series solutions of magnetohydrodynamic peristaltic flow ... · r. ellahi1,2,∗, a. riaz2, s....

Appl. Math. Inf. Sci.7, No. 4, 1441-1449 (2013) 1441

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/070424

Series Solutions of Magnetohydrodynamic PeristalticFlow of a Jeffrey Fluid in Eccentric Cylinders

R. Ellahi1,2,∗, A. Riaz2, S. Nadeem3, M. Mushtaq4

1Department of Mechanical Engineering, University of California Riverside, USA2Faculty of Basic and Applied Sciences, IIU Islamabad, Pakistan3Faculty of Natural Sciences, Quaid-i-Azam University Islamabad, Pakistan4Department of Mathematics, University of Engineering & Technology, Lahore, Pakistan

Received: 12 Dec. 2012, Revised: 1 Jan. 2013, Accepted: 2 Feb. 2013Published online: 1 Jul. 2013

Abstract: In this article, the mathematical modelling on magnetohydrodynamic peristalticflow of Jeffrey fluid in the gap between twoeccentric tubes has been discussed in the presence of applied magnetic field. Geometrically, we considered two eccentric tubes in whichthe inner tube is rigid while the tube at the outer side has a sinusoidal wave propagating along the wall. The governing equations forJeffrey fluid in a cylindrical coordinates for three dimensional flow aregiven. The approximations of low Reynolds number and longwavelength have been employed to reduce the highly nonlinear partial differential equations. The problem has been solved with thehelp of homotopy perturbation method alongwith eigen function expansion method. The graphs of pressure rise, pressure gradient andvelocity (for two and three dimensional flow) have been drawn. The streamlines have also been presented to discuss the trapping bolusdiscipline.

Keywords: Peristaltic flow, Eccentric cylinders, Jeffrey fluid, Magnetohydrodynamic (MHD),Nonlinear partial differential equations,Analytical solutions, HPM.

1 Introduction

Peristaltic pumping is a phenomenon in which fluidtransport happens when a gradually wave of areacontraction or expansion propagates along the length ofdistensible duct. It is an instinctive property of manybiological disciplines possessing smooth muscle tubeswhich helps in flowing biofluids by its propulsivemovements and is found in the transport of urine fromkidney to the bladder, the movement of chyme in thegastro-intestinal tract, intra-uterine fluid motion,vasomotion of the small blood vessels and in many otherglandular ducts. The mechanism of peristaltic transporthas been exploited for industrial applications like sanitaryfluid transport, blood pumps in heart lung machine andtransport of corrosive fluids where the contact of the fluidwith the machinery parts is prohibited. Peristaltictransport of a Newtonian fluid in a vertical asymmetricchannel with heat transfer and porous medium have beenanalyzed by Srinivas and Gayathri [1] . Kothandapani andSrinivas [2] have examined the peristaltic transport of a

Jeffrey fluid under the effect of magnetic field inasymmetric channel under the assumptions of longwavelength and low Reynolds number. A number ofanalytical and numerical studies [3,4,5,6] of peristalticflows of different fluids have been reported. Mekheimer[7] measured the effect of magnetic field on peristaltictransport of blood in a non-uniform two dimensionalchannel, when blood is characterized by a couple stressfluid. In recent years, peristaltic transport through aconcentric annulus has obtained much concentration. Alot of investigations are available in the literature to studythe effect of an endoscope on peristaltic motion ofNewtonian and non-Newtonian fluids [8,9,10,11,12] Theeccentric annulus is normally not easy to discuss evenwithout peristalsis. There are only a few studies take intoattention the effect of the eccentricity attribute [13,14].

To be more specific, only one study is available whichdiscusses the peristaltic flow of a viscous fluid througheccentric cylinders [15] To the best of author’sknowledge, the peristaltic flow of non-Newtonian fluidthrough the eccentric cylinders has not been explored so

∗ Corresponding author e-mail:[email protected], [email protected]

c© 2013 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/070424

1442 R. Ellahi et al. : Series Solutions of Magnetohydrodynamic...

for. Therefore, the main purpose of the present paper is todiscuss the effect of peristaltic transport on the Jeffreyfluid flow in the gap between two eccentric tubes, theouter tube gets the sinusoidal wave propagating down thewall and the inner tube is rigid. A motivation of thepresent work is the anticipation that such a problem willbe beneficial in many clinical applications. This analysisgives a better judgement for the speed of injection and thefluid flow characteristics within the syringe. Also, theinjection can be carried out more proficiently and pain ofthe patient can be extenuated.

