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Advances in Applied Science Research, 2012, 3 (5):2890-2899
ISSN: 0976-8610
CODEN (USA): AASRFC
Peristaltic pumping of a conducting fluid in a channel with a porous
peripheral layer
P. Lakshminarayana1, S. Sreenadh2*and G. Sucharitha3
1
Department of Mathematics, Sree Vidyanikethan Engineering College, AP, India.
Department of Mathematics, Sri Venkateswara University, Tirupati 517502, AP, India.
3
Department of Mathematics, Priyadarshini Institute of Technology, Tirupati, AP, India.
2
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ABSTRACT
Peristaltic transport of a conducting fluid in a composite region between two flexible walls is investigated under the
assumptions of long wavelength and low Reynolds number. The composite region consists of core and peripheral
layers. The core layer is a free flow region consisting of a conducting Newtonian fluid and the peripheral layer is a
porous region filled with conducting fluid. An infinite train of peristaltic waves is moving on the walls of the
channel. The fluid flow is investigated in the wave frame of reference moving with the velocity of the peristaltic
wave. Brinkman extended Darcy equation is used to model the flow in the porous layer. A shear-stress jump
boundary condition is used at the interface. The physical quantities of importance in peristaltic transport like
pressure rise etc. are discussed for various parameters of interest governing the flow like viscosity ratio, magnetic
parameter and amplitude ratio. The results found will have applications for understanding the physiological flows
in small blood vessels which can modeled as channels bounded by finite permeable layers (Fung and Tang, [5]).
Key words: peristaltic transport, conducting Newtonian fluid, porous peripheral layer, low Reynolds number.
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INTRODUCTION
Peristaltic transport of a biofluid through a channel with permeable walls is of considerable importance in biology
and medicine. Peristalsis is an inherent property of many of the smooth muscle tubes such as the gastrointestinal
tract, bile duct, ureter and other glandular ducts. The fluids present in the ducts of a living body are called biofluids.
The biofluid has to be treated as Newtonian or non-Newtonian depending on the physiological situation. Peristaltic
pumping through a tube and a channel under the assumptions of low Reynolds number and long wavelength is
discussed by Shapiro [18]. Lu [10] studied the influence of two Newtonian fluids with different viscosities on
peristaltic pumping. Kavitha et al. investigated the Peristaltic flow of a micropolar fluid in a vertical channel with
longwave length approximation. Brasseur et al. [3] discussed the influence of a peripheral layer of different viscosity
on peristaltic pumping with Newtonian fluids.
The boundary conditions to be satisfied at the interface of a two fluid system are the matching of tangential velocity,
normal velocity, shear stress and normal stress. Beavers and Joseph [2] have studied the fluid flow at the interface
between a porous medium and fluid layer experimentally and proposed a slip condition in velocity at the interface.
There exist numerous subsequent studies in the literature which suggest different boundary conditions at the
interface between porous and fluid layers (Chen and Chen [4], Neale and Nader [11], Poulikakos and Kazmierczak
[15], Saffman [17], Vafai and Kim [20]. Ochoa-Tapia and Whitaker [14] introduced a new boundary condition
which accounts for the jump in the shear stress at the interface between porous and fluid layer by applying a
sophisticated averaging volume technique. Kuznetsov [8,9] discussed the significance of the shear stress jump
condition at the interface and applied this condition to investigated the fluid flow in a channel partially filled with a
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porous medium. Alazmi and Vfai [1] investigated the fluid flow and heat transfer between porous medium and a
fluid layer by considering various types of interfacial matching of shear stress conditions reported in the literature.
The mathematical modeling of the two fluid system involves the determination of the interface between the core and
peripheral layers. Ramachandra Rao and Usha [16] analyzed the peristaltic transport of two immiscible Newtonian
fluids in a circular tube. Mishra and Ramachandra Rao [13] studied the peristaltic transport in a channel with a
porous peripheral layer. Most of the physiological fluids (for eg : blood) are observed to be electrically conducting.
Further the behavior of such fluids under a magnetic field in various organs of a human/animal body has to be
analyzed due to its applications in medical diagnosis.
Motivated by these facts, the peristaltic transport of a conducting fluid in a two-layered system with a porous and
core layers is investigated. The fluid flow is studied in the wave frame of reference moving with the velocity of the
peristaltic wave. Brinkman extended Darcy equation is used to describe the flow in the porous layer. The interface is
obtained as a part of the solution using the conservation of mass in both the porous and fluid regions independently.
