctitious

surface of the cell, in which the overlap of the electric double layers o f adjacent particles

is allowed, can be expressed as:

t= 0 : q

r =

q

= 0

, (2.5)

r = a+

: q

r =

Ucos ?, (2.6)

q =

?U sin ?+ 3

?? E

0

(2 + ?)µ

1cosh??+?

n =1 2(

?1) n

? n?(? 2

+ ?2

n

) e

?

( 2

n

+

2

) t

sin ?, (2.7)

r = b: q

r = 0

, (2.8)

r ?

?r

qr

+ 1r?q

r?? = 0

,(when Happel’s model is applied) (2.9)

1

r ??r

(

r q

)

? 1r?q

r?? = 0

,(when Kuwabara’s model is applied) (2.10)

where ?is the zeta potential of the spherical particle, ?is the uid permittivity, ?

n =

(2 n? 1) ?/ (2?), ? = 2 /?(? is the Debye screening parameter), ?= µ ?/? (? is the porosity

of the porous continuum), and r= a+

denes the outer edge of the thin double layer.

U (t) the start-up of electrophoretic velocity of the spherical partic le to be determined.

The apparent velocity slip distribution denoted by the last term of eq uation (2.7), which

has been obtained from the local electroosmosis in the double layer a nd leads to the

electrophoresis of the spherical particle, increases with increase s in the elapsed time from

zero t= 0 to the steady state values as t? ? and in the electrokinetic radius ?ato

these steady values as ?a? ? ,and also it increases with decreases in the permeability

? to the clear uid case ?= 0 22. As indicated in equations (2.8)–(2.10), Happel’s

unit cell model assumes that the radial velocity and shear stress o f the uid vanish at

the ctitious surface of the cell, whereas Kuwabara’s unit cell mode l takes this radial

velocity and the vorticity of the uid to be zero there. For the stea dy sedimentation

5

of a swarm of uncharged spherical particles, both the Happel andKuwabara models give

qualitatively the same ow elds and approximately comparable drag f orces on the sphere-

in-cell. However, Happel’s model has an advantage in that it does not require an exchange

of mechanical energy between the cell and the environment, while K uwabara’s model does

require such exchange 8. In terms of the Laplace transform (dened by a bar over the varia ble), the equation

(2.2) and using the initial condition (2.5), can be simplied as:

L?1(

L

?1 ?

?2

) ¯

= 0 , (2.11)

and the transformed boundary conditions are r = 1 +

: ¯ q

r = ¯

U cos ?, (2.12)

¯

q

=

?¯

U sin ?+ 3

?? E

0

(2 + ?)µ

1scosh ??+?

n =1 2(

?1) n

a 2

? n?(? 2

+ ?2

n

)(

?2

? + ?2

n

?

+ a2

s )

sin ?,(2.13)

r = ??

1/3

: ¯ q

r = 0

, (2.14)

r ?

?r

¯

q

r

+ 1r?

¯

q

r?? = 0

,(if Happel’s model is used) (2.15)

1

r ??r

(

r ¯

q

)

? 1r?

¯

q

r?? = 0

,(if Kuwabara’s model is used) (2.16)

where sis the transform parameter; here and in all subsequent expressio ns and param-

eters, we normalized all lengths with respect to the radius of the pa rticlea,i.e. ?=

? 2

+ a2

s/? , ? = 2/(?a ), and the non-dimensional permeability parameter ?(= a/?K

).

The general solution of (2.11) that satises the initial conditions in e quations (2.12)–

(2.16) can be expressed as

¯

=

A r 2

+ B

r+ Cr(

? r + 1) e?

r

+Dr(

? r ?1) e r

sin 2

?, (2.17)

where the four unknown coecients A, B, CandDare obtained using the boundary

conditions (2.12)–(2.16), with the result

A = ?

?

2

1

6 ¯

U ? +

2( ?2

? 3? 1

/3

) W + 3( ?1

/3

? 1) ?2

+ 3 ?1

/3

¯

U

sinh ?(1 ???

1/3

)

+

2(3 ?1

/3

? 1) ? W + ?3

? 3(3 ?1

/3

? 1) ?¯

U

cosh ?(1 ???

1/3

) ? 4W ?

