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Proc. R. Soc. A (2009) 465, 501–522
doi:10.1098/rspa.2008.0354
Published online 28 October 2008
Steady electro-osmotic flow of a micropolar
fluid in a microchannel
B Y A BUZAR A. S IDDIQUI 1,2, *
AND
A KHLESH L AKHTAKIA 2
1
Department of Basic Sciences, Bahauddin Zakariya University,
Multan 60800, Pakistan
2
Department of Engineering Science and Mechanics, Pennsylvania State
University, University Park, PA 16802, USA
We have formulated and solved the boundary-value problem of steady, symmetric and onedimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel,
under the action of a uniform applied electric field. The Helmholtz–Smoluchowski equation
and velocity for micropolar fluids have also been formulated. Numerical solutions turn out to
be virtually identical to the analytic solutions obtained after using the Debye–Hückel
approximation, when the microchannel height exceeds the Debye length, provided that the
zeta potential is sufficiently small in magnitude. For a fixed Debye length, the mid-channel
fluid speed is linearly proportional to the microchannel height when the fluid is micropolar,
but not when the fluid is simple Newtonian. The stress and the microrotation are dominant at
and in the vicinity of the microchannel walls, regardless of the microchannel height. The midchannel couple stress decreases, but the couple stress at the walls intensifies, as the
microchannel height increases and the flow tends towards turbulence.
Keywords: couple stress; electro-osmosis; micropolar fluid; microrotation;
steady flow
1. Introduction
When an electrolytic liquid comes into contact with a solid, an electric double
layer is formed in the interfacial region. The electric double layer comprises a
layer of charges of one polarity on the solid side and a layer of charges of the
opposite polarity on the liquid side of the solid–liquid interface. At the same
time, free ions in the liquid assemble into a diffuse region beyond the interfacial
charge layer. Two electric double layers are formed when a liquid is contained in
a channel with two parallel solid walls. When an electric field is applied, the free
ions in the liquid experience a force that causes bulk motion of the liquid. This
type of flow is classified as electro-osmotic.
Electro-osmosis in microchannels has attracted great interest recently, owing
to pharmaceutical, chemical, environmental and defence applications, among
others. An industrial example is furnished by reverse-osmosis membranes for
desalination of water (Murugan et al. 2006). Two other industrial examples are
* Author and address for correspondence: Department of Basic Sciences, Bahauddin Zakariya
University, Multan 60800, Pakistan ([email protected]).
Received 2 September 2008
Accepted 30 September 2008
501
This journal is q 2008 The Royal Society
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502
A. A. Siddiqui and A. Lakhtakia
the injection of detoxifying agents and the control of leakage at toxic-waste sites
(Probstein 1989, p. 191; Keane 2003). Moreover, the study of electro-osmotic
flows is useful to: (i) design channels for fluid transport in biological and chemical
instruments (Fluri et al. 1996), (ii) design micropumps, microturbines and
micromachines (Harrison et al. 1991; Fan & Harrison 1994; DeCourtye et al.
1998), and (iii) design microchannels in micro/nanoscale chips for the analysis of
DNA sequences and for drug delivery (Arangoa et al. 1999).
Reuss introduced the concept of electro-osmosis in 1809, after performing an
experiment that proved that water can be forced to move through wet clayey soil
if an external electric field is applied (Probstein 1989, p. 191). About half a
century later, Wiedemann deduced from experiments that electro-osmotic flow is
proportional to the applied current. Helmholtz developed the electric-doublelayer theory in 1879 to analytically relate electrical and flow parameters. In 1903,
Von Smoluchowski measured the electric-double-layer thickness to be much
smaller than the height of the channel and derived a slip boundary condition at
the walls of the channel. Twenty years later, Debye and Hückel calculated the
distribution of ions in a low-ionic-energy solution by using the Boltzmann
distribution for the ionic energy (Burgreen & Nakache 1964).
Thereafter, although outstanding contributions have been made to the field of
electro-osmosis, theoretically (e.g. Burgreen & Nakache 1964; Hu et al. 1999;
Yang et al. 2001; Chai & Shi 2007) as well as experimentally (e.g. Herr et al.
2000; Pikal 2001; Horiuchi et al. 2007; Kim et al. 2008), attention has been chiefly
confined to simple Newtonian fluids. However, many technoscientifically
significant liquids are not simple Newtonian fluids; instead, they are generalized
Newtonian fluids called micropolar fluids (Ariman et al. 1973; Eringen 2001). Not
only does a micropolar fluid sustain body forces and the usual (Cauchy) stress
tensor as simple Newtonian fluids do, but it also sustains body couples and the
couple stress tensor; furthermore, the stress tensor is non-symmetric in a
micropolar fluid. The additional effects arise from the presence of microscopic
aciculate elements in a micropolar fluid—whereby micropolar-fluid motion has
six degrees of freedom, three more than of a simple Newtonian fluid—because the
length scale of motion is comparable to the length scale of the aciculate elements.
Micropolar fluids are known to occur in nature and are also of technoscientific
importance. Typically, a micropolar fluid is a suspension of rigid or semi-rigid
particles that cannot only translate but also rotate about axes passing through their
centroids. Blood has often been modelled as a micropolar fluid (Eringen 1973; Turk
et al. 1973; Misra & Ghosh 2001), because it contains platelets, cells and other
particles. Modelling granular flows as micropolar fluids, Hayakawa (2000) showed
that the analytical solutions of certain boundary-value problems are topologically
very similar to relevant experimental results. We can expect rigid-rod epoxies to be
micropolar fluids as well, because their aciculate molecules exhibit rotation about
their centroidal axes (Giamberini et al. 1997; Su et al. 2000). Liquid crystals and
colloidal suspensions are also cited as examples of micropolar fluids (Eringen 2001).
Motivated by this understanding, we decided to analyse the electro-osmotic flow
of a micropolar fluid in a rectangular microchannel. Our interest lies in the crosssectional distribution of fluid speed, stress tensor, microrotation, and couple stress
tensor in the microchannel, especially at locations close to the walls and midchannel. We are also interested in a comparison with the flow of a simple Newtonian
fluid in a microchannel in order to isolate the effects of micropolarity.
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The plan of this paper is as follows: the formulation of a relevant boundaryvalue problem is presented in §2, while §3 contains detail of analytical solution
obtained by using the Debye–Hückel approximation (Li 2004, p. 19). The
Debye–Hückel approximation is not always valid. Therefore, we also need a
numerical approach for the solution of the boundary-value problem, as described
in §4. Section 5 comprises analytical and numerical results obtained and
discussions thereon. The main conclusions are summarized in §6.
