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ELECTROHYDRODYNAMIC STABILITY OF COUPLE
STRESS FLUID FLOW IN A CHANNEL OCCUPIED BY
A POROUS MEDIUM
N. Rudraiah,1,2,∗ B. M. Shankar,1 & C. O. Ng3
1
UGC-Centre for Advanced Studies in Fluid Mechanics, Department of Mathematics, Bangalore
University, Bangalore, India
2
National Research Institute for Applied Mathematics, Jayanagar, Bangalore, India
3
Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, Republic
of China
∗
Address all correspondence to N. Rudraiah E-mail: [email protected]
Original Manuscript Submitted: 3/9/2010; Final Draft Received: 7/26/2010
The linear stability of electrohydrodynamic poorly conducting couple stress viscous parallel fluid flow through a porous
channel is studied in the presence of a nonuniform transverse electric field using an energy method. Supplemented with
a single term Galerkin expansion. The sufficient condition for stability is obtained using the nature of the growth rate
as well as sufficiently small values of the Reynolds number, Re. From this condition we show that strengthening or
weakening of the stability criterion is dictated by the values of the strength of the electric field, the coefficient of couple
stress fluid, and the porous parameter. In particular, it is shown that the interaction of the electric field with couple stress
is more effective in stabilizing the poorly conducting couple stress fluid compared with that in an ordinary Newtonian
viscous fluid.
KEY WORDS: porous medium, stability, couple stress, electrohydrodynamic, nonuniform transverse electric field
1. INTRODUCTION
The effective functioning of microfluidic devices in electronics and electrical and mechanical engineering involving fluids, particularly those having vibrations and
petroleum products containing organic, inorganic, and
other microfluidics, requires the understanding and control of stability of parallel fluid flows. These substances,
dissolving in the fluid, make the fluid poorly conducting.
The electrical conductivity, σ, of such poorly conducting
fluidics, increases with the temperature and the concentration of freely suspended particles. These freely suspended
particles in fluid spin, producing microrotation, forming
micropolar fluid. According to Eringen (1966) the micropolar fluids may be regarded as non-Newtonian fluids
like fluid suspensions. The presence of dust in the atmo-
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spheric fluid, the cholesterols, RBC, WBC, and so on, in
the physiological fluid; the Hylauronic acid and nutrients
in synovial fluid in synovial joints; and the presence of
Deuterium–Tritium (DT) in Inertial Fusion Target (IFT)
may also be modeled using the micropolar fluid theory of
Eringen (1966). This theory takes care of the inertial characteristics of the substructure particles that are allowed to
spin and thus undergo microrotation (Peddieson and McNitt, 1970; Ariman et al., 1973; Lukaszewicz 1999; Eringen, 2001). A particular case of micropolar fluid theory,
when the microrotation balances the natural vorticity of a
poorly conducting fluidics in the presence of an electric
field, is called ‘electrohydrodynamic couple stress fluid
(EHDCF) (Rudraiah, 1998, 2003).
These EHDCFs exhibit a variation of electrical conductivity, ∇σ, increasing with temperature and concen-
11
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Rudraiah, Shankar, & Ng
NOMENCLATURE
qi
k
Ji
Ei
Tb
Cb
T0
C0
p
V
We
Re
c
cr
ci
l
velocity vector (u, v)
permeability of a porous
medium
current density
electric field
conduction temperature
diffusion concentration
initial temperature
initial concentration
pressure
applied uniform electric
potential
electric number,
ε0 V 2 /ρu20 h2
Reynolds number,
u0 h/ν
velocity of perturbed
quantities
phase velocity
growth rate
horizontal wavenumber
t
x, y
time
space coordinates
Greek Symbols
ρ
density
µ
viscosity
µe
effective viscosity
β
ratio of effective viscosity
to the viscosity,
µe /µ
λ
coefficient of couple stress
λe
effective coefficient of
couple stress
Λ
ratio of coefficient of effective
couple stress to the couple stress,
λe /λ
ρe
distribution of charge density
σ
electrical conductivity
αt , αc volumetric coefficient of σ
φ
electric potential
σp
porous parameter
tration of freely suspended particles, and releasing the
charges from the nuclei forming distribution of charge
~ i.
density, ρe . These charges induce an electric field, E
~
If need be, we can apply an electric field, Ea , by embedding electrodes of different potentials at the boundaries.
~ =E
~i + E
~ a , produces a current
The total electric field, E
~
~
density, J = σE, according to Ohm’s law and also pro~ This J~ acts as sensing
duces an electric force, F~e = ρe E.
~
and the force, Fe , acts as actuation. These two properties
make the poorly conducting couple stress fluid act as a
smart material (Rudraiah, 2003). This smart couple stress
poorly conducting fluid plays a significant role in controlling the stability of parallel flows, which is essential for an
effective function of machineries that are used in the practical problems mentioned above. Such smart couple stress
poorly conducting fluid also helps to develop the artificial
organs at the molecular level and will contribute to the
further advancement of physiological studies. These developments play a significant role in human joints, coronary arterial diseases, trachea (i.e., wind pipe), and so on.