, (2.18)

B = ?

1

2? 2

1

4W ? 3

? ?

2 ?4

+ 3 ?1

/3

(2 ?1

/3

? 1) ?2

? 6? W + 3 ?1

/3

(2 ?2

/3

? 6? 1

/3

+ 3) ?2

+6 ?+ ( ?1

/3

? 1) ?4

¯

U

sinh ?(1 ???

1/3

) ?

2(3 ?1

/3

? 1) ?3

+ 6 ?2

/3

(? 1

/3

? 1) ?W

+ ? ? 4

+ 3(2 ?1

/3

? 1) ( ?1

/3

? 1) ?2

? 18?2

/3

(? 1

/3

? 1) ¯

U

cosh ?(1 ???

1/3

)

, (2.19)

C = ?

1

4? 2

1

6?

2( ?? 1) W + ( ?2

? 3? + 3) ¯

U

e

+

2( ?? 1) ?3

? 3? 1

/3

(? ? 1) ?2

? 6? ? 2

/3

+ 6 ?W + 3( ?3

? 3? 2

? 1

/3

+ 6 ? ?2

/3

? 6? ) ¯

U

e

1/ 3

, (2.20)

6

D= 14

? 2

1

6?

2( ?+ 1) W?(? 2

+ 3 ?+ 3) ¯

U

e?

+

2( ?? 1) ?3

+ 3 ?1

/3

(? ? 1) ?2

? 6? ? 2

/3

? 6? W + 3(6 ? ?2

/3

+ 3 ?2

? 1

/3

+ ?3

+ 6 ?) ¯

U

e?

1/ 3

, (2.21)

for the Happel hydrodynamic model, and A= ?? B, (2.22)

B = 1

2

2

2( ?2

? ?1

/3

) W + ( ?1

/3

? 3) ?2

+ 3 ?1

/3

¯

U

sinh ?(1 ???

1/3

)

+ ?

2( ?1

/3

? 1) W + ?2

? 3(?1

/3

? 1) ¯

U

cosh ?(1 ???

1/3

)

, (2.23)

C = ?

1

4

2(

? ? ?1

/3

)

2( ?? 1) W + 3 ¯

U

e

1/ 3

, (2.24)

D = 1

4

2(

? + ?1

/3

)

2( ?? 1) W + 3 ¯

U

e?

1/ 3

, (2.25)

for the Kuwabara hydrodynamic model, where

W= 3

?? E

0

(2 + ?)µ

1scosh ??+?

n =1 2(

?1) n

a 2

? n? (? 2

+ ?2

n

) (

?2

? + ? ? 2

n

+

a2

s )

,

1= 3

? 1

/3

(? ? ?2

/3

? 1) ?2

+ ?(3 ?1

/3

? 2)

sinh ?(1 ???

1/3

)

+ ?

(? ? 1) ?2

? 3? 2

/3

(3 ?2

/3

? ?1

/3

+ 2)

cosh ?(1 ???

1/3

) + 12 ? ? ,

2=

?1

/3

( ? ? 3? 2

/3

? 1) ?2

+ 3 ? sinh ?(1 ???

1/3

)

+( ?? 1) ?3

? 3? (? 1

/3

? 1) ? cosh ?(1 ???

1/3

) .

The transform of the hydrodynamic drag force exerted on the pa rticle surfacer= 1 +

in

a cell can be calculated as 8,22

¯

F (s ) = ? µ a

0

r 4

sin 3

? ?

?r

L?1¯

r 2

sin 2

?

? ?2

r 2

sin ??

¯

?r ?

s?r

2

sin ??

¯

?r

r=1 +d?,

(2.26)

= 4

3? µ a?

2

B ?2A ? 2C (? + 1) e?

? 2D (? ? 1) e

. (2.27)

This drag force, ¯

F (s ) is equal to the rate of change of the electrophoretic particle mom en-

tum; thus

¯

F (s ) = 4

3? a

3

? p s ¯

U , (2.28)

where ?

p is the mass density of the spherical particle. Substituting equation

(2.27) into

equation (2.28), we get the transform of the time-dependent elec trophoretic velocity of

the bulk uid as

7

¯

U (s ) = ?6

W

3

? 4

+ 3 ?1

/3

(2 ?1

/3

? 1) ?2

? 6? sinh ?(1 ???

