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International Journal of Advances in Applied Mathematics and Mechanics
Velocity slip and joule heating effects on MHD peristaltic flow in a porous medium
Research Article
K. Venugopal Reddy, M. Gnaneswara Reddy∗
Department of Mathematics, Acharya Nagarjuna University Campus, Ongole-523 001, A.P., India
Received 13 November 2014; accepted (in revised version) 30 December 2014
Abstract: The effects of velocity slip and Joule heating on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties have been studied under the assumptions of longwavelength and low-Reynolds number.
The analytical solution has been derived for the stream function, temperature and heat transfer coefficient. The emerging flow parameters on the velocity, temperature and heat transfer coefficient are presented graphically and are discussed in detail.
MSC: 76S05• 35Q35
Keywords: Peristalsis• Slip flow • MHD • Joule Heating • Non-uniform channel.
2014 IJAAMM all rights reserved.c
1. Introduction
To the best of our knowledge, no investigation has been made yet to analyze the effects of velocity slip and Joule Heating on MHD Peristaltic flow in a Porous medium with wall properties. The purpose of the present paper is to provide such an attempt for a MHD Newtonian fluid through porous medium with Joule heating at convective wall conditions. The effects of different physical parameters are obtained and their salient features are discussed through graphs.
Peristaltic transport is a form of material transport induced by a progressive wave of area contraction or expansion along the length of a distensible tube, mixing and transporting the fluid in the direction of the wave propagation.
This phenomenon is known as peristalsis. It plays an indispensable role in transporting many physiological fluids in the body in various situations such as urine transport form kidney to bladder, the movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ductus efferentes of the male reproductive tract, movement of ovum in the fallopian tubes, swallowing of food through esophagus and the vasomotion of small blood vessels.
Many modern mechanical devices have been designed on the principle of peristaltic pumping to transport fluids without internal moving parts, for example, the blood pump in the heart-lung machine and peristaltic transport of noxious fluid in nuclear industry. In the past, several theoretical and experimental investigations have been made just to understand peristalsis in different situations. The literature on this topic is quite extensive. Mention may be made to some recent contributions[1–8]in the field for Newtonian and non-Newtonian fluids.
The magnetohydrodynamic (MHD) flow of a fluid in a channel with peristalsis is of interest connection with certain problems of the movement of conductive physiological fluids, e.g., the blood, blood pump machines and with the need for theoretical research on the operation of peristaltic MHD compressor. More recently different authors have investigated on the fields of MHD. One author Ebaid[9]studied the effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel and another author Srinivas et al[10]
∗ Corresponding author.
E-mail address:[email protected]
have investigated, the Influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls.
In several applications the flow pattern corresponds to a slip flow, the fluid presents a loss of adhesion at the wetted wall making the fluid slide along the wall, when the molecular mean free path length of the fluid is comparable to the distance between the plates as in Nano channels of micro channels. More recently, different authors have investigated MHD with wall properties from[11–17].Recently, Srinivas et al [18] have analyzed the influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport.
2. Mathematical Formulation
Consider the flow of a Newtonian viscous fluid through a two-dimensional channel of uniform thickness. The walls of the channel are assumed to be flexible and are taken as a stretched membrane, on which traveling sinusoidal waves of moderate amplitude are imposed.