In the human body, there are three types of joints, namely, freely movable (i.e., diarthrodial), slightly movable
(i.e., amphiarthrodial), and immovable (i.e., synarthrodial). The freely movable joints are known as the synovial
joints (SJs), and they play a significant role because of
their importance in human locomotion. The two important parts of SJs are the cartilage and the rheological nature of synovial fluid (see Fig. 1). The cartilage, a flossy,
grayish white substance of about 1–2 mm thickness covering the articulating ends of the bones, plays a significant
role in the normal functioning of joints. The synovial fluid
(SF) of high viscosity impregnates the movable joints of
the body and is contained in the capsules of the joints,
normally in volumes of about 0.2–2 mL. It serves as a
lubricant between the cartilage surfaces and carries out
metabolic functions by providing nutrients to the articular
cartilage (AC). Although SF has some resemblance compositionally to blood plasma, it lacks the clotting agents
such as fibrinogen. The most important constituent of SF
is the hyaluronic acid (HA), which makes the SF have a
high viscosity of about 1000 times greater than water. The
normal SF is non-Newtonian, exhibiting a decrease in viscosity with an increase in shear rate (Rudraiah, 1998). The
most important aspect of SJs is to understand the different
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FIG. 1: Synovial joint.
aspects, including its lubrication mechanism in human locomotion, because the results obtained may shed light on
the understanding of the degenerative changes in SJs that
directly affect the normal physiological functioning of an
individual. These changes can evolve through the following four types of arthritis: (1) Osteoarthritis, common during old age; (2) traumatic arthritis, occurs due to injuries;
(3) rheumatic arthritis, occurs due to diseases; and (4)
kinesthesia arthritis, occurs due to an erring gene. Among
the recent genetic discoveries reported by the U. S. National Institute of Arthritis is a single genetic flaw that
could cause a very common type of arthritis called kinesthesia arthritis.
According to the orthopedic surgeons, although it may
not be possible to prevent arthritis, there are steps that
can be taken to reduce the risk of developing this disease
and to slowdown or prevent permanent joint damage. The
present measures include reduction in weight (as excess
weight puts strain on the joints), doing regular exercise
to help the muscles become strong (which will protect
and support the joints), and using joint protecting devices
and techniques at work (such as proper lifting and posture) to protect muscles and joints. It is known that the
above degenerative changes in SJs may be due to physiological and mechanical aspects. One of the physiological aspects is gout, which occurs when the body cannot eliminate the natural substance called uric acid (UA).
Volume 02, Number 1, 2011
13
The excess UA forms needle-like crystals in the joints
that cause gout, namely, swelling, causing severe pain.
The mechanical aspects of degenerative changes in SJs
are mainly due to degenerative changes in the cartilage. It
is believed (Rudraiah et. al., 2006) that the degenerative
changes in ACs will not be naturally recouped using the
above-mentioned preventive measures, and when the disorder becomes severe replacement with artificial joints is
the only alternative to relieve pain. It is a common observation that artificial joints are manufactured using metals.
We know (Ng et. al., 2005) the difficulties in manufacturing metal joints, which are not biocompatible with natural joints. These metal joints will have either rough or
smooth surfaces. Both of them are dangerous to the body
because they produce stresses, which in turn produce a
force that drives the erythrocytes (i.e., RBC) in joints to
a particular region where the high concentration of RBC
leads to bursting of RBCs, releasing heamoglobin, resulting in a disease called heamolysis due to the loss of
heamoglobin. Recently, Ng et al. (2005) have suggested
a mechanism of using the smart material of nanostructure to mimic the natural joints as an alternative to metal
joints by studying the dispersion of micropolar components in a biological bearing. This force, produced by the
metal joints, may also disturb the SF, causing instabilities that in turn may contribute to the malfunctioning of
the joints. Further, we note that HA, UA, nutrients, and
so on, in the synovial fluid are freely suspended, executing microrotation. In the literature, considerable work has
been done to find the effect of this force by studying dispersion phenomena but, to our knowledge, no work has
been done on the study of the stability of SF to model it
as a couple stress fluid and to understand the degenerative
changes in arthritis caused by the instabilities produced
by the above-mentioned force. The study of the stability
of SF is the main objective of this paper. To achieve this
objective, this paper is planned as follows: The required
basic equations, corresponding boundary conditions are
given in Section 2 on mathematical formulation. The basic state and the stability equations are given in Section 3.
The stability analysis is given in Section 4. Results and
Discussion are drawn in the Section 5 and conclusions are
presented in Section 6.
2. MATHEMATICAL FORMULATION
We consider a horizontal poorly conducting couple stress fluid flow through a sparsely packed porous layer bouon both sides by electro-conducting impermeable rigid
plates embedded with segmented electrodes located at
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y = 0 and y = h having different electric potentials
φ = (V /h) x at y = 0 and φ = (V /h) (x − x0 ) at y = h,
as shown in Fig. 2.
For the sake of clarity, we first give the general form
of modified basic equations for a poorly conducting couple stress incompressible fluid through a sparsely packed
porous medium, modification in the sense of addition of
the couple stress and electric force to be obtained from
the general form of Maxwell equations. The rheological properties of physiological fluids like synovial fluid
in synovial joints and blood flow in arteries reveal that
viscosity varies nonlinearly, exhibiting either shear thinning or shear thickening behavior. This is one of the nonNewtonian fluid flow properties that has been studied
in the literature (Fung, 1981; Rudraiah, 1998), but their
work is absent of the study of microrotation due to freely
suspended particles, as explained in Section 1. These are
taken into account in this paper using the couple stress
fluid as a particular case of micropolar fluid theory, developed by Eringen (1966), as described by Rudraiah et
al. (1998).
We note that one of the limitations encountered in the
continuum theory is the lack of taking into account the microrotation of freely suspended particles in a fluid. For example, HA molecules and other nutrients present in synovial fluid, RBC, WBC, and so on, in blood, DT in (IFT)
in the extraction of inertial fusion energy (IFE), and so
on, are freely suspended, executing spin. In that case, the
microrotation of the microelements must be taken into account in deriving the required basic equations where the
microelement motions play a significant role. In such situations, the couple stress theory, a particular case of micropolar fluid, as explained above, is useful. Then, the
required basic equations for a couple stress poorly conducting fluid flowing through a sparsely packed porous
medium, following Stokes (1968) and Rudraiah et al.