Ochoa-Tapia and Whitaker [14] shear-stress jump boundary condition is used at the interface. The physical
quantities of importance in peristaltic transport like pressure rise etc. are discussed for various parameters of interest
governing the flow like viscosity ratio, amplitude ratio and magnetic parameter.
2. Mathematical formulation
Consider the peristaltic transport in a two dimensional channel, with a porous peripheral layer consisting of a
conducting Newtonian fluid of viscosity µ1 and an incompressible conducting Newtonian fluid of viscosity µ2 in
the core region. A uniform transverse magnetic field of strength
B0 is applied perpendicular to the channel walls.
We assume that the porous medium is isotropic and homogeneous. The channel wall is flexible and infinite wave
train is moving on the walls of amplitude b with wave length λ in the axial direction with a constant speed c . The
walls are taken by
y = ± H ( X − ct ) in Cartesian coordinate system ( X , Y ) with t as the time. The mean width
of the channel is 2a and the deformed interface separating the core and the peripheral regions is denoted by
y = H1 ( X − ct ) . Under the assumptions that the tube length is an integral multiple of the wavelength and the
pressure difference across the ends of the channel is constant (Shapiro et al. [18]) and an additional condition of
periodicity of the interface with the same period as the peristaltic wave (Brasseur et al. [3]), the flow becomes steady
in a wave frame of reference (x, y) moving with speed c in the direction of the wave propagation.
Fig.1: Physical Model
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The wave frame is connected to the fixed frame by the following equations:
x = X − ct , y = Y , ui = U i − c,
(1)
vi = Vi , pi ( x) = Pi ( X , t )
( ui , vi )
Here
and
(U i ,Vi )
are the velocity components in axial and transverse directions
pi and Pi are the
pressures in wave and fixed frame of references and the subscripts i takes the value 1 for the core layer and 2 for
the peripheral layer .
The governing equations of motion in the fluid region in the core layer
( 0 ≤ y ≤ H1 )
are
 ∂ 2 u1 ∂ 2 u1 
∂u1 
∂p1
 ∂u1
ρ1  u1
+ v1
+ µ1  2 + 2  − σ 1 B0 2 u1
=−
∂y 
∂x
∂x 
 ∂x
 ∂y
 ∂2v ∂2v 
∂v 
∂p
 ∂v
ρ 1  u1 1 + v1 1  = − 1 + µ1  21 + 21 
∂y 
∂y
∂x 
 ∂x
 ∂y
(2)
(3)
∂u1 ∂v1
(4)
+
=0
∂x ∂y
where µ1 , σ 1 , B0 and ρ1 are viscosity , electrical conductivity, magnetic flux and density in the fluid region. The
governing equations of the porous region in the peripheral layer
( H1 ≤ y ≤ H )
are the Brinkman extended Darcy
equations given by (Alazmi and Vafai, 2001)
∂u2 
∂p2 µ2  ∂ 2 u2 ∂ 2 u2  µ2
 ∂u2
+ v2
+
u2 − σ 2 B0 2 u2
ρ 2  u2
 2 + 2 −
=−
∂y 
∂x
ε  ∂y
∂x  k
 ∂x
∂v 
∂p
µ  ∂2v ∂2v  µ
 ∂v
ρ 2  u2 2 + v2 2  = − 2 + 2  22 + 22  − 2 v2
∂y 
∂y
ε  ∂y
∂x  k
 ∂x
∂u2 ∂v2
+
=0
∂x
∂y
ρ 2 , σ 2 , µ2 , ε
where
(5)
(6)
(7)
and k are the density, electrical conductivity, viscosity, porosity and permeability in the
porous region.
Following the experimental investigation of Gilver and Altobelli [6] the effective viscosity in the Brinkman model is
taken as
µ2
ε
.