2 Mathematical formulation of the problem

The MHD flow of an incompressible non-NewtonianJeffrey fluid is considered here. The flow geometry isdescribed as the inner tube is rigid and sinusoidal wave istravelling at the outer tube down its wall. The radius ofthe inner tube isδ but we require to consider the fluidmotion to the centre of the outer tube. The centre of theinner tube is now at the positionr = ε, z = 0, wherer andz are coordinates in the cross-section of the pipe as shownin the Fig. 1. The radially varying magnetic field isapplied normally to the direction of the flow. Then theboundary of the inner tube is described to orderε byr1 = δ + ε cosθ , whereε is the parameter that controlsthe eccentricity of the inner tube position. The geometryof the walls is visualized in Fig.1.

The equations for the radii are

r1 = δ + ε cosθ ,

r2 = a+bcos

[2πλ

(z− ct)

],

whereδ anda are the radii of the inner and outer tubes,bis amplitude of the wave,λ is the wavelength,c is thepropagation velocity andt is the time. the problem hasbeen considered in the system of cylindrical coordinates(r,θ ,z) as radial, azimuthal and axial coordinates,respectively.

The continuity and momentum equations for anincompressible Jeffrey fluid are described as follow

divV = 0, (1)

ρdVdt

=−∇p+divS+J×B, (2)

where ρ is the density, d/dt is the material timederivative,V is the velocity field,p is the pressure,S isthe stress tensor for Jeffrey fluid,J is the electric currentdensity,B is the total magnetic field,b is the body force.Note that considering the cylindrical coordinates systemfor the velocity fieldV = (v,w,u) and in the absence ofbody force, equations(1) and (2) correspondingly takethe following form

∂u∂ z

+∂v∂ r

+vr+

1r

∂w∂θ

= 0, (3)

Fig. 1: The simplified model of geometry of the problem.

ρ[

∂ v∂ t +u ∂v

∂ z + v ∂ v∂ r +

wr

∂v∂θ − w2

r

]=− ∂ p

∂ r +1r

∂∂ r (rSrr)+

1r

∂∂θ (Srθ )+

∂∂ z (Srz)−

Sθθr ,

(4)

ρ[

∂w∂ t +u ∂w

∂ z + v ∂w∂ r +

wr

∂w∂θ − wv

r

]=−1

r∂ p∂θ + 1

r2∂∂ r

(r2Srθ

)+ 1

r∂

∂θ (Sθθ )+∂∂ z (Sθz) ,

(5)

ρ[

∂u∂ t +u ∂u

∂ z + v ∂u∂ r +

wr

∂u∂θ

]=− ∂ p

∂ z +1r

∂∂ r (rSrz)+

1r

∂∂θ (Sθz)+

∂∂ z (Szz)−σB2

0(r)u,

(6)wherev, w andu are the velocity components inr, θ

andz−directions, respectively,µ is the viscosity,Srr, Srθ ,Srz, Sθθ , Sθz andSzz are stresses for Jeffrey fluid which canbe computed with the help of following stress [11]

S =µ

1+λ 1

(·γ +λ 2

··γ).

According to the flow geometry, the boundary conditionsare defined as

u = 0, at r = r2, (7)

u =V, at r = r1, (8)

whereV is the velocity of the inner tube. The velocitycomponent in the azimuthal direction is assumed to beunaffected, so the velocity field is taken as(v,0,u). Thegoverning equations are made dimensionless by using thefollowing non-dimensional parameters

p′= a2

µcλ p, u′= u

c , v′= λ

ac v, t′= c

λ t, r′

1 =r1a , φ = b

a , δ′= δ

a , ε ′= ε

a , δ 0 =aλ , z

′= z

λ ,

r′

2 =r2a , Re= ρca

µ , M2(r) =

σB20(r)a

2

µ , θ′= θ , V

′= V

c ,


Appl. Math. Inf. Sci.7, No. 4, 1441-1449 (2013) /www.naturalspublishing.com/Journals.asp 1443

whereφ is the amplitude ratio,M is the MHD parameter,Re is the Reynold’s number,δ 0 is the dimensionless wavenumber,ε is the eccentricity parameter. After using theabove dimensionless parameters, the governing equations(after dropping dashes) are reduced to the following form