(1998), are:
The conservation of mass, for an incompressible fluid:
Rudraiah, Shankar, & Ng
∂qi
=0
∂xi
The conservation of momentum:
µ
¶
∂qi
∂qi
∂p
∂ 2 qi
ρ
+ qj
=−
+ µe 2
∂t
∂xj
∂xj
∂x
µ
¶ i
4
∂ qi
µ
λ
1+
qi
− λe 4 + ρe Ei −
∂xi
k
µk
(1)
(2)
where qi = (u, v) is the velocity, ρ is the density, p is the
pressure, µe is the effective viscosity, λe is the effective
coefficient of couple stress, λ is the coefficient of couple
stress, µ is the viscosity of the fluid, k is the permeability
of a porous medium, and λ/µ has the dimension of length
square.
The conservation of charges:
∂ρe
∂Ji
+
=0
∂t
∂xj
(3)
where ρe is the distribution of charge density
Ji = ρe qi + σEi
(4)
where Ji is the current density, which is the sum of convective current, ρe qi , and conduction current, σEi , σ is
the electrical conductivity, Ei , is the electric field. These
are supplemented with the Maxwell Field equations for a
conducting medium:
Gauss law
∂Ei
ρe
(5)
=
∂xi
ε0
where εo is the dielectric constant for free space.
In a poorly conducting fluid, the induced magnetic
field is negligible and there is no applied magnetic field;
hence, the Faraday law becomes
∂Ei
∂Ej
−
=0
∂xj
∂xi
(6)
That is, the electric field is conservative, so that
Ei = −
∂φ
∂xi
(7)
where ϕis the electric potential.
Equation (3), using Eqs. (4) and (1), takes the form
∂ (σEi )
Dρe
+
=0
Dt
∂xj
FIG. 2: Physical configuration.
(8)
where D/Dt = ∂/∂t + qj (∂/∂xi ). We note that in a
poorly conducting fluid σ ¿ 1 and, hence, any perturbation on it is assumed to be negligible and increases with
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conduction temperature, Tb , and diffusion concentration,
Cb , such that
σ = σo [1 + αt (Tb − T0 ) + αc (Cb − Co )]
(9)
15
The solutions of Eqs. (13), satisfying conditions (14a)
and (14b), are
Tb =
∆T
y + T0
h
and Cb =
∆C
y + C0
h
(15)
Here, σ0 is that of σ at Tb = T0 and Cb = C0 , αt and αc where ∆T = T − T and ∆C = C − C
1
0
1
0
are the volumetric expansion coefficients of σ. The ex- Substituting the solutions given by Eq. (15) into Eq. (9),
pressions for Tb and Cb are obtained in Section 3.1.
we get
σb = σ0 (1 + αh y) ≈ σ0 eαh y
3. THE LINEAR STABILITY EQUATIONS FOR
A COUPLE STRESS POORLY
CONDUCTING FLUID
In this section, we derive the stability equations subject to
infinitesimal disturbances superposed on the basic state
given in Section 3.1 below.
3.1 Basic State
We consider a basic flow in a poorly conducting couple
stress fluid and assuming it to be fully developed and unidirectional parallel to the plates driven by a constant pressure gradient ∂pb /∂x. Then, the basic flow, ub , parallel to
the boundaries in the x-direction, satisfies the momentum
equations
∂pb
0=−
∂x
µ
µ
1+
−
k
∂pb
0=−
∂y
∂ 2 ub
∂ 4 ub
+ µe
− λe
+ ρeb Ebx
2
∂y
∂y 4
¶
λ
ub
µk
+ ρeb Eby
where the suffix b represents the basic state quantities.
The boundary conditions are the no-slip conditions
ub = 0
at y = 0
and h
(11)
at y = 0
and h
(12)
Further, Tb and Cb in Eq. (9) are the solutions of
d2 Tb
=0
dy 2
and
d 2 Cb
=0
dy 2
(17)
subject to the boundary conditions
φb =
Vx
h
at y = 0
V (x − x0 )
h
(18a)
at y = h
(18b)
where V is the applied uniform electric potential.
We make quantities in Eqs. (17), (18a), and (18b) dimensionless, using
φb
,
V
y
y∗ =
h
φ∗b =
ρ∗eb =
ε0
ρeb
¡ V ¢,
x∗ =
h2
x
,
h
(19)
∂ 2 φb
∂ 2 φb
∂φb
+
+ αh
=0
2
∂x
∂y 2
∂y
(20)
satisfying the boundary conditions
(13)
satisfying the conditions
φb = x at y = 0
φb = x − x0
Tb = T0
and Cb = C0
at y = 0
(14a)
Tb = T1
and
Cb = C1
at y = h
(14b)
Volume 02, Number 1, 2011
∂ 2 φb
∂ 2 φb
∂φb
+
+ αh
=0
2
∂x
∂y 2
∂y
where the asterisks (*) denote the dimensionless quantities. Substituting Eq. (19) into Eqs. (17), (18a), and (18b),
and for simplicity neglecting the asterisks, we get
and the couple stress conditions
∂ 2 ub
=0
∂y 2
(16)
where αh = αt ∆t/h + αc ∆C/h
In a poorly conducting fluid, the frequency of charge
distribution is smaller than the corresponding relaxation
frequency of the electric field, so that Dρe /Dt in Eq. (8)
is negligible compared to ∂ (σEi )/∂xj . Then, from
Eq. (8), after neglecting Dρe /Dt and using Eqs. (7) and
(16), we get
φb =
(10)
(∵ αh ¿ 1)
at y = 1
(21a)
(21b)
The solution of Eq. (20), satisfying boundary conditions
(21a) and (21b), is
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φb = x −
xo (1 − e−αh y )
(1 − e−αh )
(22)
The expression for ρeb can be obtained, from Eq. (5), using Eq. (22), as
ρeb = −
x0 α2h e−αh y
(1 − e−αh )
(23)
Equation (7), using Eq. (22) becomes
Ebx = −1,
Eby =
Rudraiah, Shankar, & Ng
The average velocity,
1
u0 =
2
Z1
ub dy =
0
δ3
(∆3 + ∆4 − ∆7 )
δ6
(29)
where the constants ∆i (i = 1 − 7) are given in the Appendix.