The following non-dimensional quantities are used
a 2 pi
u
v
y 1 ct
1
x = , y = , t = , pi =
, u1i = i , v1i = i
λ
a
λ
µ1cλ
c1
cδ
x
1
δ=
a
λ
ψ i1 =
1
, h=
ψi
ac
H
µ
ρ
H
b
, h1 = 1 , φ = , µ = 2 , ρ = 2
a
a
a
µ1
ρ1
, M i2 =
(8)
σ i B0 2 a 2
ca
k
, Re = ρ1 , Da = 2
µ1
µ1
a
The governing equations of motion (2) - (7) in terms of stream functions
ψi
(where
ui =
∂ψ i
∂ψ
, vi = − i )
∂y
∂x
after nondimensionalyzation become
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Re δ (ψ 1 yψ 1 yx − ψ 1xψ 1 yy ) = − p1x + ψ 1 yyy + δ 2ψ 1 yxx − M 12ψ 1 y ,
Re δ 3 (ψ 1 yψ 1xx − ψ 1xψ 1 yx ) = p1 y + δ 2 (ψ 1xyy +δ 2ψ 1xxx ) ,
0≤ y≤h
µ
µ
ψ 2 yyy + δ 2ψ 2 yxx ) −
ψ 2 y − M 2 2ψ 2 y ,
(
Da
ε
µ
µ 2
δ ψ 2 x , h1 ≤ y ≤ h
− ψ 2 xψ 2 xy ) = p2 y + δ 2 (ψ 2 xyy + δ 2ψ 2 xxx ) −
ε
Da
(9)
Re δρ (ψ 2 yψ 2 yx − ψ 2 xψ 2 yy ) = − p2 x +
Re δ ρ (ψ 2 yψ 2 xx
3
(10)
where Re is the Reynolds number, Da is the Darcy number and M is the magnetic parameter. The subscripts x and y
denote the partial differentiation with respect to that variable.
Under the assumptions of negligible inertia ( Re → o ) and long wavelength ( δ << 1 ) and eliminating pressures
p1 and p2 from the equations (9) and (10) by cross differentiations, the equations governing the motion are
ψ 1 yyyy − M 12ψ 1 yy = 0
in 0 ≤
y ≤ h1
(11)
ψ 2 yyyy − β ψ 2 yy = 0 in h1 ≤ y ≤ h
ε
ε
2
2
2
2
Where β = α + M 2 and α =
µ
Da
2
(12)
The corresponding dimensionless boundary conditions are
ψ 2 y = −1, ψ 2 = q
ψ 1 yy = 0, ψ 1 = 0
At the interface
y = h ( x)
at
at
(13)
y=0
(14)
y = h1 ( x ) :
ψ 1 = ψ 2 = q1
ψ1y = ψ 2 y
(15)
(16)
ψ 1 yy = µε (ψ 2 yy − β1ψ 2 y )
(17)
ψ 1 yyy = µε (ψ 2 yyy − α 2ψ 2 y )
where q and
µε =
µ
ε
(18)
q1 in (13) and (15) are the total and core fluxes respectively,
and
β1 =
βε
Da
In (13) the first condition at y=h(x) is the no-slip condition. The second at
y = h ( x ) in (13) and the conditions at
y = h1 ( x ) in (15) are the requirement of conservations of mass in core as well as in the peripheral layer
independently across any cross section in the wave frame. The conditions in (14) imply that the velocity attains a
maximum on the streamline y = 0 . The continuity of velocity across interface is given by (16). The shear stress
jump condition (17) is according to Ochoa-Tapia and Whitaker [14]. The continuity of normal stress implies
p1 = p2 , which means the continuity of the pressure gradient
∂p1 ∂p2
=
at the interface and it reduces to (18).
∂x
∂x
Hence the pressure remains constant across any cross section of the channel. We observe that the above governing
equations and the corresponding boundary conditions reduce to those given by Brasseur et al. [3] in the limit
Da → ∞ and M i → 0 .
SOLUTION OF THE PROBLEM
Solving equations (11) and (12) with the use of corresponding boundary conditions (13)-(18), we obtain the stream
functions in the core and peripheral layers as
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ψ 1 = c2 sinh M 1 y −
1
c1 y,
M 12
0 ≤ y ≤ h1 ,
ψ 2 = D3 cosh β y + D4 sinh β y −
1
β2
( D1 y + D2 ) ,
(19)
h1 ≤ y ≤ h,
(20)
Where
1
1
( c2 L2 + D1 L3 − D3 L5 − D4 L4 ) , c2 = ( − D1 L20 − D2 L21 − D3 L22 − D4 L23 )
L1
L19
1
1
D1 =
( −qL1 L7 − D4 L35 − D3 L34 ) , D2 = ( − D1 L29 + D3 L30 + D4 L31 )
L33
L32
c1 =
L ( qL L L + L )
1
( qL1 L7 L24 − D4 L37 ) , D4 = 24 1 7 27 36
L36