2. Basic analysis and formulation
Adopting the notation r 0 Z x 0x^0 C y 0 y^ 0 C z 0z^0 for the position vector, where
f^
x 0 ; y^ 0 ; z^0 g is the triad of Cartesian unit vectors, we are interested in examining
the steady flow of a micropolar fluid in the microchannel jy 0 j%h for x 0 [Kw,w],
when the length 2w is much greater than the height 2h of the microchannel and
there is no variation along the z 0 -axis. The walls y 0 ZGh are assumed to be
perfectly insulating and impermeable. Furthermore, we assume that: (i) neither a
pressure gradient nor a body couple is present, (ii) the effect of gravity is
unimportant, (iii) the flow is symmetric as both walls are identical, (iv) a
spatially uniform, electrostatic field is applied to the fluid, (v) the Joule heating
effects are small enough to be ignored, and (vi) the micropolar fluid is ionized,
incompressible and viscous.
Under these conditions, the three applicable equations of micropolar-fluid flow
are as follows (Eringen 2001):
V 0 $V 0 ðr 0 Þ Z 0;
ð2:1Þ
0
Kðm C cÞV 0 !½V 0 !V 0 ðr 0 Þ C cV 0 !v 0 ðr 0 Þ C re0 ðr 0 ÞE app
Z r½V 0 ðr 0 Þ$V 0 V 0 ðr 0 Þ
ð2:2Þ
and
ða C b C gÞV 0 ½V 0 $v 0 ðr 0 ÞKgV 0 !½V 0 !v 0 ðr 0 ÞK2cv 0 ðr 0 Þ C cV 0 !V 0 ðr 0 Þ
Z rjo ½V 0 ðr 0 Þ$V 0 v 0 ðr 0 Þ:
0
0
ð2:3Þ
0
E app
is
Here, V and v , respectively, are the fluid velocity and the microrotation;
the applied electric field that is spatially uniform within the microchannel; r and jo,
respectively, are the mass density and the microinertia; and a, b and g are the three
spin-gradient viscosity coefficients. The Newtonian shear viscosity coefficient and
the vortex viscosity coefficient, respectively, are denoted by m and c; these are
related by the inequality 2mCcR0, where cR0 (Eringen 2001, p. 14). Let us note
that v 0 , jo, c, a, b and g are null-valued in a simple Newtonian fluid.
In the absence of a significant convective or electrophoretic disturbance to the
electric double layers, the charge density re0 ðr 0 Þ is described by a Boltzmann
distribution, and takes the following form for a symmetric, dilute and univalent
electrolyte (Li 2004):
re0 ðr 0 Þ ZK2zo eno sinh½zo ej 0 ðr 0 Þ=kB T :
ð2:4Þ
0
Here, zo is the absolute value of the ionic valence; j is the electric potential; e is
the charge of an electron; no is the number density of ions in the fluid far away
from any charged surface; kB is the Boltzmann constant; and T is the
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A. A. Siddiqui and A. Lakhtakia
temperature. With e denoting the static permittivity of the fluid, the charge
density and the electric potential are also related by the Gauss law
V 0 2 j 0 ðr 0 Þ ZKre0 ðr 0 Þ=e;
ð2:5Þ
thus,
ð2:6Þ
V 0 2 j 0 ðr 0 Þ Z ð2zo eno =eÞsinh½zo ej 0 ðr 0 Þ=kB T:
K1
1=2
The Debye length lD Z ðzo eÞ ðekB T=2no Þ is assumed in this paper to be much
smaller than h, i.e. lD/h.
Before proceeding, let us define the non-dimensionalized quantities
9
r Z r 0 =h; V Z V 0 =U ; v Z ðh=U Þv 0 ; j Z j 0 =jo ;
>
>
>
=
ð2:7Þ
h
0 >
0
spq
and spq Z
:>
re Z ðh2 =3jo Þre0 ; mpq Z ðh 2 =gU Þmpq
>
;
ðm C cÞU
0
0
and spq
are, respectively, the components of the couple stress and the
Here, mpq
stress tensors, where p,q2{1,2,3};
ej E
U ZK o o
m Cc
ð2:8Þ
0
is the characteristic speed to be identified in §3b; Eo Z jE app
jR 0; and jo is called
the zeta potential (Li 2004) that is assumed to be temporally constant and
spatially uniform on the walls yZG1. With these quantities, equations
(2.1)–(2.3) and (2.6), respectively, simplify to
V$V ðrÞ Z 0;
ð2:9Þ
0
=Eo Z Re½V ðrÞ$VV ðrÞ;
KV !½V !V ðrÞ C k1 V !vðrÞ C re ðrÞE app
ð2:10Þ
KV !½V !vðrÞK2k 2 vðrÞ C k 3 V½V$vðrÞ C k 2 V !V ðrÞ Z Ro½V ðrÞ$VvðrÞ
ð2:11Þ
and
V2 jðrÞ Z ðm 2o =ao Þsinh½ao jðrÞ;
where
c
k1 Z
;
m Cc
Re Z
rUh
;
m Cc
Ro Z
ch2
k2 Z
;
g
rjo Uh
;
g
9
>
>
>
>
=
:
z ej >
>
and ao Z o o : >
>
kB T ;
ð2:12Þ
a Cb Cg
;
k3 Z
g
mo Z
h
lD
ð2:13Þ
Here, k1 couples the two viscosity coefficients; k 2 and k 3 are normalized
micropolar parameters; Re may be called the Reynolds number; Ro may be
called the microrotation Reynolds number (Eringen 2001); and ao is the ionicenergy parameter (Li 2004). The Gauss law (2.5) can now be written as
V2 jðrÞ ZKre ðrÞ:
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ð2:14Þ
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Steady electro-osmotic flow
505
We have already assumed that v/vzh0; now, we ignore the variations along
the x 0 -axis and set v/vxh0 consistently with the assumption that w[h.
Furthermore, the one-dimensional flow is supposed to be laminar and symmetric
with respect to the y 0 -axis. Let us therefore designate
u Z x^0 $V
and N Z z^0 $v
ð2:15Þ
for further analysis of steady flow in the microchannel. The applied electric field
0
is oriented parallel to the x 0 -axis, i.e. E app
Z x^ 0 Eo . In light of the aforementioned
assumptions, equations (2.9)–(2.11) yield the following two ordinary differential
equations:
d2
d
d2
uðyÞ C k1
ð2:16Þ
N ðyÞ C 2 jðyÞ Z 0
2
dy
dy
dy
and
d2
d
N ðyÞK2k 2 N ðyÞKk 2
uðyÞ Z 0:
2
dy
dy
ð2:17Þ
As the flow is symmetric, i.e. u(y)Zu(Ky) and j(y)Zj(Ky), the restrictions
d
d
ð2:18Þ
uðyÞ
jðyÞ
Z 0 and
Z0
dy
dy
yZ0
yZ0
must hold. The boundary conditions on u(y) and j(y) are
uðG1Þ Z 0
and jðG1Þ Z 1:
ð2:19Þ
In addition, the condition
jð0Þ Z 0
ð2:20Þ
is engendered by the assumption h[lD (Probstein 1989, p. 187).