3.2 Stability Equations
−αh y
x0 αh e
(1 − e−αh )
(24)
We make Eqs. (10)–(12) dimensionless, using
pb
y
ub
, p∗b =
, y∗ = ,
u0
ρu20
h
ρeb
x
Eb
x∗ = , ρ∗eb = ¡ V ¢ , Eb∗ = ¡ V ¢
h
ε0 h2
h
u∗b =
(25)
where u0 is the average velocity and the other quantities are as defined in Eq. (19). Substituting Eq. (25) into
Eqs. (10)–(12), using Eqs. (23) and (24), and for simplicity neglecting the asterisks, we get
In a two-dimensional, incompressible, homogeneous
poorly conducting couple stress fluid flow, with qi =
(u, v), the conservation of momentum, given by Eq. (2),
takes the form
ρe
∂u
∂u
∂u
1 ∂p µe 2
∇ u + Ex
+u
+v
=−
+
∂t
∂x
∂y
ρ ∂x
ρ
ρ
µ
¶
µ
λe 4
λ
1+
u
(30)
− ∇ u−
ρ
ρk
µk
ρe
∂v
∂v
∂v
1 ∂p µe 2
∇ v + Ey
+u
+v
=−
+
∂t
∂x
∂y
ρ ∂y
ρ
ρ
µ
¶
µ
λe 4
λ
1+
v
(31)
− ∇ v−
ρ
ρk
µk
Λ ∂ 4 ub
∂ 2 ub
− β 2 = −ReP + We ReM e−αh y
together with the conservation of mass
2
4
a ∂y
∂y
¶
µ
∂u ∂v
σ2p
+
=0
(32)
(26)
− σ2p 1 + 2 ub
∂x ∂y
a
where ∇2 = ∂ 2 /∂x2 + ∂ 2 /∂y 2 , ∇4 = ∂ 4 /∂x4 +
p
λ/µ is the coefficient of 2∂ 4 /∂ 2 x∂ 2 y + ∂ 4 /∂y 4
where a = h/m; m =
the couple stress fluid; β = µe /µ is the ratio of visTo study the electrohydrodynamic stability of coucosities (i.e., Brinkmann viscosity to viscosity of fluid); ple stress poorly conducting fluid in a saturated porous
Λ = λe /λ is the ratio of coefficient of effective couple medium, as shown in Fig. 2, we superimpose an infinitesstress to the couple stress P = ∂pb /∂x; Re = u0 h/υ is imal disturbance, denoted by the primes, over the basic
the Reynolds number; √
We = ε0 V 2 /ρu20 h2 is the elec- state denoted by suffix ”b” of the form
tric number; σp = h/ k is the porous parameter; and
u = ub + u0 , v = v 0 , p = pb + p0 ,
M = x0 α2h /(1 − e−αh ). Physically We represents the
Ex = Ebx + Ex0 , Ey = Eby + Ey0
(33)
ratio of electric energy to kinetic energy. The required
boundary conditions are
Substituting Eq. (33) into Eqs. (30)–(32), and linearizing by neglecting the product and higher order of prime
2
∂ ub
ub =
=
0
at
y
=
0
and
1
(27)
quantities compared with the basic state, we obtain
∂y 2
∂u0
∂u0
∂ub
1 ∂p0
µe
+ ub
+ v0
=−
+
Solving Eq. (26), using the boundary conditions (27), we
∂t
∂x
∂y
ρ ∂x
ρ
get
µ 2 0
¶
2 0
λe
∂ u
∂ u
1
×
+
+ (ρeb Ex0 + ρ0e Ebx ) −
ub = C1 eδ1 y + C2 e−δ1 y + C3 eδ2 y + C4 e−δ2 y
∂x2
∂y 2
ρ
ρ
µ 4 0
¶
−αh y
2
2
4
0
4
0
We ReP a M e
ReP a
∂ u
∂ u
∂ u
µ
+
−
(28)
×
+2 2 2 +
−
ΛK2
Λ (α4h − K1 α2h + K2 )
∂x4
∂x ∂y
∂y 4
ρk
¶
µ
λ
where the constants Ki (i = 1 and 2), δi (i = 1 − 6), and
u0
(34)
× 1+
µk
Ci (i = 1 − 4) are given in the Appendix.