1/3

) + ?

(3 ?1

/3

? 1) ?2

+6( ?? ?2

/3

)

cosh ?(1 ???

1/3

) ? 2? 3

?

, (2.29)

for Happel’s model, and ¯

U (s ) = ?6

W ? 2

4

(? 2

? ?1

/3

) sinh ?(1 ???

1/3

) + ( ?1

/3

? 1) ?cosh ?(1 ???

1/3

)

, (2.30)

for Kuwabara’s model, where 3= 24

? ?(? 2

? ?) + 3

(2 ?4

/3

+ ?1

/3

? 2? ? 3) ?4

+ 2(3 ??) ? 4

/3

? 18?2

/3

+2( ?+ 1) ?+ (2 ?+ 9) ?1

/3

? 2

? 6? ? 4

/3

+ 2(2 ?+ 9) ?

sinh ?(1 ???

1/3

)

+ ?

(2 ?+ 1) ?4

+ 2(3 ??) ? + 2 ?+ 9 ?18?4

/3

+ 6 ?2

/3

? 27?1

/3

? 2

+63 ? ?4

/3

+ (2 ?+ 9) ?2

/3

? (9 + ?) ?

cosh ?(1 ???

1/3

) ,

4=

(2 ?4

/3

? 6? + ?1

/3

? 9) ?4

+ 2(3 ??) ? 4

/3

+ 6 ? ?+ (2 ?+ 9) ?1

/3

? 2

? 6? ? 4

/3

× sinh ?(1 ???

1/3

) + ?

(2 ?+ 1) ?4

+ 2(3 ??) ? ? 6? 4

/3

? 9? 1

/3

+ 2 ?+ 9 ?2

+6 ?(? 4

/3

? ?)

cosh ?(1 ???

1/3

) ,

with ?= ?(a 2

s/? ) and ?= ?

p?/?

is the ratio of particle density to porous uid density.

Clearly, the spherical particle velocity tends to zero as ?? ? ,while it has a large value at

? = 0. The innite series in equation (2.29) or (2.30) is the contribution f rom the dynamic

response of the electroosmotic porous ow at the particle surfac e after the electric eld is

suddenly applied.

The unsteady electrophoretic velocity of the porous uid can be ob tained from the

inverse Laplace transform of the equation (2.29) or (2.30) by using the Talbot method or

other numerical approaches 24–27. The calculated results of nor malized particle velocity

µU/ (?? E

0) as functions of the permeability parameter

?,electrokinetic particle radius ?a

of the constitutive spheres, particle-to-medium density ratio ?,particle volume fraction

? , and dimensionless elapsed time ? t/a2

will be presented in Sec. 4.

When ?? ? ,the electrophoretic mobility vanishes for two models. In the limiting

situation as ?= 0, the Happel and Kuwabara unit cell models lead to the same start ing

electrophoretic velocity, which can be analytically obtained in closed f orm 22.

As ? t/a 2

? ? ,the electrophoretic velocity attains its steady value, i.e.

µU

?? E 0=

? 18 sech

??(? + 2)

5

? 4

+ 3 ?1

/3

(2 ?1

/3

? 1) ?2

? 6?

sinh ?(1 ???

1/3

)

+

(3 ?1

/3

? 1) ?3

+ 6( ?? ?2

/3

) ?

cosh ?(1 ???

1/3

) ? 2? 3

?

, (2.31)

8

for the Happel model, andµU

?? E 0=

?18 sech

??(? + 2)

6

(? 2

? ?1

/3

) sinh ?(1 ???

1/3

)+ ?(? 1

/3

? 1) cosh ?(1 ???

1/3

)

, (2.32)

for the Kuwabara model, with 5= 24

?3

? + 3

(? 1

/3

? 3 + 2 ?4

/3

? 2? )? 4

+ (9 ?1

/3

? 18?2

/3

+ 6 ?4

/3

+ 2 ?)? 2

+ 18 ?

× sinh ?(1 ???

1/3

) + ?