∂v∂ r

+vr+

∂u∂ z

= 0, (9)

Reδ 30

[∂v∂ t +u ∂v

∂ z + v ∂v∂ r

]=− ∂ p

∂ r +δr

∂∂ r (rSrr)+

δr

∂∂θ (Srθ )+δ 2 ∂

∂ z (Srz)−δ Sθθr ,

(10)

0=−1r

∂ p∂θ + δ

r2∂∂ r

(r2Srθ

)+ δ

r∂

∂θ (Sθθ )+δ 2 ∂∂ z (Sθz) ,

(11)

Reδ 0

[∂u∂ t +u ∂u

∂ z

]=− ∂ p

∂ z +1r

∂∂ r (rSrz)+

1r

∂∂θ (Sθz)+δ ∂

∂ z (Szz)−M2(r)u.

(12)The components of non-dimensional stresses for Jeffreyfluid are evaluated as

Srr =2δ

1+λ 1

(1+

λ 2cδa

(∂∂ t

+ v∂∂ r

+u∂∂ z

))∂v∂ r

,

Srθ =δ

1+λ 1

(1+

λ 2cδa

(∂∂ t

+ v∂∂ r

+u∂∂ z

))1r

∂v∂θ

,

Srz =1

1+λ 1

(1+ λ 2cδ

a

(∂∂ t + v ∂

∂ r +u ∂∂ z

))(δ 2 ∂ v

∂ z +∂u∂ r

),

Sθθ =2δ

1+λ 1

(1+

λ 2cδa

(∂∂ t

+ v∂∂ r

+u∂∂ z

))vr,

Sθz =1

1+λ 1

(1+

λ 2cδa

(∂∂ t

+ v∂∂ r

+u∂∂ z

))1r

∂u∂θ

,

Szz =2δ

1+λ 1

(1+

λ 2cδa

(∂∂ t

+ v∂∂ r

+u∂∂ z

))∂u∂ z

.

(13)Using the long wavelength approximation (δ 0 → 0) andtaking M(r) = M

r , the governing equations(10−12) aresimplified to the following form

∂ p∂ r

= 0, (14)

∂ p∂θ

= 0, (15)

(1+λ 1)∂ p∂ z

=∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u

∂θ 2 − (1+λ 1)M2

r2 u

or

(1+λ 1)∂ p∂ z

=∂ 2u∂ r2 +

1r

∂u∂ r

+1r2

∂ 2u

∂θ 2 −N2

r2 u, (16)

whereN = M√

1+λ 1. Eqs. (14) and (15) show thatpis not a function ofr andθ . The corresponding boundaryconditions in non-dimensional form are

u = 0, at r = r2 = 1+φ cos[2π(z− t)] , (17)

u =V, at r = r1 = δ + ε cosθ . (18)

3 Solution of the problem

Solution of the above boundary value problem isevaluated by using homotopy perturbation method[16−17]. The homotopy equation for the given problemis defined as

H(u,q) = (1−q)(L [u]−L [u0])+q(L [u]+ 1

r2∂ 2u∂θ2 − (1+λ 1)

d pdz

)= 0,

(19)whereL , the linear operator is assumed to beL = ∂ 2

∂ r2 +1r

∂∂ r −

N2

r2 . We define the following initial guess

u0 =V sinh(N (log[r]− log[r2]))csch(N (log[r1]− log[r2])) .(20)

Now we describe

u(r,θ ,z, t,q) = u0+qu1+ ... (21)

Using the above equation into Eq.(19) and then taking theterms of first two orders, we get the following problemsalongwith corresponding boundary conditions

Zeroth order system

L [u0]−L [u0] = 0, (22)

u0 = 0, at r = r2, (23)

u0 =V, at r = r1. (24)

The solution of the above zeroth order system can beobtained by using Eq.(20) and is directly written as

u0 (r,θ ,z, t,q) = u0 =V sinh(N (log[r]− log[r2]))csch(N (log[r1]− log[r2])) .