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∂v 0
∂v 0
1 ∂p0
µe
+ ub
=−
+
∂t
∂x
ρ ∂y
ρ
µ 2 0
¶
2 0
¡
¢ λe
∂ v
∂ v
1
ρeb Ey0 + ρ0e Eby −
×
+
+
2
2
∂x
∂y
ρ
ρ
µ 4 0
¶
4 0
4 0
∂ v
∂ v
∂ v
µ
×
+2 2 2 +
−
∂x4
∂x ∂y
∂y 4
ρk
¶
µ
λ
v0
(35)
× 1+
µk
∂u0
∂v 0
+
=0
(36)
∂x
∂y
These perturbed equations are made dimensionless using
the quantities
u
v
y
x
, v∗ =
, y ∗ = , x∗ = ,
u∗ =
u0
u0
h
h
ρe
p ∗
t
E
¡ V ¢ , Ex∗ = ¡ Vx¢ ,
, t = ³ ´ , ρ∗e =
p∗ =
h
ρu20
ε0 h2
h
u0
Ey
Ey∗ = ¡ V ¢
(37)
h
where c = cr + ici is the velocity of perturbed quantities,
cr is the phase velocity, ci is the growth rate, and l is
the horizontal wavenumber, which is real and positive. If
ci > 0, the system is unstable and ci < 0, the system
is stable. Equations (38)–(40), using Eq. (41), and after
simplification, take the form
"
Λ
β(D2 − l2 ) − ilRe(ub − c) − 2 (D2 − l2 )2
a
µ
¶#
σ2p
σ2p
1+ 2
u = ilRep + ReDub v
−
Re
a
− We Re(ρeb Ex + ρe Ebx )
"
β(D2 − l2 ) − ilRe(ub − c) −
σ2p
−
Re
µ
σ2p
1+ 2
a
¶#
∂u
∂u
∂p
β
+ ub
+ Dub v = −
+
∂tµ
∂x
∂x Re
¶
∂2u ∂2u
×
+ 2 + We (ρeb Ex + ρe Ebx )
∂x2
∂y
µ 4
¶
∂ u
∂4u
∂4u
Λ
+2 2 2 + 4
− 2
a Re ∂x4
∂x ∂y
∂y
µ
¶
2
2
σp
σp
1+ 2 u
−
Re
a
∂v
∂v
∂p
β
+ ub
=−
+
∂t
∂x
∂y Re
µ 2
¶
∂ v
∂2v
×
+ 2 + We (ρeb Ey + ρe Eby )
∂x2
∂y
µ 4
¶
∂ v
Λ
∂4v
∂4v
− 2
+2 2 2 + 4
a Re ∂x4
∂x ∂y
∂y
µ
¶
2
2
σp
σp
1+ 2 v
−
Re
a
Volume 02, Number 1, 2011
y) eil(x−ct)
Λ
(D2 − l2 )2
a2
ilu + Dv = 0
(43)
(44)
Eliminating pressure p between Eqs. (42) and (43), by
operating D on Eq. (42), multiplying Eq. (43) by il, and
then subtracting and expressing u = (i/l)(Dv) and Ey =
(−i/l)(DEx ) using, respectively, Eqs. (44) and (5), we
get the stability equation
(D2 − l2 )2
D2 ub
v + iβ
v
(ub − c)
(ub − c)
(Dρeb Ex + Dρe Ebx )
Λ (D2 − l2 )3
v + We lRe
− 2
a (ub − c)
(ub − c)
¢
¶¡ 2
µ
2
2
D −l
σp
v=0
− iσ2p 1 + 2
(45)
a
(ub − c)
¡
¢
Equation (45), using Ebx = −1, Dρe = − D2 − l2
× Dφ and retaining terms up order α2h since αh ¿ 1,
takes the form
lRe(D2 − l2 )v − lRe
(38)
(39)
∂u ∂v
+
=0
(40)
∂x ∂y
where the parameters a, We , σp , β, Λ, and Re are defined
in Eq. (26). To discuss the stability of systems (38)–(40),
we use the normal mode solution of the form
(function of
(42)
v = ReDp
− We Re(ρeb Ey + ρe Eby )
Substituting Eq. (37) into Eqs. (34)–(36), and for simplicity neglecting the asterisks (*) and the primes, we get
17
(41)
(D2 − l2 )2
D2 ub
v + iβ
v
(ub − c)
(ub − c)
¡ 2
¢
D − l2 Dφ
iΛ (D2 − l2 )3
v + We lRe
− 2
a (ub − c)
(ub − c)
¢
¶¡ 2
µ
2
2
D −l
σp
v=0
− iσ2p 1 + 2
(46)
a
(ub − c)
lRe(D2 − l2 )v − lRe
Equation (46) is the required stability equation, which
is called the modified form of Orr–Sommerfeld equation
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Rudraiah, Shankar, & Ng
(modified in the sense of incorporating the contribution The second term on the left-hand side of Eq. (48) is the
~ and the couple stress fluid) contribution of the couple stress fluid and the term involvfrom the electric force, ρe E
and satisfying the boundary conditions
ing We in the first term on the right-hand side of Eq. (48)
is the effect of the electric field.
v = Dv = D3 v = 0 at y = 0 and 1
(47)
Equating the real and imaginary parts of Eq. (48) to
zero we respectively, get
The conditions on Dv and D3 v are obtained from
Qr
Eq. (44) using the no-slip and couple stress conditions on
(51)
cr = 2
(I1 + l2 I02 )
u.
