The impact of compliant walls on magneto hydrodynamics

Scientia Iranica B (2018) 25(2), 741{750

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

The impact of compliant walls on magnetohydrodynamics peristalsis of Je�rey material in acurved con�guration

T. Hayata,b, S. Farooqa;�, and B. Ahmadb

a. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan.b. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

Received 23 July 2016; received in revised form 9 December 2016; accepted 28 August 2017

KEYWORDSJe�rey uid;Radial magnetic �eld;Compliant wallconditions;Soret and dufoure�ects;Curved channel.

Abstract. The primary aim of the current attempt is to analyze the peristaltic ow of non-Newtonian material in a curved channel subject to two salient features,namely the Soret and Dufour and radial magnetic �eld. Channel walls are of compliantcharacteristics. The problem formulations for constitutive equations of Je�rey uid aremade. The lubrication approach is implemented to simplify the mathematical analysis.Dimensionless problems of stream function, temperature, and concentration are computednumerically. Characteristics of distinct variables on the velocity, temperature, coe�cientof heat transfer, and concentration are examined. Besides, the graphical results indicatethat the velocity pro�le enhances the compliant wall parameters signi�cantly, primarily dueto the resistance characteristics of Lorentz force velocity pro�le decays. Furthermore, itis noted that the temperature pro�le enhances larger Dufour number; however, reversebehavior is noticed in the concentration pro�le when Soret and Schmidt numbers areincreased.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

The peristalsis regarding ows of viscous and inviscidmaterials has attracted the attention of recent re-searchers in view of their physiological and engineeringapplications. Physiological processes involve peristalsisconsisting of urine movement through kidney to blad-der, chyme transportation in intestine, spermatozoatransport, food swallowing via esophagus, and vasomo-tion in tiny blood vessels. Peristalsis is also quite preva-lent in heart-lung machine, roller and �nger pumps,sanitary and toxic liquid transport, and food, paper,

*. Corresponding author. Tel.: + 92 51 90642172;E-mail address: [email protected] (S. Farooq)

doi: 10.24200/sci.2017.4322

and cosmetic industries. Pioneering research studieson peristalsis involving viscous materials were theoret-ically and experimentally analyzed by Latham [1] andShapiro et al. [2]. Afterwards, ample investigationshave been conducted into peristalsis subject and thediverse aspects of rheological characteristics, heat andmass transfer, magneto hydrodynamics, wave shapes,geometries, etc. (see [3-21] and many studies therein).Much importance in the past was directed to theperistalsis of Newtonian and non-Newtonian liquids ina straight channel which seems not realistic in severalapplications relevant to physiological and engineeringprocesses. In addition, the Magneto Hydro Dynamic(MHD) e�ect, in the existing attempts, is mostlythrough the consideration of applied constant magnetic�eld. However, there is no doubt that MHD ows areof great interest in the MHD power generators, pumps,

742 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 741{750

accelerators, hyperthermia, bleeding minimization dur-ing surgeries, treatment of cancer tumor, blockagetreatment in the arteries, magnetic endoscopy, Mag-netic Resonance Imaging (MRI), puri�cation of moltenmetals from non-metallic inclusions, and several others.Keeping in mind the above-mentioned uses of MagnetoHydro Dynamic (MHD), Ellahi et al. [22] reported Ionslip and Hall aspects in MHD peristalsis of Je�rey uidin an irregular rectangular duct. In addition, Hayatet al. [23] described the MHD peristalsis in Je�reymaterial through curved channel with convective con-straints. Few recent developments in MHD peristaltic ow can be seen through [24-27]. Hence, the objectiveof the present communication is to predict the in uenceof radial magnetic �eld on the peristalsis of Je�reymaterial in a channel. Plus, the e�ect of curvatureis studied. Moreover, the compliant wall properties ofcurved channel are analyzed. Heat and mass transferare examined in the presence of Dufour and Soretphenomenon. Dimensionless problems subject to lubri-cation approach are studied. The outcome in discussionsection consists of salient features of sundry parametersentering into the formulation. This work is structuredas follows.

The �rst section contains introduction. Sections 2and 3 consist of formulation and technique for the so-lutions. Section 4 discusses various pertinent variables.Conclusions are given in Section 5.