(2 ?+ 1) ?4

+ 3(2 ?? 6? 4

/3

+ 2 ?2

/3

? 9? 1

/3

+ 3) ?2

? 54( ?? ?2

/3

)

cosh ?(1 ???

1/3

) ,

6=

(2 ?4

/3

? 6? + ?1

/3

? 9) ?2

+ 3 ?1

/3

(2 ?+ 3)

sinh ?(1 ???

1/3

)

+ ?

(2 ?+ 1) ?2

? 3(?1

/3

? 1) (2 ?+ 3)

cosh ?(1 ???

1/3

) .

In the limit ?= 0 ,the normalized electrophoretic velocity reduces to the results obt ained

by Keh and Huang 21.

When ?= 0 ,the steady state electrophoretic velocity for both models given by 22

µU

?? E 0=

9(

?+ 1) sech ???2

+ 9 ?+ 9 .

(2.33)

3. Time-dependent translation of spherical particles caus ed by a suddenly

applied body force

Initially at the time t= 0 ,a constant force F

A is applied to a homogeneous suspension

of dielectric spherical particles in the zdirection and kept acting on the particles after-

wards. When the Reynolds number is low, the stream function of the uid ow through

the charged porous medium in a unit cell model is governed by the equ ation (2.2). The

initial and boundary conditions of this problem are still given by equat ions (2.5)–(2.10),

but with E

0= 0 in equation (2.7), and

U(t) is the starting settling (dynamic translational)

velocity of the particle to be determined.

The time-dependent drag force Facting on the translating spherical particle by the

porous uid can also be determined from equation (2.27). The sum of this drag force and

the applied force is equal to the rate of change of the spherical pa rticle momentum with

respect to time, and in the Laplace transform

¯

F A + ¯

F = 4

3? a

3

? p s ¯

U (s ). (3.1)

With substitution equation (2.27) (with W= 0) into (3.1), and after considerable alge-

braic manipulation, the expression for the start-up of velocity usin g the Happel boundary

condition (stress free cell surface) yields

9

¯

U (s ) = U

0?

2

s ? 1

7

3

(? 4

/3

? ?1

/3

? ?)? 2

+ 3 ?4

/3

? 2?

sinh ?(1 ???

1/3

)

+ ?

(? ? 1) ?2

+ 3( ?? 3? 4

/3

? 2? 2

/3

)

cosh ?(1 ???

1/3

) + 12 ? ?

, (3.2)

U 0? ?

3F

A?

1

2? aµ ?

2,

(3.3)

and also the corresponding expression for the time-dependent se ttling velocity obtained

using the Kuwabara boundary condition (zero vorticity at the cell b oundary) is found to

be

¯

U (s ) = U

0?

4

s ? 3

8

(? 4

/3

? ?1

/3

? 3? )? 2

+ 3 ?4

/3

sinh ?(1 ???

1/3

)

+ ?

(? ? 1) ?2

? 3(?4

/3

? ?)

cosh ?(1 ???

1/3

)

, (3.4)

U 0? ?

3F

A?

3

2? aµ ? 2

? 4,

(3.5)

where

7= 3

(2 ?4

/3

+ ?1

/3

? 2? ? 3) ?4

+ 2(3 ??) ? 4

/3

+ 2( ?+ 1) ?? 18?2

/3

+ (2 ?+ 9) ?1

/3

? 2

? 6? 4

/3

? + 2(2 ?+ 9) ?

sinh ?(1 ???

1/3

) + ?

(2 ?+ 1) ?4

+ (6 ?? 2? ? ?18?4

/3

+ 2 ?

+6 ?2

/3

? 27?1

/3

+ 9) ?2

+ 63 ?4

/3

? + (2 ?+ 9) ?2

/3

? (? + 9) ?

cosh ?(1 ???

1/3

)

+24 ? ?(? 2

? ?),

8=

(2 ?4

/3

? 6? + ?1

/3

? 9) ?4

+ (6 ?4

/3

? 2? 4

/3

? + 2 ?1

/3

? + 6 ? ?+ 9 ?1

/3

) ? 2

? 6? 4

/3

?

× sinh ?(1 ???

1/3

) + ?

(2 ?+ 1) ?4

+ (6 ?? 6? 4

/3

? 2? ? ?9? 1

/3

+ 2 ?+ 9) ?2

+6 ?(? 4

/3

? ?)

cosh ?(1 ???