L27 L37 − L28 L36
1
1
L1 =
, L2 = M 1 cosh M 1 h1 , L3 = 2 , L4 = β cosh β h1 , L5 = β sinh β h1 , L6 = h L3
2
M1
β
D3 =
β1
2
, L9 = µε ( ββ1 sinh β h1 − β cosh β h1 )
2
β
2
L10 = µε ( ββ1 co sh β h1 − β sinh β h1 ) , L11 = β sinh β h , L12 = β co sh β h
L7 = M 12 sinh M 1 h1 , L8 = µε
L13 = M 13 co sh β h1 , L14 = µε
α2
2
2
, L15 = µε β ( β − α ) sinh β h1
β2
L16 = µε β ( β 2 − α 2 ) co sh β h1 , L17 = h1 L1 ,
L18 = h1 L3 , L19 = L1 sinh M 1 h1 − L2 L17
L20 = L1 L18 − L3 L17 , L21 = L1 L3 , L22 = L5 L17 − L1 cosh β h1 , L23 = L4 L17 − L1 sinh β h1
L24 = L7 L14 − L8 L13 , L25 = L7 L15 + L9 L13 , L26 = L7 L16 + L10 L13 , L27 = L3 L25 + L11 L24
L28 = L3 L26 + L12 L24 , L29 = L8 L19 + L7 L20 , L30 = L9 L19 − L7 L22 , L31 = L10 L19 − L7 L23
L32 = L7 L21 = L7 L1 L3 , L33 = L1 L7 L6 − L29 , L34 = L30 − L1 L7 cosh β h ,
L35 = L31 − L1 L7 sinh β h , L36 = L25 L33 − L24 L34 , L37 = L26 L33 − L24 L35
Using the relations (19) and (20) in the momentum equations (9) or (10) we obtain the pressure gradient as
dp
1
= c1 = ( c2 L2 + D1 L3 − D3 L5 − D4 L4 )
dx
L1
(21)
At any axial station, the non-dimensional flux Q in the fixed frame is connected with the flux q in the wave frame by
h
Q = ∫ (u + 1) dy = q + h
(22)
0
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=
The mean flow rate over one period ( T
T
λ
c
) of the peristaltic wave is given by
T
1
1
Q = ∫ Q dt = ∫ (q + h( x, t )) dt
T 0
T 0
(23)
1
= q + ∫ h dx = q + 1
0
The peristaltic wave propagating on the walls in fixed frame is govern by
H ( X , t ) = a + b sin
h( x) = 1 + φ sin 2π x .
its non-dimensional form in wave frame is
2π
λ
( X − ct ) , and
The interface between the porous and fluid regions which is not known ‘a priori’, and it should be a streamline in
order to satisfy the conservation of mass in both the regions. From the equations (15) and (20), we find an equation
governing the interface h1 ( x ) as
f ( h1 ) = q1 − D3 cosh β h1 − D4 sinh β h1 +
1
β2
( D1 h1 + D2 ) = 0
(24)
q and q1 are independent of x . By prescribing h1 = γ at x = 0 in (24), we get
1
q1 = E3 cosh βγ − E4 sinh βγ + 2 ( E1γ + E2 )
The constants
β
Where
((
)
1
− Q − 1 S1 S7 − E4 S35 − E3 S34
S33
1
E2 =
( − E1 S29 + E3 S30 + E4 S31 )
S32
1
E3 =
Q − 1 S1 S7 S 24 − E4 S37
S36
E1 =
E4 =
S 24
(( )
((Q − 1) S S S
1
7
)
)
27
+ S36
)
S 27 S37 − S 28 S36
S1 = L1 , S 2 = M 1 cosh M 1γ , S3 = L3
S 4 = β cosh βγ , S5 = β sinh βγ , S6 = S3
S7 = M 12 sinh M 1γ , S8 = L8
S10 = µε ( ββ1 co sh βγ − β sinh βγ )
2
S15 = µε β ( β − α
2
S16 = µε β ( β 2 − α 2
S17 = γ S1 ,
) sinh βγ
) co sh βγ
S 22 = S5 S17 − S1 cosh βγ
S 23 = S 4 S17 − S1 sinh βγ
S 24 = S7 S14 − S8 S13
S 25 = S7 S15 + S9 S13
S 26 = S7 S16 + S10 S13
S 27 = S3 S 25 + S11 S 24
S 28 = S3 S 26 + S12 S 24
S 29 = S8 S19 + S7 S 20
S31 = S10 S19 − S 7 S 23
S32 = S7 S 21 = S7 S1 S3
S33 = S1 S 7 S 6 − S 29
S11 = β sinh β , S12 = β co sh β
2
S 20 = S1 S18 − S3 S17 , S 21 = S1 S3
S30 = S9 S19 − S7 S 22
S9 = µε ( ββ1 sinh βγ − β 2 cosh βγ )
S13 = M 13 co sh βγ , S14 = µε
S19 = S1 sinh M 1γ − S 2 S17
α
= L14
β2
2
S34 = S30 − S1 S7 cosh β
S35 = S31 − S1 S7 sinh β
S36 = S 25 S33 − S 24 S34
S37 = S 26 S33 − S 24 S35
S18 = γ S3
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We assume that γ > φ to avoid the intersection of the interface with x-axis (Mishra and Ramachandra Rao, [12]).