(25)First order system

L [u1]+L [u0]+1r2

∂ 2u0

∂θ 2 − (1+λ 1)d pdz

= 0 (26)

or

∂ 2u1

∂ r2 +1r

∂u1

∂ r−

N2

r2 u1 = (1+λ 1)d pdz

−1r2

∂ 2u0

∂θ 2 , (27)

u1 = 0, at r = r2, (28)

u1 = 0, at r = r1. (29)

The solution of the above linear ordinary differentialequation is found as

u1 =1

4N(N2−4)

(csch

(N log

[r1r2

])((−4N d p

dz r22+ f

(N2−4

)log

[rr1r22

])×

sinh(

N log[

rr1

])− f

(N2−4

)log

[rr1

]sinh

(N log

[rr1r22

])+4N d p

dz

(r21 sinh

(N log

[rr2

])−

r2sinh(

N log[

r1r2

]))−4N d p

dz

(r22 sinh

(N log r

r1

)− r2

1 sinh(

N log rr2

)+

r2sinh

(N log

r1

r2

))λ 1

)), (30)


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where

f = 1(δ+ε cosθ)2

(NV ε csch

(N log

[δ+ε cosθ

r2

])((ε +δ cosθ)coth

(N(

log[

δ+ε cosθr2

]))+

Nε(

1+2csch2(

N

(log

[δ + ε cosθ

r2

])))sin2 θ

)).

Now for q → 1, we approach the final solution. So fromEq.(21) , we get

u(r,θ ,z, t) = u0+u1, (31)

whereu0 andu1 are defined in Eqs.(25) and (30) . Theinstantaneous volume flow rateQ(z, t) is given by

Q(z, t) = 2π∫ r2

r1

rudr, (32)

Q/2π = 1

4N(N2−4)2

((N2+4

)(2 f r2

2+N d pdz

(r41− r4

2

))+8N

(N2−4

)r21V −2 f

(N2+4

)r21×

cosh(

N log[

r1r2

])+2N csch

(N log

[r1r2

])(2Nr2

2

(2d p

dz r21+

(N2−4

)V)− f

(N2−4

)r21×

log[

r1r2

])−2N coth

(N log

[r1r2

])(2N

(d pdz

(r41+ r4

2

)+(N2−4

)r21V

)− f

(N2−4

)r22×

log[

r1r2

])+8 f Nr2

1 sinh(

N log r1r2

)+N d p

dz

((N2+4

)(r41− r4

2

)−4N

(r41+ r4

2

)coth

(N log r1

r2

)+

8Nr21r2

2 csch

(N log

r1

r2

))λ 1

). (33)

The mean volume flow rateQ over one period is given asMekheimer et. al. [15]

Q(z, t) = Qπ − φ2

2 +2φ cos[2π (z− t)]+φ2cos2[2π (z− t)],(34)

whereQ is the time average of the flow over one period ofthe wave. Now we can evaluate pressure gradientd p/dzby solving Eqs.(33) and(34) and is obtained as

d pdz =

(2(−4 f πr2

2+N(

2(N2−4

)2Q+π

(f Nr2

2−4(N2−4

)r21V

))+N

(N2−4

)2 πφ×

(4cos[2π (z− t)] +φ cos[4π (z− t)])+ f(N2+4

)πr2

1 cosh(

N log[

r1r2

])+

Nπ csch(

N log r1r2

)(2 f r2

1−2N(N2−4

)r22V + f r2

1

(2cosh

(2N log r2

r1

)+

(N2−4

)log r1

r2

)+(N2−4

)cosh

(N log

[r1r2

]) (2Nr2

1V + f r22 log

[r2r1

]))))/

(Nπ

((N2+4

)(r41− r4

2

)−4N

(r41+ r4

2

)coth

(N log

[r1r2

])+

8Nr21r2

2 csch

(N log

[r1

r2

])(1+λ 1)

). (35)

The pressure rise∆ p(t) in non-dimensional form isdefined as

∆ p(t) =∫ 1

0

∂ p∂ z

dz. (36)

4 Results and discussions

The analytical and numerical results obtained above forthe given analysis are discussed graphically in thissection. The comparison table and graph of the resultsfound in the present case with that of the previous one arepresented. The graphical treatment for the data ofpressure rise∆ p, pressure gradientd p/dz and velocity

profile u(r,θ ,z, t) with the variation of all emergingdimensionless parameters like timet, flow rate Q,amplitude ratioφ , the velocity of the inner tubeV, theeccentricity parameterε , Jeffrey fluid parameterλ 1 andthe MHD parameterM has been analyzed. In the end, thestream lines observing the peristaltic flow are drawn forthe parametersM, Q, λ 1 and φ while other parametersremain fixed.