To find the conditions for stability or instability of the basic flow, we find (following Rudraiah, 1962, 1963; Drazin
and Reid, 2004; Shubha et al., 2008) the nature of c using
the energy method. For this, we multiply Eq. (46) by v̄,
the complex conjugate of v, and integrating the resulting
equation with respect to y from 0 to 1 and using boundary
conditions (47) and after simplification, we get
¡
¢ Λ ¡
¢
β I22 +2l2 I12 +l4 I02 + 2 I32 +3l2 I22 +3l4 I12 +l6 I02
a
¶
µ
2
¡
¢
σ
p
I12 + l2 I02 = −ilReQ
+ σ2p 1 + 2
a
+ ilRec(I12 + l2 I02 )
(48)
where
Z
In2
1
=
0
Z 1
Q=
Z
+
2
|Dn v| dy
(n = 0 − −3)
h
i
2
2
ub |Dv| + (l2 ub + D2 ub ) |v| dy
0
1
Z
v̄DvDub dy − We
0
"
µ 2
¶
3l Λ
1 Λ 2
I +
+ β I22
ci = Qi −
lRe a2 3
a2
Ã
!
3l4 Λ + σ2p
2
2
+ (2l β + σp ) I12
+
a2
! #!
Ã
l2 (l4 Λ + σ4p )
2 2
2
+ l (l β + σp ) I02
+
a2
Ã
4. STABILITY ANALYSIS
0
1
¡ 2
¢
D − l2
÷ (I12 + I 2 I02 )
(52)
The nature of stability is determined from Eq. (52) in the
following two cases.
Case 1: A Sufficient condition for stability, in terms of
growth rate ci is obtained, following Joseph (1968), in
Result 1.
Result 1: Let c be the eigenvalue of Eq. (48). Then, a
sufficient condition for stability is
N1 ≥
max (We F1 + F0 )
F2
Proof: Eq. (52) can be written as
ci =
F0 + We F1 − N1 F2
(I12 + l2 I02 )
(53)
× Dφv̄ = Qr + iQi
R1
where F0 = 0 (vr Dvi − vi Dvr )Dub dy, which is the
¶
µ
Z 1(
1
first integral on the right-hand side of Eq. (50); F1 =
Qr = Re (Q) =
ub |Dv|2 + l2 ub + D2 ub
¢
R1¡ 2
2
0
D − l2 (vi Dφr − vr Dφi ) dy, which is the second
0
)
integral on the right-hand side of Eq. (50); and
¡ 2
¢
2
2
× |v| − We D − l (vr Dφr + vi Dφi ) dy (49)
¶
µ 2
3l Λ
Λ
+
β
I22
F2 = 2 I32 +
a
a2
¶
µ 4
and
¢ 2
3l Λ + σ2p ¡ 2
2
(
+
2l
β
+
σ
I1
+
p
Z 1
a2
!
à ¡
¢
Qi = Im (Q) =
(vr Dvi − vi Dvr ) Dub
¡
¢ 2
l2 l4 Λ + σ4p
0
2 2
2
+ l l β + σp
I0
+
)
a2
¡ 2
¢
+ We D − l2 (vi Dφr − vr Dφi ) dy
(50)
1
N1 =
lRe
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Electrohydrodynamic
19
Here, F2 is always positive, and F0 and F1 may be posi- and using Schwarz’s inequality, we get
tive or negative depending on the signs of the arguments
|Im (Q)| ≤ I1 I0 q + B1
in their integral:
If N1 ≥ max
(F0 + We F1 )
F2
where
(54)
0≤y≤1
ci is always negative; hence, the motion is always stable.
Inequality (54) gives the sufficient condition for stability;
hence, Result 1.
In particular, if the basic velocity is uniform, that is,
ub = constant, so that Dub = 0 implying F0 = 0, then
Eq. (53) becomes
ci =
We F1 − N1 F2
(I12 + l2 I02 )
If N1 ≥ max
We F1
F2
q = max |Dub |
(55)
Z
1
B1 = We
¯¡ 2
¯
¢
¯ D − l2 (vi Dφr − vr Dφi )¯ dy
0
This gives the upper bound for ci
"
µ 2
¶
3l Λ
1 Λ 2
I +
+ β I22
ci ≤ (qI0 I1 + B1 ) −
lRe a2 3
a2
à ¡
¢
¶
µ 4
¢ 2
l2 l4 Λ + σ4p
3l Λ + σ2p ¡ 2
2
+ 2l β + σp I1 +
+
a2
a2
! #
¡
¢
¡
¢
+ l2 l2 β + σ2p I02 / I12 + l2 I02
(57)
ci is always negative; hence, the motion is always stable.
Inequality (55) gives the sufficient condition for stability From Eq. (57), it follows that a sufficient condition for
for uniform basic velocity.
stability is
Case 2: In this case, a sufficient condition for stability is
"
µ 2
¶
3l Λ
1
Λ 2
determined in terms of the Reynolds number, Re, as given
I
+
+
β
I22
Re <
in the Result 2.
l(qI0 I1 + B1 ) a2 3
a2
!
Ã
Ã
Result 2: A sufficient condition for stability is
4
2
¡ 2
¢
l2 (l4 Λ + σ4p )
3l
Λ
+
σ
p
2
2
"
+
2l
β
+
σ
I
+
+
µ 2
¶
p
1
a2
a2
3l Λ
1
Λ 2
I3 +
+ β I22
Re <
!
#
2
2
l (qI0 I1 + B1 ) a
a
¶
µ 4
+ l2 (l2 β + σ2p ) I02
(58)
2
¢ 2
3l Λ + σp ¡ 2
2
+
2l
β
+
σ
I
+
p
1
a2
! #
à ¡
¢
Hence, Result 2.
¡
¢ 2
l2 l4 Λ + σ4p
2 2
2
+
l
l
β
+
σ
I
+
p
0
a2
5. RESULTS AND DISCUSSION
Proof: Following Drazin and Reid (2004), we prove Result 2.