2. Problems formulation

In Figure 1, we consider the peristaltic ow of anincompressible Je�rey liquid in curved channel of awidth of 2a0, twisted in a circle of radius, R�, and acenter at O. We consider �v1 as the velocity along radial(�r) and �v2 as the velocity along axial (�x) directions.A radially imposed magnetic �eld of strength, B0, isapplied along radial direction. In addition, heat andmass transfer is studied. Mathematical analysis alsoinvolves the Soret and Dufour e�ects.

The channel walls' temperatures are denoted byT0 and T1. The concentrations at the walls are C0

Figure 1. Physical diagram.

and C1. The uid ow here is due to propagation ofthe channel walls. Mathematical description of wallsurfaces is as follows:

�h(�x; �t) = a0 + a1 sin

2��c

(�x� c�t)!: (1)

Here, c represents the wave speed, a0 the half width ofthe channel, a1 the wave amplitude, �c the wavelength,and �t the time in �xed frame. Velocity, �V, can bede�ned as follows:

�V = (�v1(�x; �r; �t); �v2(�x; �r; �t); 0): (2)

extra stress tensor �S for Je�rey uid model is given by[9,10,20-23,28]:

�S =�

1 + �1(�A1 + �2

dd�t

�A1): (3)

In the above relation, �, �1, and �2 denote dynamicviscosity, ratio of relaxation to retardation time, andretardation time, respectively. The present study canbe reduced to viscous material when �1 = �2 = 0.Moreover:

�A1 = (grad �V) + (grad �V)�; (4)

dd�t

�A1 =@@�t

(�A1) + ( �V:r)�A1: (5)

The ow under consideration is governed by the follow-ing expressions:

@@�rf(R� + �r)�v1g+R� @�v2

@�x= 0; (6)

��@�v1

@�t+ �v1

@�v1

@�r+

R��v2

�r +R�@�v1

@�x� �v2

2�r +R�

�=

� @�p@�r

+1

R� + �r@@�r

�(�r +R�) �Srr

�+

R�

�r +R�

!@ �Srx@�x� �Sxx

�r +R� ; (7)

��@�v2

@�t+ �v1

@�v@�r

+R��v2

�r +R�@�v@�x� �v1�v2

�r +R��

=

��

R��r+R�

�@�p@�x

+1�

�r+R��2

@@�r

��r+R�

�2�Srx�

+�

R��r+R�

�@ �Sxx@�x� �B2

0�r +R� �v2;

(8)

T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 741{750 743

�cp�@T@�t

+ �v1@T@�r

+�v2R�

�r +R�@T@�x

�=��@2T@�r2 +

1R� + �r

@T@�r

+R�2

(�r +R�)2@2T@�x2

�+ �

��@�v2

@�r� �v2

�r +R� +R�

�r +R�@�v1

@�x

�Srx+

@�v2

@�r(Srr � Sxx)

�+DKT

Cs

�@2

@�r2 +1

�r +R�@@�r

+R�2

(�r +R�)2@2

@�x2

�C;(9)�

@@�t

+ �v1@@�r

+�v2R�

�r +R�@@�x

�C

=D�@2

@�r2 +1

�r +R�@@�r

+R�2

(�r +R�)2@2

@�x2

�C

+DKT

Tm

�@2

@�r2 +1

�r +R�@@�r

+R�2

(�r +R�)2@2

@�x2

�T:(10)

In the above equations, �p denotes the pressure, � the uid density, � the thermal conductivity, cp the speci�cheat, �t the time, �Sxx, �Srr, �Srx, the stress components,T and C the temperature and concentration of the uid, respectively. The subjected boundary conditionsare:

�v2 = 0 at �r = ��h; (11)�� @3

@�x3 +m@3

@�x@�t2+ d

@2

@�t@�x

��h

=1

R�(�r +R�)@@�r

n(�r +R�)2Srx

o+@Sxx@�x� �(�r +R�)

��@�v2

@�t+ �v1

@�v2

@�r+

R��v2

�r +R�@�v2

@�x� �v1�v2

R� + �r

�� Ha2�v2

R�(�r +R�) at �r = ��h;(12)

T = T1; T = T0 at �r = ��h and

C = C1; C = C0 at �r = ��h: (13)

We denote the dimensionless parameters as follows:

x =2��x�c

; r =�ra0; v1 =

�v1

c; v2 =

�v2

c;