The geometry of the channel wall is given by
y= η(x , t ) = d (x ) + a sin2π
λ(x − c t ) (1)
where d(x ) = d + m0x , m0<< 1
∂ u
∂ x +∂ v
∂ y = 0 (2)
ρ∂ u
∂ t + u∂ u
∂ x + v ∂ u
∂ y
= −∂ p
∂ x + µ
∂2u
∂ x2+∂2u
∂ y2
− σ B02u−µ
ku (3)
ρ
∂ v
∂ t + u∂ v
∂ x + v ∂ v
∂ y
= −∂ p
∂ y + µ
∂2v
∂ x2+∂2v
∂ y2
−µ
kv (4)
ζ∂ T
∂ t + u∂ T
∂ x + v ∂ T
∂ y
=k ρ
∂2T
∂ x2+∂2T
∂ y2
+ v
2
∂ u
∂ x
2
+∂ v
∂ y
2 +∂ u
∂ y +∂ v
∂ x
2
+ σ B02u2 (5)
Where u , v are the components of velocity along x and y directions respectively,ρ is the density, µ is the coefficient of viscosity of the fluid, p is pressure, d is the mean half width of the channel , a is the amplitude,λ is the wave- length , c is the phase speed of the wave, m0is the dimensional non-uniformity of the channel ,σ is the electrical conductivity of the fluid, Bo is the applied magnetic field,ζ is the specific heat at constant volume , γ is kinematic viscosity of the fluid, k is thermal conductivity of the fluid and T is temperature of the fluid
The governing equation of motion of the flexible wall may be expressed as
L∗(η) = p − p0, (6)
where L∗is an operator, which is used to represent the motion of stretched membrane with viscosity damping forces such that
L∗= −τ ∂2
∂ x2+ m1
∂2
∂ t2+ C ∂
∂ t (7)
Continuity of stress at y = ±η and using x -momentum equation, yield
∂
∂ xL∗(η) =∂ ρ
∂ x = µ
∂2u
∂ x2+∂2u
∂ y2
− ρ
∂ u
∂ t + u∂ u
∂ x + v ∂ u
∂ y
− σB02u−µ
ku (8)
u= ∓h∂ u
∂ ya t y= ±η = ±
d+ m0x+ a sin2π
λ(x − c t )
(9)
T± β2
∂ T
∂ y =hT
1 T
i
a t y= ±η (10)
Where L∗= −τ∂ x∂22+ m1 ∂2
∂ t2+ C∂ t∂ . Hereτ is the elastic tension in the membrane, m is the mass per unit area, C is the coefficient of viscous damping forces, p0is the pressure on the outside surface of the Wall due to the tension in the muscles and h is the dimensional slip parameter. We assumed p0= 0.
Introducingψ such that u =∂ ψ∂ y,v= −∂ ψ∂ x and the following non-dimensional quantities
x0=x λ, y0=y
d,ψ0= ψ
c d, t0=c t
λ ,θ = T− T0
T1− T0
,η0=η
d, p0= d2
cλµp , K= k
d2 (11)
In eqs. (1) to (10) , we finally get (after dropping primes) Rδ
∂2ψ
∂ t ∂ y +∂ ψ
∂ y
∂2ψ
∂ x ∂ y −∂ ψ
∂ x
∂2ψ
∂ y2
= −∂ p
∂ x + δ2 ∂3ψ
∂ x2∂ y +∂3ψ
∂ y3− M2∂ ψ
∂ y − 1 K
∂ ψ
∂ y (12)
Rδ3
∂2ψ
∂ t ∂ y +∂ ψ
∂ y
∂2ψ
∂ x2 −∂ ψ
∂ x
∂2ψ
∂ x ∂ y
= −∂ p
∂ y + δ2
δ2∂3ψ
∂ x3+ ∂3ψ
∂ x ∂ y3
−δ2 K
∂ ψ
∂ y (13)
Rδ
∂ θ
∂ t +∂ ψ
∂ y
∂ θ
∂ x−∂ ψ
∂ x
∂ θ
∂ y
= 1 P r
δ2 ∂2
∂ x2+ ∂2
∂ y2
θ + E
¨ 4δ2
∂2ψ
∂ x ∂ y
2
+
∂2ψ
∂ y2 − δ2∂2ψ
∂ x2
2«
(14)
The corresponding boundary conditions are
∂ ψ
∂ y = ∓β∂2ψ
∂ y2a t y= ±η = ±[1 + m x + εsin2π(x − c t )] (15)
δ2 ∂3ψ
∂ x2∂ y+∂3ψ
∂ y3−R δ
∂2ψ
∂ t ∂ y +∂ ψ
∂ y
∂2ψ
∂ x ∂ y −∂ ψ
∂ x
∂2ψ
∂ y2
−M2∂ ψ
∂ y −1 K
∂ ψ
∂ y =
E1 ∂3
∂ x3+ E2
∂3
∂ x ∂ t2+ E3
∂2
∂ x ∂ t
(η) (16)
θ ± β2
∂ θ
∂ y =1
0
a t y= ±η (17)
whereε(=a d),δ(=d
λ) are geometric parameters, R(=c dρ
µ ) is the Reynolds number, the Hartman number M =
v tσ
µB0d , E1
= − τd3 λ3µc
, E2
=m1c d3 λ3µ
, E3
=C d3 λ2µ
are the non-dimensional elastic- ity parameters, P r = ρυζ
k is the Prandtl number,E = c2
ζ(T1− T0) is the Eckert number, m = λm0
d is non-uniform parameter,β1is the Knudsen number(Slip parameter) andβ2is the thermal slip-parameter.