Another boundary condition is evident in the literature: vCboV!VZ0 at the
walls, where bo2[K1,0] is some constant. This boundary condition simplifies in
the present case to
d
ð2:21Þ
N ðG1ÞKbo uðyÞ
Z 0:
dy
yZG1
There are two major schools of thought on the boundary condition (2.21). One
school ignores microrotation effects near solid walls and sets boZ0 (Papautsky
et al. 1999; Eringen 2001). But the second school holds that bo!0 because the
shear and couple stresses at the walls must be high in magnitude in comparison
with locations elsewhere, as can be reasoned from the existence of boundary
layers (Ramachandran et al. 1979; Chiu & Chou 1993; Rees & Bassom 1996;
Hegab & Liu 2004). This argument is held valid when thermal transfer and
magnetic effects are to be accounted for (Hegab & Liu 2004). In the present case,
electric double layers are present; hence, bo!0 may be reasonable if electroosmosis occurs. Analytical and numerical results are provided in this paper for
zero, as well as non-zero, bo. In addition,
N ð0Þ Z 0:
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ð2:22Þ
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A. A. Siddiqui and A. Lakhtakia
3. Solution of boundary-value problem
(a ) Analytical solution of the Poisson–Boltzmann equation
Under the assumptions made in §2, (2.12) reduces to
d2
mo2
jðyÞ
Z
sinh½ao jðyÞ:
ao
dy2
ð3:1Þ
The solution of equation (3.1) satisfying boundary conditions (2.18)2, (2.19)2 and
(2.20) is (Dutta & Beskok 2001)
a n
o
4
jðyÞ Z tanhK1 tanh o exp½m o ðjyjK1Þ ; y 2½K1; 1:
ð3:2Þ
ao
4
An equivalent form derived by us is
2
K1 1 C Uo exp½2m o ðjyjK1Þ
jðyÞ Z cosh
;
ao
1KUo exp½2m o ðjyjK1Þ
where
y 2½K1; 1;
coshðao =2ÞK1 :
Uo Z coshðao =2Þ C 1 ð3:3Þ
ð3:4Þ
(b ) Micropolar Helmholtz–Smoluchowski equation and velocity
As the counterparts of the Helmholtz–Smoluchowski equation and the
Helmholtz–Smoluchowski velocity for simple Newtonian fluids (Probstein 1989,
p. 192) are not available for steady flows of micropolar fluids, let us derive both in
this section. We begin by assuming that the gradient of microrotation is much
less than the Laplacian of the velocity, i.e.
2
d
d
k1
ð3:5Þ
dy N ðyÞ / dy2 uðyÞ:
Making use of equation (2.14), we see that equation (2.10) reduces to
(d2/dy2)[u(y)Cj(y)]Z0, which is merely a relation between viscosity and
electromagnetism. After integrating both sides of this equation with respect to
y and using the conditions (2.18), we get (d/dy)[u(y)Cj(y)]Z0. On integrating
again with respect to y and using the boundary conditions (2.19), we obtain
u(y)ZKj(y)C1, whence
u 0 ðy 0 Þ Z
eEo j 0 ðy 0 Þ
CU
m Cc
ð3:6Þ
follows. Thus, U can be called the micropolar Helmholtz–Smoluchowski velocity,
while equation (3.6) is the micropolar Helmholtz–Smoluchowski equation. Setting
cZ0 in equations (2.8) and (3.6), we revert to the Helmholtz–Smoluchowski
velocity and equation, respectively, for simple Newtonian fluids (Probstein 1989,
p. 192).
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Steady electro-osmotic flow
(c ) Analytical solutions for the Debye–Hückel approximation
Let us now derive an analytical solution of the boundary-value problem for
steady flow, subject to the Debye–Hückel approximation (Li 2004), as follows. If
the electric potential energy is small compared with the thermal energy of the
ions, i.e. jzoejojj/jkBT j, then jaojj/1; accordingly,
ð3:7Þ
sinhðao jÞ zao j;
which is called the Debye–Hückel approximation. Making this approximation on
the right-hand side of equation (3.1), we get
d2
jðyÞ zmo2 jðyÞ;
dy 2
ð3:8Þ
whose solution is given by
ð3:9Þ
jðyÞ zcoshðm o yÞ=coshðm o Þ:
The Debye–Hückel solution (3.9) satisfies the symmetry condition jðyÞZ jðKyÞ;
as well as the boundary conditions (2.18)2 and (2.19)2.
Let us integrate both sides of equation (2.16) with respect to y and use the
boundary conditions (2.18) and (2.22) to obtain
d
d
uðyÞ ZKk1 N ðyÞK jðyÞ:
ð3:10Þ
dy
dy
Next, on eliminating du/dy from equation (2.17), we get
d2
d
N ðyÞKð2Kk1 Þk 2 N ðyÞ C k 2
jðyÞ Z 0:
ð3:11Þ
dy
dy2
Using the Debye–Hückel solution (3.9) in equation (3.11), we reduce it to
d2
N ðyÞKð2Kk1 Þk 2 N ðyÞKk4 sinhðm o yÞ Z 0;
ð3:12Þ
dy 2
where
k4 ZKk 2 m o =coshðm o Þ:
ð3:13Þ
As equation (3.12) is a linear, second-order, non-homogeneous, ordinary
differential equation, its solution is found as (Kreyszig 2006)
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i
k4
ðN Þ
sinhðm o yÞ C C1 exp ð2Kk1 Þk 2 y
N ðyÞ Z 2
mo Kk 2 ð2Kk1 Þ
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i
ðN Þ
C C2 exp K ð2Kk1 Þk 2 y ;
ð3:14Þ
ðN Þ
ðN Þ
where C1 and C2 are constants. The boundary condition (2.22) requires that
ðN Þ
ðN Þ
C 1 C C 2 Z 0. Furthermore, after using equations (3.10) and (3.14) in the
boundary condition (2.21), the solution of equation (3.12) is obtained as
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i
ð3:15Þ
N ðyÞ Z k5 sinh ð2Kk1 Þk 2 y C k 6 sinhðm o yÞ;
where
k5 Z ½Nw Kk 6 sinhðm o Þ=sinh½
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Kk1 Þk 2 and
k 6 Z k4 = m 2o Kð2Kk1 Þk 2 ;
Proc. R. Soc. A (2009)
Nw ZKbo m o tanhðm o Þ=½1 C k1 bo 9
>
>
=
:
>
>
;
ð3:16Þ
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508
A. A. Siddiqui and A. Lakhtakia
Finally, on using equation (3.15) in equation (3.10), integrating with respect to
y, and using the boundary condition (2.19)1, we obtain the fluid speed as
coshðm o yÞ k1 k 6
C
½coshðmo ÞKcoshðm o yÞ
coshðm o Þ
mo
h
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i
k1 k5
C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
coshð ð2Kk1 Þk 2 ÞKcoshð ð2Kk1 Þk 2 yÞ :
ð2Kk1 Þk 2
uðyÞ Z 1K
ð3:17Þ
Thus, equations (3.15) and (3.17) constitute the solution of the boundaryvalue problem for steady flow of a micropolar fluid when the Debye–Hückel
approximation (3.7) holds, whether boZ0 or bos0. On setting cZ0, we get
k1Zk2Zk4Z0, which implies from equation (3.15) that N(y)h0; simultaneously,
from equation (3.17), we recover
uðyÞ Z 1Kcoshðm o yÞ=coshðm o Þ
ð3:18Þ
for a simple Newtonian fluid (Probstein 1989).