Table.1 is shown to see the matching of results for thecurrent case and the previously discussed case.

Table 1: Variation of velocity distribution forr.Mekheimer et al. [15] Present work

r u(r,θ ,z, t) u(r,θ ,z, t) f orM = 0,λ 1 = 0 u(r,θ ,z, t) f orM = 0.5,λ 1 = 1.00.20 0.1000 0.1000 0.10000.25 0.1093 0.1081 0.09690.30 0.1119 0.1116 0.09440.35 0.1119 0.1120 0.09180.40 0.1096 0.1099 0.08870.45 0.1054 0.1057 0.08480.50 0.0995 0.0997 0.08010.55 0.0919 0.0920 0.07430.60 0.0829 0.0829 0.06740.65 0.0724 0.0723 0.05930.70 0.0606 0.0604 0.05010.75 0.0474 0.0472 0.03950.80 0.0329 0.0327 0.02770.85 0.0171 0.0170 0.01450.90 0.0000 0.0000 0.0000

The comparison graph for the values obtained inpresent work with the results of Mekheimer et. al. [15] isdisplayed in Fig.2. The graphs for the pressure rise∆ p(t) versus flow rateQ under the effects of givenparameters are drawn in Figs.3-7. These graphs show thepumping regions, that is, the peristaltic pumping(Q > 0, ∆ p > 0) , the augmented pumping(Q > 0, ∆ p < 0) and the retrograde pumping(Q < 0, ∆ p > 0) . The pressure gradientd p/dz againstthe the coordinatez with the variation of pertinentparameters are shown in Figs.8-12. The velocity fieldu(r,θ ,z, t) versus the radial coordinater is plotted inFigs. 13-17 for both two and three dimensions. Thestream line graphs are shown in Figs.18-21.

Fig. 2: Variation of velocity distribution withr for fixedδ = 0.1,θ = 0.01, φ = 0.1, z = 0, t = 0.5, V = 0.1, ε = 0.1, Q = 0.6.

It is observed from Fig.2 that the results obtained inthe present case are in good agreement with that of



previous calculations in most part of the geometry. It isalso observed that the presence of magnetic field forJeffrey fluid causes to slow down the flow.

Fig. 3: Variation of pressure rise∆ p with Q for fixed δ = 0.1,φ = 0.2, t = 0.1, M = 0.5, V = 0.5, λ 1 = 1.5.

Fig. 3 is plotted to see the variation of pressure risefor different values of the eccentricity parameterε and theangle θ while all other parameters are kept fixed. It isobserved that peristaltic pumping region is in betweenQ ∈ [0,0.9] , augmented pumping is inQ ∈ [0.9,2] andretrograde pumping part isQ ∈ [−1,0]. It is also observedfrom this graph that the pressure rise increases with thevariation ofε but decreases with the angleθ in betweenthe regionQ ∈ [−1,0.9] and opposite behavior is seen inthe remaining part. The graph of pressure rise for theparameterM andδ is plotted in Fig.4.

Fig. 4: Variation of pressure rise∆ p with Q for fixed ε = 0.01,φ = 0.2, t = 0.1, θ = 500, V = 0.5, λ 1 = 1.5.

The peristaltic pumping occurs in the regionQ ∈ [0,0.6] , augmented pumping is inQ ∈ [0.6,2] andretrograde pumping part isQ ∈ [−1,0] . It is clear that thesimilar behavior is seen in this case for the parameterMbut the opposite attitude is observed with the variation ofδ as compared to that ofε andθ . It tells that the flow ratedecreases withM while increases withδ , so this showsthat the back flow increases and decreases withM andδ ,respectively.

Fig. 5 shows that the peristaltic pumping part isQ ∈[0,0.3] , while augmented and retrograde pumping regionsareQ∈ [0.3,2] andQ∈ [−1,0] , respectively. The variation

Fig. 5: Variation of pressure rise∆ p with Q for fixed δ = 0.1,M = 0.5, t = 0.1, θ = 500, V = 0.5, λ 1 = 1.5.

of pressure rise∇p for V is similar to that ofM (See Fig.6).