Since the equation (24) is a transcendental equation and it may have many positive real roots. But the interface is
well defined only when there exists a single root
given x,
h1 of f ( h1 ) = 0 in the interval 0 ≤ h1 ≤ h . In this way for a
h1 (x) is computed. Integrating equation (21) over one wave length, we get the pressure rise as
∆P = p1 − p2
1
=∫
0
1
( c2 L2 + D1 L3 − D3 L5 − D4 L4 ) dx
L1
(25)
Fig.2: The Shape of the interface for different µ with fixed M = 0.5, M =1, γ = 0.6, φ = 0.4 , ε = 0.5,Da=0.5, Q = 0.5.
1
2
Fig.3: The Shape of the interface for different µ with fixed M =1.5, M =1, γ = 0.6, φ = 0.4 , ε = 0.5,Da=0.5, Q = 0.5.
1
2
Fig.4: The Shape of the interface for different M 1 with fixed M =1, γ = 0.6, φ = 0.4 , ε = 0.5, Da=0.5, µ = 2, Q = 0.5.
2
Fig.5: The Shape of the interface for different M 2 with fixed M =1, γ = 0.6, φ = 0.4 , ε = 0.5, Da=0.5, µ = 2, Q = 0.5.
1
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Fig.6: The Shape of the interface for different φ with fixed M = 0.5, M =1, γ = 0.6 , µ = 2, ε = 0.5,Da=0.5, Q = 0.5.
1
2
Fig.7: The Shape of the interface for different φ with fixed M =1.5, M =1, γ = 0.6 , µ = 2, ε = 0.5,Da=0.5, Q = 0.5.
1
2
Fig.8: Variation of pressure rise versus mean Flow rate for different µ ( M = 0.5, M =1 ).
1
2
Fig.9: Variation of pressure rise versus mean flow rate for different µ ( M = 1.5, M =1 ) with fixed γ = 0.4 , φ = 0.6 ,
1
2
ε = 0.4 , Da=0.6.
RESULTS AND DISCUSSION
The interface which is a stream line in the wave frame is obtained from (24) and plotted in figures (2) - (7) to study
the effects of different parameters on the shape of interface. From figures (2) and (3) we observe that the thickness
of the peripheral layer decreases slightly in the dilated region with the increase in viscosity ratio. From figures (4)
and (5) it is noticed that lower magnetic field gives rise to a thin peripheral layer in the dilated region. The uniform
shape is never obtained. From (6) and (7) we observe that the thickness of the peripheral layer decreases in the
dilated region with the increase in φ .
The equation (25) gives the expression for the pressure rise
∆P in terms of the mean flow Q . Figures (8) and (9)
shows that the pressure rise decreases with the increase in Q . We find that for fixed Q pressure rise increases
with increasing
µ . We also notice that for a given ∆P
mean flow rate increases with the increase in
µ.
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Fig.10: Variation of pressure rise versus mean flow rate for different M 1 ( M =1).
2
Fig.11: Variation of pressure rise versus mean flow rate for different M 2 ( M =1) with fixed γ = 0.4, φ = 0.6 , ε = 0.4 ,
1
Da=0.6, µ = 0.6.
Fig.12: Variation of pressure rise versus mean flow rate for different φ
Fig.13: Variation of pressure rise versus mean flow rate for different φ with fixed M =1.5, M =1, γ = 0.4 , µ = 0.6, ε = 0.4 ,
1
2
Da=0.6.
The variation of pressure rise with the mean flow for different values of
that for fixed Q pressure rise increases with the increase in
increases with increasing
M 1 is shown in figure (10). We observe
M 1 . It is noticed that for a given ∆P mean flow rate
M1 .
The variation of pressure rise with the mean flow for different values of
M 2 is shown in figure (11). We observe
that the pumping curves are intersect at appoint between 0.05 and 0.1, above this point for fixed Q pressure rise
increases with the increase in
M 2 and below the intersecting point opposite behavior can be observed.
The variation of pressure rise with the mean flow for different values of
notice that for fixed Q pressure rise increases with increasing
rate increases with increasing
φ.
φ
is shown in figures (12) and (13). We
φ . It is observed that for a given ∆P
mean flow
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