4. Numerical approach
The analytical solution for steady flow of a micropolar fluid in a rectangular
microchannel with impermeable walls presented in §3c is premised on the
Debye–Hückel approximation. An analytical solution appears impossible to
find, if the solution (3.3) of the Poisson–Boltzmann equation has to be used
instead of the Debye–Hückel solution (3.9); therefore, we resorted to the
following numerical approach. Furthermore, in the absence of experimental data
on micropolar fluids, the numerical approach provides a check against analytical
results obtained using the Debye–Hückel approximation.
The interval y2[K1,1] was divided into 2I segments of size DyZ1/I. We
used the fourth-order finite-difference method (Mancera & Hunt 1997),
followed by the central difference method (in which the y-derivative is
approximated by the central difference formula). Accordingly, the differential
equations (2.16) and (2.17) transform into difference equations. The difference
equations were solved iteratively by the successive overrelaxation (SOR)
method (Hoffman 1992, p. 56). For interpolation, the Richardson method
was used (Hoffman 1992, p. 195). The iterative procedure was repeated
until convergence was obtained according to the following criterions (Hoffman
1992, p. 425):
i 1K ðmC1Þ ! 10K5
u
max i
i 1K ðmC1Þ ! 10K5 :
N
ðmÞ
ui
and
max ðmÞ
Ni
ð4:1Þ
i
Here, uihu(iDy) and NihN(iDy), i2[KI,I ], and the superscript mR1
represents the iteration number. Different values of I were used until
convergent solutions were obtained. Solutions obtained in §3c after using
the Debye–Hückel approximation (3.7) were compared against the numerical
results obtained.
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Steady electro-osmotic flow
509
5. Results and discussion
All calculations were made with the following material properties fixed: noZ
6.02!1022 mK3; zoZ1; eZ10eo; eoZ8.854!10K12 F mK1; mZ3!10K2 Pa s; and
gZ10K4 kg m sK1. The temperature was fixed at TZ290 K. Since the Boltzmann
constant kBZ1.38!10K23 J KK1 and the electron charge eZ1.6!10K19 C, the
Debye length lDZ10.72 nm. Consistent with the assumption that h[lD, we chose
h2[107.2,5360] nm so that mo2[10,500]. Most results are presented for j0zK25!
10K3 V, which is about the upper limit for the Debye–Hückel approximation to be
valid at approximately room temperature (Hunter 1988, p. 25; Li 2004, p. 19), but
some results are also presented for higher magnitudes of jo in order to transcend the
limitations of the Debye–Hückel approximation. The parameter k12[0,0.95] was
kept as a variable, after noting that k1/1 as c/N. The magnitude of the applied
electric field was fixed at EoZ104 V mK1, which is a reasonable practical value.
The numerical approach described in §4 was implemented with computations
made for IZ250, 500 and 1000. The dependences of the relevant components of
the fluid velocity, microrotation, stress tensor and couple stress tensor on mo, k1
and bo were investigated for steady flow.
In addition to the speed u 0 (y), the only non-zero components of the stress
tensor and the couple stress tensor for the steady flow of an incompressible
micropolar fluid in the absence of pressure are (Eringen 2001)
9
d
d
0
0
>
ðy 0 Þ Z m 0 u 0 ðy 0 ÞKcN 0 ðy 0 Þ; s21
ðy 0 Þ Z ðm C cÞ 0 u 0 ðy 0 Þ C cN 0 ðy 0 Þ >
s12
>
>
dy
dy
>
=
and
>
>
>
d
>
0
0
0 0
>
;
m 23 ðy Þ Z g 0 N ðy Þ;
dy
ð5:1Þ
where N 0 Z z^0 $v 0 . On normalizing the foregoing quantities, equations (5.1) take
the following form:
9
d
d
>
s12 ðyÞ Z ð1Kk1 Þ
s21 ðyÞ Z
uðyÞKk1 N ðyÞ;
uðyÞ C k1 N ðyÞ >
>
>
dy
dy
>
=
and
ð5:2Þ
>
>
>
d
>
>
;
N ðyÞ:
m 23 ðyÞ Z
dy
Whereas s12Zs21 for a simple Newtonian fluid, let us note that the equality does
not hold for a micropolar fluid.
As mentioned in §2, the choice of bo affects fluid flow. Graphs from the
Debye–Hückel approximation and from the numerical approach show that u 0 (y)
does not depend on k1 when boZ0, and is the same as for a simple Newtonian
fluid. If jboj is increased, the effects of micropolarity grow and become dominant.
Therefore, all the numerical results presented in this section are for bo!0.
(a ) Electric potential
Let us begin with the electric potential j 0 (y) when joZK25!10K3 V.
Figure 1 contrasts the solution (3.3) of the Poisson–Boltzmann equation with the
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510
A. A. Siddiqui and A. Lakhtakia
(a) 0.025
(b) 9
8
7
6
5
4
3
2
1
0
0.020
0.015
0.010
0.005
0
0
± 0.2
± 0.4
± 0.6
± 0.8
±1.0
0
±0.05
±0.10
±0.15
±0.20
Figure 1. (a) Profiles of j 0 (y) with y, as predicted independently by the analytical solution (3.3)
and the Debye–Hückel solution (3.9), for joZK25!10K3 V. Values from (3.3) are denoted by
down-triangles (moZ10), diamonds (moZ50), crosses (moZ100) and circles (moZ500).
Values from (3.9) are indicated by the solid curve (moZ10), the dotted curve (moZ50), the
dashed curve (moZ100) and the dashed dotted curves (moZ500). (b) Magnified view of (a) in the
mid-channel region.