3. Solution of the Problem
Using the long wavelength approximation and neglecting the wave number along with low-Reynolds number, one can find from Eqs.(12) to (16) that
0= −∂ p
∂ x +∂3ψ
∂ y3 − M2∂ ψ
∂ y − 1 K
∂ ψ
∂ y (18)
0= −∂ p
∂ y (19)
Eq.(18) shows that p is not function of y
∂2θ
∂ y2+ Br
∂2ψ
∂ y2
2
+ BrM2
∂ ψ
∂ y + 1
2
= 0 (20)
On differentiating Eq.(19) with respect to y, the compatibility equation as follows
∂4ψ
∂ y4 − N2∂2ψ
∂ y2 = 0 (21)
where N =q M2+K1 Eq.(16) gives
∂3ψ
∂ y3 − N2∂ ψ
∂ y =
E1 ∂3
∂ x3+ E2
∂3
∂ x ∂ t2+ E3
∂2
∂ x ∂ t
(η) (22)
The closed form solution for Eq.(21) with boundary conditions (15),(17) and (22) is
ψ = −8επ3(E1+ E2)cos2π(x − t ) −2πE3sin 2π(x − t )
N2 ×
sinh N y
N(coshN η + N β sinhN η)− y
(23) Substituting Eq.(23) into Eq.(20) and the temperature ,subject to the condition (18) is
θ =1
2[1 + L1∗ L2] + BrM2
η2+ L2
(L∗)2∗ L3+ L2η2− 4L2
L∗N2cosh(N η) +4L L∗ ∗ L4
− β2
(−L1∗ L5) − BrM2
2η + ( L2
(L∗)2∗ L6) + 2L2η − 4L2
L∗N(sinhN η) +4L L∗∗ L7
+ y
2(β + η)−L1 2 ∗ L8− BrM2 y2
2 + L2
2(L∗)2∗ L9+L2y2 2 − 2L2
L∗N2cosh(N y ) +2L L∗∗ L10
(24) where
L= −8"π3(E1+ E2)cos2π(x − t ) −2πE3sin 2π(x − t ) N2
L∗= cosh(N η) + N β sinh(N η) L1=BrL2N2
(L∗)2 , L2= cosh(2N η) 4N2 −η2
2
, L3= cosh(2N η) 4N2 +η2
2
, L4= cosh(N η) N2 −L∗η2
2
, L5=
sinh(2N η)
2N − η
, L6=
sinh(2N η)
2N + η
, L7=
sinh(N η) N − L∗
, L8= cosh(2N y )
4N2 −y2 2
, L9= cosh(2N y ) 4N2 +y2
2
, L10= cosh(N y ) N2 −L∗y2
2
, L11=
sinh(N η) N − L∗η
The coefficient of heat transfer at the wall is given by Z= ηxθy
Z= m + 2π" cos2π(x − t ) ∗ 1
2(β + η)−L1∗ L5
2 − BrM2
η +L2∗ L6
2(L∗)2 + L2η − 2L2
L∗N sinh(N η) +2L L∗ ∗ L11
4. Results and discussion
To study the behavior of the distributions of the axial velocity (u ), numerical calculations for several values of velocity slip parameter (β1), Thermal slip parameter (β2), Non-uniform parameter (m ) ,Permeability parameter (K ) and Hartmann number (M ) and Elasticity parameter E1, E2and E3are carried out.
The variation of the axial velocity (u ), Heat transfer (θ ) , Heat transfer coefficient ( Z ) and Streamlines studied with fixed constants are adopted for numerical computations unless specified in the graph
K = 2 , M = 1 , β1= 0.2 , β2= 0.2 , ε = 0.1 , x = 0.2 , t = 0.1, E1= 1 , E2= 0.5 , E3= 0.1 , Br= 2.0
The effect ofβ1on the velocity distribution can be seen through Fig.1. It reveals that the axial velocity increases with increasingβ1. The effect of M on the velocity distribution is observed from Fig.2.It is founded that the axial velocity decreases with increasing M .The effect of K on the velocity distribution is shown that from Fig.3.It gives that the axial velocity increases with increasing K .From Fig.4,it is clear that the velocity increases with an increase in E1and E2while it decreases with an increase in E3. From Fig.5, it depicts that the velocity for a divergent channel (m> 0)
Fig. 1. Effect of Axial Velocity (u ) against y with different values ofβ1
Fig. 2. Effect of Axial Velocity (u ) against y with different values of M
is higher compared to its value for a uniform channel ( m= 0 ) , where as it is lower for a convergent channel (m < 0).