1/3

) ,

? 1= 3

(? 4

/3

? ?1

/3

? ?)? 2

+ 3 ?4

/3

? 2?

sinh ?(1 ???

1/3

)

+ ?

(? ? 1) ?2

+ 3( ?? 3? 4

/3

? 2? 2

/3

)

cosh ?(1 ???

1/3

) + 12 ? ?,

? 2= 3

(2 ?4

/3

+ ?1

/3

? 2? ? 3) ?4

+ (6 ?4

/3

? 18?2

/3

+ 9 ?1

/3

+ 2 ?)? 2

+ 18 ?

× sinh ?(1 ???

1/3

) + ?

(2 ?+ 1) ?4

+ 3(2 ?2

/3

? 6? 4

/3

? 9? 1

/3

+ 2 ?+ 3) ?2

+54( ?2

/3

? ?)

cosh ?(1 ???

1/3

) + 24 ?3

? ,

? 3=

(? 4

/3

? ?1

/3

? 3? )? 2

+ 3 ?4

/3

sinh ?(1 ???

1/3

)

+ ?

(? ? 1) ?2

? 3(?4

/3

? ?)

cosh ?(1 ???

1/3

) ,

? 4=

(2 ?4

/3

+ ?1

/3

? 6? ? 9) ?2

+ 3 ?1

/3

(2 ?+ 3)

sinh ?(1 ???

1/3

)

+ ?

(2 ?+ 1) ?2

+ 3(2 ?? 2? 4

/3

? 3? 1

/3

+ 3)

cosh ?(1 ???

1/3

) .

For ?= 0 ,we get the same result for the settling velocity as obtained by Keh an d

Huang 21. The equations (3.3) and (3.5) give the steady Brinkman tr anslational velocity

10

of the spherical particle in a unit cell; for the case when the porous uid becomes clear

? = 0 ,the particle velocity reduces to the classical result as in 16,17,23. Pa rticularly, as

? t/a 2

? ? , U=U

0,

which is independent of the particle-to-medium density ratio ?and,

in the limit of maximal porosity ( ?= 0), reduces to Brinkman’s model in the absence of

the electric eld ( U

0 =

F

A/

{ 6?µa (? 2

/ 9 + ?+ 1) }). The unsteady translational velocity

of the particle can also be numerically obtained from the inverse Lapla ce transform of

equation (3.2) or (3.4). Again, the particle velocity vanishes as ?? ? .For the case of

? = 0 ,both cases of the Happel and Kuwabara unit cell model result in the same starting

particle velocity, whose analytical result available in the literature fo r limiting case, as

reported by Saad and Faltas 22.

The results in equations (3.2)–(3.5) might be extended to the limiting c ase of time-

dependent Brinkman electrophoretic motion of a swarm of dielectric spheres with thick

electric double layers ( ?a?0) 22. As expected, the Debye-H¨uckel approximation for a

dielectric spherical particle in the unit cell for permeable medium leads to zero charge on

the particle surface with a xed value of zeta potential as long as ?6

= 0 and ?has a xed

nite value 28,29. Thus, the unsteady/steady electrophoretic re sponse of the sphere in a

cell to an imposed step function electric eld vanishes in this limit ( ?a?0, ? 6

= 0 and ?

is a nite value).

4. Results and discussion

4.1. Time-dependent electrophoretic velocity

The results of the normalized starting electrophoretic velocity µU/(?? E

0) in a homo-

geneous assemblage of spherical particles with thin electric double la yers (say,?a?2)

caused by a suddenly applied electric eld, are derived from equation s (2.29) and (2.30) for

both Happel’s and Kuwabara’s unit cell models, are plotted against th e non-dimensional

elapsed time ? t/a2

, the permeability parameter ?,and the electrokinetic particle radius

?a in Figs. 2–5 for various values of the mass density ratio ?,and particle volume frac-

tion ? .The results are presented up to the maximum possible volume fractio n for an

assemblage of identical spherical particles, ?= 0 .74 30,31. When the spherical particle

is suspended in aqueous solutions and has a radius about a micromete r, the viscous relax-