Fig. 6: Variation of pressure rise∆ p with Q for fixed ε = 0.01,M = 0.5, t = 0.1, θ = 500, φ = 0.2, λ 1 = 1.5.

Fig. 7: Variation of pressure rise∆ p with Q for fixed ε = 0.01,M = 0.5, t = 0.1, θ = 500, φ = 0.2, V = 0.5, λ 1 = 1.5.

Fig. 7 indicates the effect of the parametersλ 1 andδupon pressure rise. This plot reveals that the peristalticpumping area lies in betweenQ ∈ [0,0.6] , the retrogradepumping appears in the partQ ∈ [−1,0] and theaugmented pumping region isQ ∈ [0.6,2] . The pressuregradientd p/dz for the parametersM and δ is drawn inFig. 8.

It is measured from this figure that pressure gradientis in linear relation with both of the parameters in




Fig. 8: Variation of pressure gradientd p/dz with z for fixed ε =0.01, V = 0.3, t = 0.3, θ = 500, φ = 0.1, Q = 0.5, λ 1 = 1.5.

narrowest parts of the cylinders but inverse relation isseen in the wider parts. The variation of pressure gradientwith the parametersφ andε is very much similar to thatof the parametersM andδ and is shown in Fig.9.

Fig. 9: Variation of pressure gradientd p/dz with z for fixed δ =0.1, V = 0.3, t = 0.3, θ = 500, M = 0.5, Q = 1, λ 1 = 1.5.

The only difference is that the pressure gradient isminimum on the left and right sides of the cylinder whileappears maximum at the centre. It means that flow caneasily pass without imposition of large pressure gradientin the two sides of the channel while much pressuregradient is required to maintain the flux in the central partnearz = 0.8. This is in good agreement with the physicalcondition.

Fig. 10: Variation of pressure gradientd p/dz with z for fixedε = 0.01,V = 0.3, t = 0.3, θ = 500, M = 0.5, φ = 0.1, λ 1 = 0.5.

Fig. 11: Variation of pressure gradientd p/dz with z for fixedε = 0.01, Q = 0.5, t = 0.3, θ = 500, M = 0.5, φ = 0.1, λ 1 = 0.5.

It can be observed from Figs.10 and 11 that thepressure gradient increases with the parametersQ andV ,while whenδ is increased the pressure gradient decreaseson the left and right sides but increases at the centre of thecylinders. It is also seen that the variation of pressuregradient remains same in the two sides of the channel andbecome different at the central part with changingV butthis variation remains same throughout for the parameterQ. The pressure gradient graph for the parametersλ 1 andδ is drawn in the Fig.12.

Fig. 12: Variation of pressure gradientd p/dz with z for fixedε = 0.01, Q = 0.5, t = 0.3, θ = 500, M = 0.5, φ = 0.1, V = 0.3.

It is seen here that the pressure gradient increases withδ at the middle but decreases at the two sides of thecylinders. However, the effect of the parameterλ 1 istotally opposite with that ofδ .

Fig. 13: Variation of velocity profileu with r for fixed ε = 0.1,Q = 0.5, t = 0.5, z = 0, V = 0.1, θ = 500, φ = 0.1, λ 1 = 1.5.



The Fig. 13 shows that the velocity field is anincreasing function of the parameterδ while decreasingwith the parameterM. The velocity field is in inverserelation withQ but have a direct variation withε (See Fig.14).

Fig. 14: Variation of velocity profileu with r for fixed δ = 0.1,M = 0.5, t = 0.5, z = 0, V = 0.1, θ = 500, φ = 0.1, λ 1 = 1.5.

Fig. 15: Variation of velocity profileu with r for fixed Q = 0.1,M = 0.5, ε = 0.01, z = 0, V = 0.1, θ = 500, λ 1 = 1.5.

It is observed from Fig.15that the velocity distributionis increasing withδ andφ while reducing fort.

Fig. 16: Variation of velocity profileu with r for fixed Q = 0.1,M = 0.5, t = 0.5, ε = 0.1, z = 0, δ = 0.2, θ = 500, λ 1 = 1.5.

Fig. 16 shows that the velocity profile is linearlychanging withφ andV .

From Fig.17, it is measured that velocity is lessenedwith the increasing effect of the parameterλ 1. It is alsoobserved that velocity is decreasing function ofδ in theregion r ∈ [0.2,0.4], while increasing on the rest of thedomain.