We first write Eq. (50) in the form
Z
i 1
(vDv̄ − v̄Dv)Dub dy
Im (Q) =
2 0
Z 1
¢
¡ 2
+ We
D − l2 (vi Dφr − vr Dφi ) dy
0
From Eq. (56) it follows that
Z
|Im (Q)| ≤
Z
+ We
1
|v| |Dv| |Dub | dy
0
1
¯¡ 2
¯
¢
¯ D − l2 (vi Dφr − vr Dφi )¯dy
0
Volume 02, Number 1, 2011
(56)
A sufficient condition for stability (ci < 0) is obtained
from Eq. (52) as well as in terms of the Reynolds number, Re, given by Eq. (57). The growth rate given by
Eq. (52) is also computed numerically using the singleterm Galerkin expansion and the results are depicted
graphically.
Figure 3 shows that the plot of growth rate ci as a function of horizontal wavenumber l for different values of
couple stress parameter, a for fixed values of the electric number, We = 10, Re = 0.8, and porous parameter
σp = 0.5. From the Fig. 3, it is clear that an increase in
the value of a increases the value of ci because an increase
in a is to decrease the viscosity, implying a decrease in
resistance to the flow, which in turn promotes instability
much faster.
20 Begell House Inc., http://begellhouse.com Downloaded 2011-3-13 from IP 147.8.84.141 by Dr. Chiu-On Ng (cong)
FIG. 3: Variation of ci with l for different values of a
when We = 10, Re = 0.8, and σp = 0.5.
The variation of ci with l for different values of σp
with We = 10, Re = 0.8, and a = 10 is shown in Fig. 4.
From Fig. 4 it is clear that the effect of σp is to decrease
the growth rate, which makes the system more stable. The
plausible reason for this is that the decrease in growth rate
is due to the resistance offered by the solid particles in the
porous media to the fluid.
Figure 5 shows that the plot of ci with l for different
values of We for fixed values of Re = 0.8, σp = 0.5,
and a = 10. From Fig. 5, it may be inferred that for an
increase in the value of We decreases the value of ci , and
FIG. 4: Variation of ci with l for different values of
σp when We = 10, Re = 0.8, and a = 10.
Rudraiah, Shankar, & Ng
FIG. 5: Variation of ci with l for different values of We
when Re = 0.8, σp = 0.5, and a = 10.
thus make the system more stable. The reason being that
an increase in We is to decrease the kinetic energy and,
hence, makes the system more stable.
6. CONCLUSIONS
It is known that in the stability of classical Poiseuille flow
a sufficient condition for stability is the existence of the
point of inflexion. This stability of Poiseuille flow was
extended to the electrohydrodynamic stability of an inviscid poorly conducting parallel fluid flow in the presence
of an electric field and in the absence of couple stress
fluid by Shubha et al. (2008). They have shown that the
electrohydrodynamic stability is determined in terms of
the electric number rather than the point of inflexion of
the basic velocity profile. In contrast to this, in our paper, a sufficient condition for stability is obtained using
the nature of growth rate ci as well as a sufficiently small
value of Reynolds number, Re. From this we found that
strengthening or weakening of a sufficient condition for
stability depends on the electric number, We , the coefficient of couple stress fluid, m, and the porous parameter,
σp . From these, we conclude that the interaction of the
electric field with the couple stress is more effective in
stabilizing a poorly conducting couple stress fluid compared with that of ordinary Newtonian viscous fluid. This
conclusion on stability analysis helps to understand the
degenerative changes in arthritis caused by the instability
produced by the force generated by metal joints discussed
in the last paragraph of Section 1.
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2011-3-13
Stability of Couple
Stress Fluid
Flow from IP 147.8.84.141 by Dr. Chiu-On Ng (cong)
Electrohydrodynamic
ACKNOWLEDGMENTS
21
nano and smart materials, Proc. of the Workshop on Modeling
of Nano and Smart Materials, Jayanagar, Bangalore: Book
Paradise, Chap. 2, pp. 4–46, 2003.
This work is supported by ISRO under Research Project Nos. ISRO/ RES/2/338/2007-08 and ISRO/RES/2 Rudraiah, N. and Shankar, B. M., Stability of parallel couple
/335/2007-08. The financial support of ISRO is gratefully
stress viscous fluid flow in a channel, Int. J. App. Math., vol.
acknowledged. B. M. S. gratefully acknowledges ISRO
1, pp. 67–78, 2009.
for providing JRF under the above projects. The work of Rudraiah, N., Kantha, S., and Manonmani, M. N., Anatomy and
C. O. N. was partly supported by the Research Grants
biomechanics of synovial joints. Part I, Recent Trends in BaCouncil of the Hong Kong Special Administrative Resic and Applied Anatomy, eds. Thomas, I. M., Srinivasa, K.,
gion, China, through Project No. KHU 7156/09E.
Rajeshwari, T., and Rajangam, S., Karnataka, Bangalore: Rajiv Gandhi University of Health Sciences, pp. 1–9, 1998.
REFERENCES
Ariman, T., Truk, M. A., and Sylvester, N. D., Microcontinuum
fluid mechanics—a review, Int. J. Eng. Sci., vol. 11, pp. 905–
930, 1973.
Drazin, P. G., Introduction to Hydrodynamic Stability, Cambridge, U. K.: Cambridge University Press, 2002.
Drazin, P. G. and Reid, W. H., Hydrodynamic Stability, Cambridge, U. K.: Cambridge University Press, 2004.
Eringen, A. C., Theory of Micropolar Fluids, J. Math. Mech.,
vol. 16, pp. 1–18, 1966.