ation time a2

/? is about a microsecond. For xed values of ?, ?, ?and?a, the spherical

particle velocity increases monotonically and quickly with the time from zero att= 0 to

its steady state magnitude as t? ? for both the cell models. Fig. 2 exhibits the eect of

the particle volume fraction ?on the development of the time-dependent electrophoretic

mobility µU/(?? E

0) with

? t/a2

is important and a much smaller ? t/a2

will be needed for

µU/ (?? E

0) to reach a specied percentage of its steady value for a porous r

egion with

a greater ?a,a smaller ?, or a larger ?,as expected. Our results indicate that the time

needed for the particle velocity to reach 50% of its steady value for the case of particles

with ?= 4 , ?= 1 and ?a? ? in aqueous solutions would be about 0.18 µs for ?= 0 ,

and 0.02 µs for ?= 0 .5 as shown in Fig. 2(b). The steady state is reached faster at large r

? because less ambient uid needs to grow in a velocity eld by the inuen ce of the un-

steady porous electroosmotic ow in the thin electric double layer (a nd electrophoretic

11

motion) of each spherical particle for an assemblage of larger particle volume fraction. For

specied values of ?, ?and?a, µU/ (?? E

0) at the steady state (

t? ? ) decreases with an

increase in ? ,while this normalized electrophoretic velocity at small nite values of ? t/a2

may be increased with an increase in ?if the value of ?is relatively large, but µU/(?? E

0)

at any instant is a decreasing function of ?if? is almost small.

Interestingly, for constant values of ? t/a2

, ?, ?a, ?, and? ,the Kuwabara unit cell

model results in a lower value (a stronger particle concentration e ect) for the normal-

ized time-dependent electrophoretic mobility than the Happel unit m odel does, but the

dierence in general is small and disappears at ?= 0 as expected. This dierence occurs

because the Kuwabara zero vorticity model yields a larger energy d issipation in the cell

than that due to the spherical particle drag alone, owing to the add itional work done

by the stress at the outer surface of the cell 8. Also note that be cause the values of

the electrophoretic velocity at the steady state decreases signi cantly with? ,the dier-

ence between the two cases of the cell model appearances to be n on-monotonic with?as

illustrated in Fig. 2.

For xed values of ? t/a2

, ?, ?a, and?as shown in Fig. 3(a), the dimensionless start-

up of electrophoretic mobility does not depend on the particle-to-m edium density ratio at

the steady state, but is a monotonic decreasing function of ?,meaning that a high-density

particle has a longer relaxation time than a low-density one in the deve lopment of the

particle velocity. In the limit of ?? ? or?? ? , µU/ (?? E

0) disappears regardless

of the values of ? t/a2

, ?a, and?as expected. For any value of the density ratio, it is

interesting to observe that for very thin double layer and a specie d value of? t/a2

, the

electrophoretic velocity decreases as the permeability parameter increases. The particle

velocity µU/(?? E

0) is plotted versus the dimensionless elapsed time

? t/a2

, permeability

parameter ?and electrokinetic radius ?aas shown in Fig. 3(b) for specied values of ?

and ?.Again, this velocity grows continuously with the time from zero to its s teady value

for a constant value of ?aand Kuwabara’s unit model results in a smaller velocity than

Happel’s unit model does. For nite values of ? t/a2

and ?, µU/ (?? E

0) increases (and the

starting time scale decreases) with an increase in the value of ?a.This outcome reects the

fact that the start-up of electroosmotic velocity at the spherica l surface given by equation

(2.7), which drives the bulk electroosmosis, is an increasing function of?a. Fig. 4 exhibits

the variation of the normalized electrophoretic velocity versus the permeability parameter

? for dierent values of ? t/a2

with given values of ?a/b, ?and?.It indicates that the

maximum relative values of µU/(?? E

0) occur in the Stokes limit. For low permeability

in the range of Darcian limit, the particle velocity decreases rapidly to zero, regardless

of ? t/a 2

for both cases of Happel and Kuwabara unit cell model. For consta nt values

of ? , ? and?as shown in Fig. 5, the normalized electrophoretic mobility µU/(?? E

0)

decreases monotonically with a decrease in ?a.Furthermore, at a nite value of ?a, the

time-dependent electrophoretic velocity of the particle increases monotonically and quickly

with the time from zero at t= 0 to its steady state value as t? ? .