(b) 0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
(a) 0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
± 0.2
± 0.4
± 0.6
± 0.8
±1.0
0
±0.2
±0.4
±0.6
±0.8
±1.0
Figure 2. Same as figure 1a, except that: (a) joZK100!10K3 V and (b) joZK500!10K3 V.
Debye–Hückel solution (3.9). Both solutions are virtually identical for all y and
mo, except at and in the vicinity of yZ0. Whereas (3.3) correctly yields j(0)Z0
in conformity with (2.20), the Debye–Hückel solution yields j(0)Zsech(mo). The
latter agrees with the former increasingly better as mo/N. Moreover, if jjoj
increases, then the difference between the solutions (3.3) and (3.9) increases, as
shown in figure 2 for joZK100!10K3 V and K500!10K3 V. Furthermore,
these two figures show that j 0 (y) for all ys0 increases with jjoj.
(b ) Fluid speed
Next, we examined the fluid speed u 0 (y) for different values of mo, k1 and bo,
using the numerical approach as well as the Debye–Hückel approximation. Some
representative results are provided in figure 3 for j oZK25!10 K3 V,
mo2{50,500}, k12{0,0.5,0.95} and bo2{K0.1,K0.1,K0.5,K1}. The numerical
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Steady electro-osmotic flow
(b) 3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
(c) 3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
(d) 1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
(e)
(f)
5
(a) 3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
6
6
5
4
4
3
3
2
2
1
0
1
0
(g) 3.5
3.0
2.5
2.0
1.5
1.0
0.5
(h) 1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
4
5
5
0
511
± 0.2 ± 0.4 ± 0.6 ± 0.8 ±1.0
0
±0.2 ±0.4 ±0.6 ±0.8 ±1.0
Figure 3. Profiles of u 0 (y) with y, as predicted by the Debye–Hückel solution (3.17), as well as calculated
by the numerical approach, for joZK25!10K3 V and (a–d ) moZ50 or (e–h) moZ500. Values from the
numerical approach are denoted by down-triangles (k1Z0), diamonds (k1Z0.5) and circles (k1Z0.95).
The Debye–Hückel solutions are indicated by the solid curves (k1Z0), the dotted curves (k1Z0.5) and
the dashed curves (k1Z0.95). (a,e) boZK0.01, (b,f ) boZK0.1, (c,g) boZK0.5 and (d,h) boZK1.
and the Debye–Hückel solutions in both figures are virtually identical, the
greatest difference being less than 2.6858 per cent when yZ0, moZ500, k1Z0.95
and boZK0.01; moreover, this difference decreases if jboj increases. Near the
walls (yZG1), the two solutions are indistinguishable from each other.
Figure 3 clearly shows that u 0 (y) depends on k1, bo and mo. The dependence on k1 is
almost absent in the vicinity of the walls because the boundary condition (2.19)1 was
enforced, but that dependence intensifies significantly at locations away from the
walls. Furthermore, u 0 (y) increases with jboj for all mo[1 and/or for all k1O0. As jboj
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512
A. A. Siddiqui and A. Lakhtakia
(a) 25
(b) 250
20
200
15
150
10
100
5
50
0
–1.0
– 0.8
– 0.6
– 0.4
– 0.2
0
0
–1.0
o
–0.8
–0.6
–0.4
–0.2
0
o
Figure 4. Variations of the ratio u 0 ð0Þ=u N0 ð0Þ with bo predicted by the Debye–Hückel solution
(3.17) for k1Z0 (solid curves), k1Z0.5 (dotted curves) and k1Z0.95 (dashed curves), when
joZK25!10K3 V. The subscript N in uN indicates that the quantity holds for a simple Newtonian
fluid (i.e. k1Z0). (a) moZ50 and (b) moZ500.
increases, the speed in an increasingly larger central portion of the microchannel
exceeds the speed of a simple Newtonian fluid (k1Z0), and the proclivity of the speed
to exceed the speed of the simple Newtonian fluid decreases as k1 increases.
The fluid speed increases in the vicinity of the walls as mo increases, whether
the fluid is simple Newtonian or micropolar. Mid-channel, the fluid speed
increases with mo for all k1 and bo, but it decreases as k1 increases for all mo and
low values of jboj, as shown in figure 4. Incidentally, u 0 (0) is identical for all
values of k1O0 and mo[1 when boZ0.
In order to further assess the influence of mo on u 0 (y), we examined the
derivative du 0 (0)/dmo when the Debye–Hückel approximation is valid. Equation
(3.17) yields
U K1
du 0 ð0Þ
Z ½1 C c1 k1 sechðm o Þtanhðm o Þ C c 2 c3 tanhðm o Þ
dm o
!½cschðc 2 m o Þcothðc 2 m o ÞKcsch2 ðc 2 m o Þ C c3 sech2 ðm o Þ
!½cothðc 2 m o ÞKcschðc 2 m o Þ;
ð5:3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where c1 Z c5 =ðc 22 K1Þ; c 2 Z ð2Kk1 Þc5 ; c3 Z k1 c4 =c 2 ; c4 ZK½c1 C bo =ð1C k1 bo Þ;
and c5 Z l2D c=g. Graphical analysis of this equation indicates that du 0 ð0Þ=dm o is
independent of mo for any specific k12(0,1); hence, u 0 ð0Þ wm o ; k1 2ð0; 1Þ, which
is in agreement with figure 3. By contrast, du 0 ð0Þ=dm o Z 0 for all mo[1 when
k1Z0, which means that u 0 (0) is independent of mo for a simple Newtonian
fluid. Furthermore, for fixed k1, du 0 ð0Þ=dm o has a maximum with respect to
bo2[K1,0], which is in conformity with figure 3. The value of bo for maximum
du 0 ð0Þ=dm o lies in the neighbourhood of K0.945 when joZK25!10K3 V.
Finally, using (2.8) and (5.3), we also conclude that du 0 ð0Þ=dm o wjo , because
u(y) is independent of jo according to (3.17).
For higher values of jjoj, the Debye–Hückel solution (3.17) deviates
considerably from the numerical solution, as shown in figure 5. This deviation
intensifies with increasing jjoj for all k12[0,1). Furthermore, u 0 (y) c y2[K1,1]
increases with jjoj for all k12[0,1).
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513
Steady electro-osmotic flow
(a)
(b)
2.0
1.5
4
5
2.0
1.0
1.0
0.5
0
± 0.2
± 0.4
± 0.6
± 0.8
±1.0
0
±0.2
±0.4
±0.6
±0.8
±1.0
fluid speed (×10 – 6 m s–1)
fluid speed (×10 –7m s–1)
Figure 5. Profiles of u 0 (y) with y, as predicted by the Debye–Hückel solution (3.17), as well as
calculated by the numerical approach, for moZ50 and boZK0.5. Values from the numerical
approach are denoted by down-triangles (k1Z0), diamonds (k1Z0.5) and circles (k1Z0.95). The
Debye–Hückel solutions are indicated by the solid curves (k1Z0), the dotted curves (k1Z0.5) and
the dashed curves (k1Z0.95). (a) joZK100!10K3 V and (b) j0ZK500!10K3 V.