Fig. 17: Variation of velocity profileu with r for fixed Q = 0.5,M = 0.5, t = 0.5, ε = 0.1, z = 0, θ = 500, φ = 0.1, V = 0.1.

Fig. 18: Stream lines for different values ofM. (a) for M = 0.3,(b) for M = 0.4, (c) for M = 0.5, (d) for M = 0.6. The otherparameters areε = 0.4, V = 0.3, t = 0.2, θ = 500, φ = 0.05,Q = 0.6, δ = 0.05, λ 1 = 1.

Fig. 18 is drawn to see the stream lines for theparameterM. It is measured from this figure that numbersof bolus are not changing but size is increasing with theincreasing effects ofM in the bottom of the cylinder,while bolus are lessened in number when seen in theupper part. The boluses are reduced both in size andnumber when seen for the parameterQ in both parts ofthe geometry (see Fig.19).

Fig. 19: Stream lines for different values ofQ. (a) for Q = 0.6,(b) for Q = 0.7, (c) for Q = 0.8, (d) for Q = 0.9. The otherparameters areε = 0.4, V = 0.3, t = 0.2, θ = 500, φ = 0.05,M = 1, δ = 0.05, λ 1 = 1.




Fig. 20: Stream lines for different values ofφ . (a) for φ = 0.05,(b) for φ = 0.1, (c) for φ = 0.15, (d) for φ = 0.2. The otherparameters areε = 0.4,V = 0.3, t = 0.2, θ = 500, Q = 0.6, M =0.4, δ = 0.05, λ 1 = 1.

It is seen from Fig.20 that the numbers of bolus aredecreasing with different values of the parameterφ inboth sides of the cylinder but in the lower half of thechannel, the bolus becomes smaller with increasingmagnitude of the parameterφ . Fig. 21 reveals the fact thatwhen we increase the value of the parameterλ 1, theboluses decreased in number but expanded in size.

Fig. 21: Stream lines for different values ofλ 1. (a) for λ 1 = 0.5,(b) for λ 1 = 0.7, (c) for λ 1 = 0.9, (d) for λ 1 = 1.1. The otherparameters areε = 0.4,V = 0.3, t = 0.2, θ = 500, Q = 0.6, M =1, δ = 0.05, φ = 0.05.

Acknowledgment

R. Ellahi would like to thanks Higher EducationCommission of Pakistan and PCST to award him withNRPU and Productive Scientists of Pakistan Awards. A.Riaz also thanks to Higher Education Commission ofPakistan to award him Indigenous Scholarship for hisPHD studies.

References

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[2] M. Kothandapani and S. Srinivas, Peristaltic transport ofa Jeffrey fluid under the effect of magnetic field in anasymmetric channel, Int. J. Nonlinear. Mech.,43, 915 (2008).

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Dr. R. Ellahi is aFulbright Fellow in theUniversity Of CaliforniaRiverside, USA and alsofounder Chairpersonof the Department ofMathematics and Statisticsat IIUI, Pakistan. He hasseveral awards and honors onhis credit: Fulbright Fellow,

Productive Scientist of Pakistan, Best University TeacherAward by higher education commission (HEC) Pakistan,Best Book Award, and Valued Reviewer Award byElsevier. He is also an author of six books and editor oftwo international journals.

Mr. Arshad Riazis working as a PhDScholar in the department ofMathematics at InternationalIslamic University, Islamabadunder the supervisionof Dr. Rahmat Ellahi.He obtained his MS degreeswith distinction and publishedhis research articles in

internationally refereed journals.

Dr. Sohail Nadeemis an Associate Professorat Quaid-i-AzamUniversity Islamabad. Heis recipient of Razi-ud-Dingold medal by PakistanAcademy of Sciencesand Tamgha-i-Imtiaz by thePresident of Pakistan. He isyoung fellow of TWAS, Italy.

Dr. M. Mushtaqis an Assistant Professorin the Department ofMathematics, University ofEngineering & Technology,Lahore Pakistan. His fieldof research is ComputationalFluid Dynamics. His researcharticles have been publishedin national and international

journals.



series solutions of magnetohydrodynamic peristaltic flow ... · r. ellahi1,2,∗, a. riaz2, s....

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