4.2. Time-dependent fal l velocity Corresponding calculation results of the normalized starting fall ve locity 6?µaU/F

A

of the particles in a swarm obtained from equations (3.2) and (3.4) fo r both Happel’s

12

and Kuwabara’s cell models, respectively, are plotted versus the dimensionless elapsed

time ? t/a2

in Figs. 6 and 7 for various values of ? , ?,and?.As expected, this unsteady

settling velocity develops continuously with the time from zero to its s teady value for

constant values of ?, ?and?; the Kuwabara unit model results in a smaller value for

the time-dependent settling velocity than the Happel unit model do es. The eect of the

parameters ?, ?and?on the growth of 6 ?µaU/F

Awith

? t/a2

is important and a smaller

elapsed time is needed for the unsteady settling velocity to be within a certain percentage

of the steady state results for particles with a smaller ?,a greater ?or an exceeding ?

as illustrated in Fig. 6. For constant values of ?and ? t/a 2

, 6?µaU/F

Ais a monotonically

increasing as ?decreases. Again, this velocity decreases with an increase in ?for given

values of ?and ?at a nite time, showing that a heavier particle has a longer relaxation

time than a lighter one in the development of the settling velocity. As e xpected, in the

limiting case of ?? ? or?? ? ,6?µaU/F

Avanishes regardless of the values of

? t/a2

and ? .Again, the maximum relative values of the start-up of settling velocit y appear in

the Stokes limit ( ?= 0) for various values of ? t/a2

with xed values of ?and ?as shown

in Fig. 7. For low permeability in the range of Darcian limit, the settling ve locity decreases

quickly to zero, regardless of ? t/a2

for both cases of Happel and Kuwabara model.

An important result of some of these investigations is that the sphe rical particle inter-

action eect in a porous suspension (eect of ?) on the unsteady/steady electrophoresis

is substantially weaker than that on the unsteady/steady sedimen tation of the particles

(as shown in Fig. 6, and Figs. 2 and 3), because the disturbance to t he ambient uid

velocity eld caused by an electrophoretic spherical particle with th in electric double

layer in the charged porous medium decays much faster than that p roduced by a settling

sphere 32,33. Another interesting nding of the eects of the mas s density ratio?on

electrophoresis and sedimentation are similar with each other (altho ugh the eect of?on

the start-up of electrophoresis is somewhat stronger than that on time-dependent sedimen-

tation). On the other hand, the inuence of the medium permeability on electrophoresis

and sedimentation in a porous assemblage can be very signicant. Th e development of the

electrophoretic mobility with the permeability ?is substantially slower than of a settling

velocity. Obviously, for particles with the same values of ?, ?and?, a smaller elapsed

time ? t/a2

is generally needed for the normalized electrophoretic mobility µU/(?? E

0)

than for the normalized settling velocity 6 ?µaU/F

Ato reach a denite percentage of their

corresponding steady state values.

5. Conclusion

The start-up of electrophoresis in a homogeneous swarm of spher ical particles in a

charged porous medium, under the time-dependent Darcy–Brinkm an model with arbi-

trary values of the particle-to-medium density ratio ?and the particle volume fraction

? in response to a step application of an electric eld (a body force) at any values of

the nondimensional elapsed time ? t/a2

and the permeability of the porous medium ?,is

analyzed in this work by employing unit cell models with various boundar y conditions at

the outer surface of the cell. The electric double layer surrounding each spherical particle

is assumed to be thin, but the electrokinetic radius ?aof the sphere can be nite. In a

13

10-310-210-1100101102

? t/a2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

U

?? E0

?= 0

? = 0 .1

? = 0 .3

? = 0 .5

? = 0 .7Happel’s modelKuwabara’s model

(a)

10-310-210-1100101102

? t/a2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

U

?? E0

?= 0

? = 0 .1

? = 0 .3

? = 0 .5

? = 0 .7Happel’s modelKuwabara’s model

(b)

Fig. 2. Variation of the normalized electrophoretic veloci tyµU / (?? E

0) versus the dimensionless

elapsed time ? t/a2

for the cases of ?= 1 and ?a? ? with?as the parameter: (a) ?= 1; (b)

? = 4 .