(a)
8
7
6
5
4
3
2
1
0
–1
(b)
(c)
(d)
(e)
(f)
14
12
10
8
6
4
2
0
±0.992
±0.996
±1.000
±0.992
±0.996
±1.000
±0.992
±0.996
±1.000
Figure 6. Profiles of u 0 (y) (solid curves) obtained by the numerical approach and U (dashed curves)
predicted by (2.8) with y in the vicinity of the walls, when moZ50 and boZK0.5. (a,d ) k1Z0,
(b,e) k1Z0.5 and (c,f ) k1Z0.95. (a–c) j0ZK25!10K3 V and (d–f ) joZK500!10K3 V.
Next, figure 6 shows a comparison of the fluid speed u 0 (y) and the micropolar
Helmholtz–Smoluchowski velocity U, near the walls, for three different values of
k1 and two different values of jo. Near the walls (regions that include the electric
double layers), we see that u 0 ðyÞ/ U as k1/1, for all jo. Furthermore, the
difference ju 0 ðyÞKU j decreases as y/0.
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514
A. A. Siddiqui and A. Lakhtakia
(a)
(c)
(e)
(g)
(b) 5
4
3
2
1
0
–1
–2
–3
–4
–5
0.5
0.4
0.3
0.2
0.1
0
– 0.1
– 0.2
– 0.3
– 0.4
– 0.5
(d )
60
60
40
40
20
20
0
0
–20
–20
– 40
– 40
– 60
– 60
(f)
0.5
0.4
0.3
0.2
0.1
0
– 0.1
– 0.2
– 0.3
– 0.4
– 0.5
5
4
3
2
1
0
–1
–2
–3
–4
–5
60
(h) 60
40
40
20
20
0
0
–20
–20
– 40
– 40
– 60
–1.0
– 0.5
0
0.5
1.0
– 60
–1.0
–0.5
0
0.5
1.0
Figure 7. Profiles of N 0 (y) with y predicted by the Debye–Hückel solution (3.15), when
joZK25!10K3 V and (a–d) moZ50 or (e–h) moZ500, for k1Z0.25 (solid curves), k1Z0.5 (dotted
curves) and k1Z0.95 (dashed curves). (a,e) boZK0.01, (b,f ) boZK0.1, (c,g) boZK0.99 and
(d,h) boZK1.
(c ) Microrotation
When the Debye–Hückel approximation holds, the microrotation is given by
equation (3.15). According to this expression, N 0 (y) is an odd function of y, in
contrast to u 0 (y) and j 0 (y). In a simple Newtonian fluid k1Z0 and N 0 (y)h0.
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515
Steady electro-osmotic flow
(a) 200
150
100
50
0
–50
–100
–150
–200
–1.0
(b)
2000
1500
1000
500
0
–500
–1000
–1500
–2000
– 0.5
0
0.5
1.0
–1.0
–0.5
0
0.5
1.0
Figure 8. Profiles of N 0 (y) with y obtained by the numerical approach, when moZ50 and boZK0.5,
for k1Z0.25 (solid curves), k1Z0.5 (dotted curves) and k1Z0.95 (dashed curves). (a) joZK100!
10K3 V and (b) joZK500!10K3 V.
Plots of N 0 (y) versus y when joZK25!10K3 V, for different values of mo, k1
and bo are presented in figure 7. This figure shows that: (i) jN 0 ðyÞj increases as
jboj increases for all mo[1 and k1O0 and (ii) jN 0 ðyÞj decreases as k1 increases
for all bo2(K1,0) and bo2[10,500]. Unlike u 0 (y), jN 0 ðyÞj increases with jyj for
all mo, k1 and bo. Furthermore, the highest value of jN 0 ðyÞj occurs at the walls
(yZG1) for all mo, k1 and bo, such that
N 0 ðG1Þ zH
U bo
;
ð1 C k1 bo ÞlD
ð5:4Þ
because tanh(mo)z1 for all mo2[10,500]. This equation indicates that the
microrotation at the walls is independent of mo. Moreover, for boZK1, which is
representative of turbulent boundary layers (Rees & Bassom 1996), we get
N 0 ðG1Þ zHejo Eo =mlD , which is independent of both mo and k1, in agreement
with figure 7d,h. Consequently, the microscopic aciculate elements near the walls
have a greater proclivity to rotate about their centroids, regardless of the height
h of the microchannel and the coupling parameter k1.
In §5b, we observed significant differences in the predictions of the fluid speed
by the numerical approach and the Debye–Hückel approximation to analyse the
effect of higher values of jjoj on N 0 ðyÞ. Figure 8 indicates that, just like j 0 ðyÞ and
u 0 ðyÞ, jN 0 ðyÞj increases as jjoj increases for all k12[0,1). Furthermore, whether
jjoj is low or high, the highest value of jN 0 ðyÞj occurs at the walls for all k1.
(d ) Stress tensor
0
0
Analogous to N 0 ðyÞ, the components s12
ðyÞ and s21
ðyÞ of the stress tensor are
also odd functions of y, by virtue of equations (3.17), (3.15) and (5.2).
0
The profiles of s12
ðyÞ with y, at and in the vicinity of the wall yZK1, are
presented in figure 9, when joZK25!10K3 V, for different values of mo, k1 and
0
bo. This figure shows that js12
ðyÞj decreases as k1 increases for all mo2[10,500],
0
k12(0, 1) and bo2(K0.5, 0); however, js12
ðyÞj increases with k1 when
0
bo2[K1,K0.5]. Furthermore, js12 ðyÞj increases with jboj for all mo and k1. The
0
maximum value of js12
ðyÞj in the boundary layer always occurs at the walls for
all mo, k1 and bo. These observations can be confirmed as follows.
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A. A. Siddiqui and A. Lakhtakia
12
(N m–2)
(a) 2.0
(b)
(c)
(e)
(f)
(h)
(i)
1.5
1.0
0.5
0
12
(N m–2)
(d) 2.0
1.5
1.0
0.5
0
12
(N m–2)
(g)
45
40
35
30
25
20
15
10
5
0
–1.00
– 0.96
– 0.94
–1.00
– 0.96
–0.94
–1.00
–0.96
–0.94
0
Figure 9. Profiles of s12
ðyÞ with y predicted after the substitution of the Debye–Hückel solution
(3.15) and (3.17) in (5.1)1, at and in the vicinity of the wall yZK1, when joZK25!10K3 V, for
k1Z0 (solid curves), k1Z0.5 (dotted curves) and k1Z0.95 (dashed curves). (a,d,g) moZ10,
(b,e,h) moZ50 and (c,f,i) moZ500. In addition, (a–c) boZK0.1, (d–f ) boZK0.5 and (g–i) boZK1.