The effects of heat transfer on peristalsis is illustrated in figs.6to10, numerical calculations for several values of slip parameters (β1) and (β2) ,Permeability parameter ( K ) and Hartmann number (M ) and wall parameters E1 , E2and E3 are carried out. The effect of velocity slip parameterβ1 on the Temperature distribution can be seen through Fig.6.It reveals that the temperature increases with increasingβ1. The effect of Thermal slip parameterβ2
on temperature distribution is observed from Fig.7.It is shown that the temperature increases with the increasing values ofβ2.The effect of Hartmann number M on the temperature distribution is observed from Fig.8.It is founded that the temperature decreases with increasing M .The effect of Permeability parameter K on the temperature distribution is shown that from Fig.9.It gives that the temperature increases with increasing K from Fig.10.It is clear that the temperature increases with an increase in E1and E2while it decreases with an increase in E3.
Fig. 3. Effect of Axial Velocity (u ) against y with different values of K
Fig. 4. Effect of Axial Velocity (u ) against y with different values of E1, E2a n d E3
The variation in coefficient of heat transfer Z for various values of the pertinent parameters can be analyzed through figs.11to14 .It is observed that due to peristalsis, the heat transfer coefficient is in oscillatory behavior.
The absolute value of heat transfer coefficient increases with increase of K while it decreases with increasing m and M .
An interesting phenomenon of peristaltic transport is trapping. The effect of M on trapping can be seen in Fig.15 We observe that the size of the bolus reduces with an increase in M .The effectof K on the trapping is analyzed in Fig.16. It reveals that the volume of the trapped bolus increases with increasing K and more trapped bolus with increasing K .To see the effects of E1, E2and E3on the stream lines, we have sketched Fig.17.
It depicts that an increase in E1, E2and E3results in the increase in the size of the trapped bolus and more trapped bolus appears. The damping nature of the wall ( E3) has very insignificant influence on the trapping. We observe that streamlines closed loops creating a cellular flow pattern in the channel and more trapped bolus appears with
Fig. 5. Effect of Axial Velocity (u ) against y with different values of m
Fig. 6. Effect of Temperature(θ ) against y with different values of β1
increasing slip parameter. The streamlines for uniform and non- uniform channels are shown in Fig.18.
It can be concluded that the size of trapped bolus in large in the left-hand side of the channel for convergent channel while it has opposite behavior for divergent channel. Further, the size of bolus is symmetric for uniform channel.
Fig. 7. Effect of Temperature(θ ) against y with different values of β2
Fig. 8. Effect of Temperature(θ ) against y with different values of M
Fig. 9. Effect of Temperature(θ ) against y with different values of K
Fig. 10. Effect of Temperature(θ ) against y with different values of E1, E2a n d E3
Fig. 11. Effect of Heat Transfer (Z ) against X with different values M
Fig. 12. Effect of Heat Transfer(Z ) against X with different values of K
Fig. 13. Effect of Heat Transfer(Z ) against X with different values of E1, E2a n d E3
Fig. 14. Effect of Heat Transfer(Z ) against X with different values of m
(a) M= 1 (b) M= 2 (c) M= 3
Fig. 15. Stream lines for various M
(a) K= 1 (b) K= 2 (c) K→ ∞
Fig. 16. Stream lines for various K
(a) E1= 1, E2= 0.5, E3= 0.1 (b) E1= 2, E2= 0.5, E3= 0.1 (c) E1= 1, E2= 0.5, E3= 1 Fig. 17. Stream lines for various E0s
(a) m= 0 (b) m= 0.2 (c) m= −0.2
Fig. 18. Stream lines for various m
5. Conclusions
The important findings of the present study are:
1. The velocity field increase with an increase in the velocity slip parameter..
2. Hartmann number cause to weaken the slip at the wall whereas the effect of the Permeability parameter is to strengthen the slip.
3. The absolute value of heat transfer coefficient increases with decreasingβ1,β2and M 4. The size of the trapped bolus increases with increasing K , E1, E2and E3
5. The results of the influence of slip conditions, wall properties and heat transfer on MHD Peristaltic transport can be captured as a limiting case of our analysis by choosingβ1= β2= 0 in the heat transfer.
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