10-310-210-1100101102

? t/a2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

U

?? E0

? = 0

? = 2

? = 5

?= 0

? = 1

? = 4

? = 25?

Happel’s modelKuwabara’s model

(a)

10-310-210-1100101102

? t/a2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

U

?? E0

? = 0

? = 3

?a = 2?a = 4?a ?

Happel’s modelKuwabara’s model

(b)

Fig. 3. Variation of the normalized electrophoretic veloci tyµU / (?? E

0) versus the dimensionless

elapsed time ? t/a2

for the case of ?= 0 .125: (a) ?a? ? with?and ?as the parameters; (b)

? = 4 with ?and ?aas the parameters.

14

0 5 10 15 20 25?0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

µ U

?? E0

?t/a 2

= 0 .005

? t/a2= 0 .01

? t/a 2

= 0 .03

? t/a2= 0 .1

? t/a

2

Stokes ow Darcian

ow

Happel’s modelKuwabara’s model

Fig. 4. Variation of the normalized electrophoretic veloci tyµU / (?? E

0) versus the permeability

parameter ?for dierent values of ? t/a2

with ?= 0 .3 , ?a = 10 and ?= 2.

100

101

102

? a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

µ U

?? E0

?t/a2= 0 .01

?

t/a

2= 0 .05

?

t/a

2= 0 .1

?

t/a

2= 0 .4

?

t/a

2

Happel’s modelKuwabara’s model

Fig. 5. Variation of the normalized electrophoretic veloci tyµU / (?? E

0) versus the electrokinetic

radius ?afor dierent values of ? t/a2

with ?= 0 .125 , ? = 2 and ?= 3.

15

10-310-210-1100101102

? t/a2

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

6

?µ aU

FA

?= 0

? = 0 .01

? = 0 .1

? = 0 .3

? = 0 .5Happel’s modelKuwabara’s model

(a)

10-310-210-1100101102

? t/a2

0

0.05 0.1

0.15 0.2

0.25

0.3

6

?µ aU

FA

?= 0

? = 2

?= 0?= 1?= 4? = 25

? or?

Happel’s modelKuwabara’s model

(b)

Fig. 6. Variation of the normalized time-dependent settlin g velocity 6?µaU /F

Aversus the di-

mensionless elapsed time ? t/a2

: (a) ?= 1 and ?= 4 with ?as the parameter; (b) ?= 0 .125

with ?and ?as the parameters.

0 5 10 15

?0

0.05 0.1

0.15 0.2

0.25 0.3

6 ?µ aU

FA

?t/a2= 0 .01

? t/a2= 0 .05

? t/a2= 0 .1

? t/a2= 0 .2

? t/a

2

Stokes ow Darcian

ow

Happel’s modelKuwabara’s model

Fig. 7. Variation of the normalized time-dependent settlin g velocity 6?µaU /F

Aversus the per-

meability parameter ?for dierent values of ? t/a2

with ?= 0 .125 and ?= 1.

16

unit cell, the time-evolving electroosmotic velocity at the outer edgeof the double layer

is taken as a slip condition at the sphere surface to solve the Darcy– Brinkman equation

for the uid ow outside the double layer. Expressions of the start -up electrophoretic and

settling velocities of the particle in the Laplace transform are obtain ed explicitly. Com-

bined analytical-numerical technique solutions for the development of the electrophoretic

velocity with ? t/a2

are obtained for various values of ?, ?a, ?and? .Our results indicate

that the time scale for the starting electrophoresis and sedimenta tion decreases substan-

tially with an increase in ? ,a decrease in ?or an exceed in ?, and the electrophoretic

mobility increases with an increase in ?a.The eect of permeability of the charged porous

medium on the electrophoretic mobility and settling velocity yield intere sting and signif-

icant results. The normalized time-dependent electrophoretic velo city decreases with an

increase in ?if ? is almost small, but it may increase from a minimum with an increase

in ?if ? is relatively large. The eect of ?on the starting electrophoresis of the spherical

particles is substantially weaker than that on the starting sediment ation.

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19