0
0
Note that s12
ðKyÞZKs12
ðyÞ.
Using equations (2.21) and (5.2)1, we obtain
0
s12
ðG1Þ
½1Kk1 ð1 C bo Þðm C cÞ d 0 Z
;
u ðyÞ
m o lD
dy
yZG1
ð5:5Þ
0
ðG1Þ is directly proportional to the velocity gradient at the
which shows that s12
wall. As the magnitude of the velocity gradient must be large near the walls
owing to the no-slip boundary condition (2.19)1 (Currie 1974, p. 276), and
because figure 3 indicates that the velocity gradient does have maximum
0
magnitude at the walls, it is not surprising that the maximum value of js12
ðyÞj
occurs at the walls yZG1.
Next, on using equations (3.9), (3.10), (5.4) and (5.5), with the argument that
tanhðm o Þ z1 for all mo2[10,500], we get
0
ðG1Þ zG
s12
Proc. R. Soc. A (2009)
½1Kk1 ð1 C bo Þejo Eo
;
½1 C k1 bo lD
ð5:6Þ
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517
Steady electro-osmotic flow
12
(N m–2)
(a) 15
(b) 150
10
100
5
50
0
–1.0 – 0.8
– 0.6
– 0.4
– 0.2
0
0
–1.0 –0.8
–0.6
–0.4
–0.2
0
0
Figure 10. Profiles of s12
ðyÞ with y obtained by the numerical approach, when moZ50 and
boZK0.5, for k1Z0.25 (solid curves), k1Z0.5 (dotted curves) and k1Z0.95 (dashed curves).
0
0
(a) joZK100!10K3 V and (b) joZK500!10K3 V. Note that s12
ðKyÞZKs12
ðyÞ.
0
accordingly, s12
ðG1Þ is independent of mo for all k1 and bo, which is in agreement
with figure 9. In addition, for simple Newtonian fluids (k1Z0), equation (5.6) yields
0
s12
ðG1Þjk1Z0 zG
ejo Eo
;
lD
ð5:7Þ
which also agrees with figure 9.
0
ðG1Þ zGejo Eo =lD ,
Furthermore, if we set boZK1/2 in equation (5.6), we get s12
0
which means that s12 ðG1Þ is independent of k1, in conformity with figure 9d–f. This
0
explains the divergent dependences of s12
ðyÞ on k1 in the regimes bo2(K0.5, 0) and
0
bo2[K1,K0.5). In addition, when boZK1, equation (5.6) reduces to s12
ðG1Þ zG
0
ejo Eo =ð1Kk1 ÞlD , which shows that s12 ðG1Þ is inversely proportional to (1Kk1) for
boZK1, in agreement with figure 9g–i.
0
0
In contrast to s12
ðG1Þ, s12
ð0Þ h 0, which can be proved by using the boundary
0
conditions (2.18)1 and (2.22) in equation (5.1)1. Accordingly, s12
ðyÞ in the
micropolar fluid is significant only at locations close to the walls of the
microchannel, just as in a simple Newtonian fluid.
For higher values of jjoj (i.e. jjojO25!10K3 V), the maximum value of
0
js12 ðyÞj occurs at the walls yG1, similar to the case when jjojZ25!10K3 V, for
0
all k12[0,1), as shown in figure 10. This figure also indicates that js12
ðG1Þj
increases as jjoj increases for all k1.
For the other non-null component of the stress tensor, on using equations (2.7)
and (3.10) in equation (5.2)2, we obtain
ej E d
0
s21
ðyÞ Z o o
jðyÞ:
ð5:8Þ
h dy
0
ðyÞ does not depend on k1 and bo, and is the same as that in a simple
Hence, s21
Newtonian fluid, regardless of jo. However, the graphs in figure 11, plotted for
0
joZK25!10K3 V, show that s21
ðyÞ does depend on mo. This figure also shows
0
that js21 ðyÞj is maximum at the walls yG1 and rapidly drops to a null value as
y/0 in accordance with equation (2.18)2.
On using equations (2.8) and (3.9) in equation (5.8), consistently with the
0
argument that tanhðm o Þ z1 for all m o 2½10; 500, we obtain s21
ðG1Þ zG
ejo Eo =lD , which is independent of mo, k1 and bo. This expression agrees with
figure 11 and it also coincides with equation (5.7).
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518
A. A. Siddiqui and A. Lakhtakia
(b) 2.5
1.6
2.0
1.2
1.5
0.8
1.0
0.4
0.5
21
(N m–2)
(a) 2.0
0
–1.0
– 0.8
– 0.6
– 0.4
– 0.2
0
–1.00
0
–0.98 –0.96 –0.94 –0.92 –0.90
0
Figure 11. (a) Profiles of s21
ðyÞ with y predicted after the substitution of the Debye–Hückel
solution (3.9) in (5.8), when joZK25!10K3 V, for moZ10 (solid curves), moZ50 (dotted curves),
moZ100 (dashed curves) and moZ500 (dashed-dotted curves). (b) Magnified view of (a) near the
0
0
wall yZK1. Note that s21
ðKyÞZKs21
ðyÞ.
21
(N m–2)
(a) 8
7
6
5
4
3
2
1
0
–1.0
(b)
35
30
25
20
– 0.8
– 0.6
– 0.4
– 0.2
0
15
10
5
0
–1.0
–0.8
–0.6
–0.4
–0.2
0
Figure 12. Same as figure 11a, except that (a) joZK100!10K3 V and (b) joZK500!10K3 V.
0
Figure 12 shows the profiles of s21
ðyÞ with y for jjo jO 25 !10K3 V. This figure
0
indicates that js21 ðyÞj increases with jjo j for all mo. In addition, the maximum
0
value of js21
ðyÞj occurs at the walls. These characteristics hold whether or not the
Debye–Hückel approximation is valid.
(e ) Couple stress tensor
A high magnitude of a component of the couple stress tensor indicates
a greater tendency of the microscopic aciculate elements in a micropolar
fluid to rotate about their respective centroids. Under the validity of the
0
ðyÞ
Debye–Hückel approximation, we find that the sole non-null component m 23
of the couple stress tensor is an even function of y, by virtue of equations (3.15)
and (5.2)3. This conclusion turned out to be valid even when the Debye–Hückel
approximation is not.