Eringen, A. C., Microcontinuum Field Theories. II: Fluent Media, New York: Springer, 2001.
Fung, Y. C., Biomechanics, New York: Springer, 1981.
Rudraiah, N., Ng, C. O., Nagaraj, C., and Nagaraj, H. N., Electrohydrodynamic dispersion of macromolecular components
in biological bearing, J. Energy Heat Mass Transfer, vol. 28,
pp. 261–280, 2006.
Stokes, G. G., On some cases of fluid motion, Trans. Cambridge
Philos. Soc., vol. 8, pp. 287–319, 1843.
Stokes, V. K., Effects of couple stress in fluid on hydromagnetic
channel flow, Phys. Fluids, vol. 11, pp. 1131–1133, 1968.
Shivakumara, I. S. and Venkatachalappa, M., Stability of Flows,
3-I, New Delhi, Indis: Tata McGraw-Hill, 2004.
Shubha, N., Rudraiah, N., and Chow, K. W., Electrohydrodynamic stability of poorly conducting parallel fluid flow in the
presence of transverse electric field, Int. J. Non-Linear Mech.,
vol. 43, pp. 643–649, 2008.
Joseph, D. D., Eigenvalue bounds for the Orr–Sommerfeld
equation, J. Fluid Mech., vol. 33, pp. 617–621, 1968.
Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge,
U.K.: Cambridge University Press, 1955.
Lukaszewicz, G., Micropolar Fluids: Theory and Application,
Basel, Switzerland: Birkhauser, 1999.
Ng, C. O., Rudraiah, N., Nagaraj, C., and Nagaraj, H. N., Electrohydrodynamic dispersion of macromolecular components
in nanostructured biological bearing, J. Energy Heat Mass
Transfer, vol. 27, pp. 39–64, 2005.
Peddieson, J. and McNitt, R. P., Boundary layer theory for a
micropolar fluid, Recent Adv. Eng. Sci., vol. 5, pp. 405–476,
1970.
Rudraiah, N., Magnetohydrodynamic stability of heterogeneous
incompressible nondissipative conducting liquids, Appl. Sci.
Res., Sect. B, vol. 11, pp. 105–117, 1962.
Rudraiah, N., Magnetohydrodynamic stability of heterogeneous
dissipative conducting liquids, Appl. Sci. Res., Sect. B, vol.
11, pp. 118–133, 1963.
Rudraiah, N., Anatomy and biomechanics of synovial joints.
Part II: Mathematical modeling, Recent Trends in Basic and
Applied Anatomy, eds. Thomas, I. M., Srinivasa, K., Rajeshwari, T., and Rajangam, S., Karnataka, Bangalore: Rajiv
Gandhi University of Health Sciences, pp. 37–105, 1998.
Rudraiah, N., Instabilities of importance in the manufacture of
Volume 02, Number 1, 2011
APPENDIX
¡
¢
σ2 a2 + σ2
βa2
, K2 =
,
K1 =
Λ
rΛ
q
1
K1
−
K12 − 4K2 ,
δ1 =
2
2
r
q
1
K1
+
K12 − 4K2 ,
δ2 =
2
2
δ3 = a2 e−αh P Re, δ4 = K2 M We ,
δ5 = K2 − K1 α2h + α4h , δ6 = 2K2 δ5 Λ,
·
µ ½
αh
δ1
2
C1 = δ3 (e − e )δ4 αh + eαh (δ5 − δ4 )
¸ ¾·
¸¶
2
δ1
αh
− 1 + coth(δ1 )
+ e (δ4 − e δ5 ) δ2
µ
¶
÷ δ6 (δ21 − δ22 ) ,
µ
½
·
αh +δ1
2
C2 = − δ3 (e
− 1)δ4 αh + δ4
¸ ¾
¶
αh +δ1
2
αh
(δ5 − δ4 ) δ2 cos ech(δ1
− e δ5 + e
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µ
¶
2
2
÷ δ6 (δ1 − δ2 ) ,
µ ½
·
αh
δ2
2
C3 = δ3 (e − e )δ4 αh + eαh (δ5 − δ4 )
¸ ¾·
¸¶
δ2
αh
2
+ e (δ4 − e δ5 ) δ1
− 1 + coth(δ2 )
µ
¶
÷ δ6 (δ22 − δ21 ) ,
µ
½
·
αh +δ2
2
C4 = − δ3 (e
− 1)δ4 αh + δ4 − eαh δ5
¸ ¾
¶
αh +δ2
2
+e
(δ5 − δ4 ) δ1 cos ech(δ2 )
µ
¶
2
2
÷ δ6 (δ2 − δ1 ) ,
Rudraiah, Shankar, & Ng
∆1 = δ4 + eαh (δ4 − 2δ5 ) ,
¡
¢
∆2 = 1 + eδ2
¡
¢
× δ22 − δ21 ,
¡
¢
eδ2 − 1 ∆1 δ21
,
∆3 =
∆ 2 δ2
¡ δ
¢£
¤
e 1 − 1 (eαh + 1) δ4 α2h − ∆1 δ22
,
∆4 =
∆ 2 δ1
µ ¶
δ2
,
∆5 = (eαh + 1) δ4 α3h sinh
2
µ ¶
δ2
αh
∆6 = [δ4 + e (αh δ5 − δ4 )] cosh
2
¡
¢
× δ2 δ22 − δ21 ,
∆7 =
2eδ2 /2 (∆5 + ∆6 )
αh δ2 ∆2
Special Topics & Reviews in Porous Media — An International Journal