Unlike the stress tensor, the couple stress tensor is not null-valued in the
middle of the microchannel for micropolar-fluid flow. Consistent with the
0
ð0Þ can be obtained after
argument that tanhðm o Þ z1 for all mo2[10,500], m 23
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519
6
4
2
0
–2
–4
–6
–8
–10
(c) 9
8
7
6
5
4
3
2
1
0
8
7
6
5
4
3
2
1
0
(f) 9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
(i) 9
8
7
6
5
4
3
2
1
M23 )(× 10 –10)
(h)
0
M23 )(×10 –10)
(g) 3.5
3.0
2.5
2.0
1.5
1.0
0.5
M23 )(×10 –11)
(d ) 1.5
1.0
0.5
0
– 0.5
–1.0
–1.5
–2.0
–2.5
(e)
M23 )(× 10 –11)
–20
± 0.2
± 0.6
±1.0
0
M23 × 10 – 8)
M23 )(× 10 –8)
–10
(b)
M23 )(×10 –8)
0
M23 )(× 10 –10)
(a)
M23 )(× 10 –11)
Steady electro-osmotic flow
± 0.2
±0.6
±1.0
0
±0.2
±0.6
±1.0
0
Figure 13. Profiles of M23
ðyÞ with y predicted after the substitution of the Debye–Hückel solution
(3.15) in (5.2)3, when joZK25!10K3 V, for k1Z0.25 (solid curves), k1Z0.5 (dotted curves)
and k1Z0.95 (dashed curves). (a,d,g) moZ10, (b,e,h) moZ50 and (c,f,i) moZ500. In addition,
(a–c) boZK0.01, (d–f ) boZK0.1 and (g–i) boZK1.
using equations (3.15) and (5.2)3 as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c 7 k1
bo
0
k1 ð1Kk1 Þð2Kk1 Þ;
C
m 23 ð0Þ zc 6
c 7 k1 ð2Kk1 ÞKð1Kk1 Þg 1 C k1 bo
ð5:9Þ
where c 6 Z ðe jo Eo =lD Þðg=mÞ1=2 , c 7 Z l2D m and the Debye–Hückel approximation is
valid. Furthermore, the couple stress tensor is not null-valued at the walls either,
0
0
0
as shown in figure 13, which contains the profiles of M23
ðyÞ Z ½m 23
ðyÞKm 23
ð0Þ=
0
K3
m 23 ð0Þ with respect to y, for jo ZK25 !10 V, k 1 2{0.25,0.5,0.95},
mo2{10,50,500} and bo 2fK0:01;K0:1;K1g. This figure clearly indicates that,
0
for low values of jboj and mo, the magnitude of m23
ðyÞ mid-channel is greater than at
0
any other location. As jboj, as well as mo, increases, a decrease in jm 23
ð0Þj and an
0
increase in jm 23 ðyÞj at and in the vicinity of the walls are observed. Furthermore,
0
0
the difference between m 23
ðyÞ and m 23
ð0Þ increases if: (i) mo increases for all k1 and
bo, (ii) k1 increases for all k1 and bo, or (iii) jboj increases for all mo and k1.
Finally, when the Debye–Hückel approximation does not hold, we examined
0
0
M23
ðyÞ for two different values of jo in figure 14. This figure shows that M23
ðyÞ
does not alter with jjoj. It is not surprising because, if we use equations (2.7),
0
(3.15), (5.1)3 and (5.9) to calculate M23
ðyÞ, we get an expression that is
0
independent of jo. But, m 23 ðyÞ increases c y 2½K1; 1 as jjoj does.
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520
A. A. Siddiqui and A. Lakhtakia
(a) 3.0
(b)
M23 )(× 10 –11)
2.5
2.0
1.5
1.0
0.5
0
± 0.2
± 0.4
± 0.6
± 0.8
±1.0
0
±0.2
±0.4
±0.6
±0.8
±1.0
0
Figure 14. Profiles of M23
ðyÞ with y obtained by the numerical approach, when moZ50 and
boZK0.5, for k1Z0.25 (solid curves), k1Z0.5 (dotted curves) and k1Z0.95 (dashed curves).
(a) joZK100!10K3 V and (b) joZK500!10K3 V.
6. Concluding remarks
We formulated the boundary-value problem of steady electro-osmotic flow of a
micropolar fluid in a rectangular microchannel, and we solved it numerically as
well as analytically using the Debye–Hückel approximation, when the Debye
length is no more than 5 per cent of the height of the microchannel and the length
of the microchannel is much larger than its height. The numerical results turned
out to be virtually identical to the Debye–Hückel results for low magnitudes of
the zeta potential jo. By contrast, the results differ significantly for higher
magnitudes of jo, and the differences intensify with increasing jjoj. We also
defined the micropolar Helmholtz–Smoluchowski equation and velocity U.
The micropolar-fluid speed depends significantly not only on the height of the
microchannel in relation to the Debye length (moZh/lD), analogous to the speed
of a simple Newtonian fluid, but also on the micropolar nature of the fluid as
expressed through: (i) the boundary parameter bo that mediates the velocity
gradient and the microrotation at the walls of the microchannel and (ii) the
viscosity coupling parameter k1 relating the Newtonian shear viscosity coefficient
and the (micropolar) vortex viscosity coefficient. The fluid speed in the central
portion of the microchannel exceeds the speed of a simple Newtonian fluid
(k1Z0). For a fixed Debye length, the mid-channel fluid speed depends on the
microchannel height when the fluid is micropolar, but not when the fluid is
simple Newtonian. Moreover, in the regions near the walls (including the electric
double layers), u 0 ðyÞ/ U as k1/1, regardless of the magnitude of jo. In contrast
to the fluid speed, the microrotation is null-valued mid-channel, but is dominant
at the walls. The microrotation at the walls does not depend on the microchannel
height. Furthermore, if the boundary layers are turbulent, the microrotation at
the walls also becomes independent of the viscosity coupling parameter.
The stress at the walls is dominant at and in the vicinity of the walls, whether
the fluid is micropolar or simple Newtonian. However, as the micropolarity
parameter jboj, increases, the stress at the walls intensifies. The mid-channel
couple stress decreases, but the couple stress at the walls intensifies, as the
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Steady electro-osmotic flow
521
microchannel height, as well as jboj, increases. Finally, the electric potential,
fluid speed, microrotation and stresses are enhanced as the zeta potential
increases in magnitude.
We have begun to examine non-steady electro-osmotic flows of micropolar
fluids in microchannels, wherein the work reported here will be incorporated into
the solution of an initial-boundary-value problem.
We gratefully acknowledge the very useful comments of the anonymous reviewers. We thank
Mr Ambuj Sharma and Prof. Charles E. Bakis of the Pennsylvania State University for discussions.
A.A.S. is grateful to the Higher Education Commission of Pakistan for a grant to enable him to
visit Penn State. A.L. thanks the Charles Godfrey Binder Endowment at Penn State for partial
financial support.
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