� =2�a0

�c; � = � �h

a0; p =

2�a20 �p

c��c;

Ha =

��

!1=2

B0a0; Re =�ca0

�; k =

R�a0;

Pr =�cp�; t =

c�t�c; � =

T � T0

T1 � T0;

� =C � C0

C1 � C0; Ec =

c2

(T0 � T1)cp; Br = Pr Ec;

Sc =��D

; Sr =�DKT (T1 � T0)�(C1 � C0)

;

Du =DKT (C1 � C0)Cs�cp(T1 � T0)

; E1 =��a3

0�3c�c

;

E2 =mca3

0�3c�

; E3 =da3

0�3c�: (14)

Here, k represents the curvature parameter, � the wave,Ha the Hartman, Re the Reynolds, Pr the Prandtl, Ecthe Eckert, Br the Brinkman, Du the Dufour, Sr theSoret, Sc the Schmidt as dimensionless numbers, andE1, E2, and E3 the wall tension, mass characterizing,and wall damping parameters, respectively.

The dimensionless representation of the boundaryconditions (11)-(13) yields:

v2 = 0 at r = ��; (15)"E1

@3

@x3 + E2@3

@x@t2+ E3

@2

@t@x

#�

=1

k(r + k)@@r

�(r + k)2Srx

�+�k@Sxx@x

� Re(r + k)k

�"�@v2

@t+ v1

@v2

@r

+k�v2

k + r@v2

@x� v1v2

k + r

#� Ha2v2

k(r + k)

at r = ��; (16)

� = 1; � = 0; at r = �� and

� = 1; � = 0 at r = ��; (17)

where boundary condition in Eq. (15) represents theno-slip conditions at walls, and condition for the com-pliant wall at upper and lower walls is given in Eq. (16),


respectively. Boundary condition for temperature andconcentration pro�les is represented in Eq. (17).

Velocity components in terms of stream function, , can be de�ned by:

v1 = �k

r + k x; �2 = � r: (18)

In large peristaltic wavelength, low Reynolds numberapproximations play a vital role (see [2,12,29,30]). Theexistence of such assumptions in physiology is justi�edthrough the transportation of chyme in small intestine[31]. In such conditions, the wavelength (�c = 8:01) ofperistaltic wave is very large in comparison to the half-width (a0 = 1:25) of the channel/tube i.e. (a0=�c =0:156). Further, Lew et al. [32] examined liquidsin small intestine with low Reynolds number. Urinetransport in the human ureter is also the applicationof low Reynolds number approximation. In view ofthe above-mentioned applications of long wavelengthand low Reynolds number assumption in peristalsis,Eq. (6) is satis�ed identically and Eqs. (7)-(10) lead tothe following expressions:

dpdr

= 0; (19)

kr + k

dpdx

=1

(r + k)2@@r

�(r + k)2Srx

�� Ha2

(r + k)2 ( r); (20)

�rr +1

r + k�r + Br

"(� rr +

1r + k

r)Srx

#+ Du

��rr +

1r + k

�r�

= 0; (21)

�rr +1

r + k�r + Sc Sr

��rr +

1r + k

�r�

= 0; (22)

Srx =1

1 + �1

�� rr +

1r + k

r�; (23)

where � and � are the non-dimensionlized temperatureand concentration pro�les. Now, the dimensionlessboundary conditions are:

r = 0; at r = ��; (24)�E1

@3

@x3 + E2@3

@x@t2+ E3

@2

@t@x

��

=1

k(r + k)@@r

�(r + k)2Srx

�� Ha2 rk(r + k)

at r = ��; (25)

� = 1; � = 0; at r = ��; and

� = 1; � = 0; at r = ��; (26)

�(x) =�1 + sin 2�(x� t)

�: (27)

Results for planner channel can be obtained whencurvature parameter k is large, i.e. k !1.

3. Solution procedure

The non-dimensionlized boundary value problem givenin Eqs. (20)-(22) corresponding to boundary conditions(24)-(26) is calculated for stream function, , tem-perature pro�le, �, and concentration pro�le, �. Thenumerical solution is presented utilizing the built-inshooting algorithm in Mathematica software.

4. Physical interpretation

The prime objective in this section is to scrutinize theoutcome of involved physical variables on di�erent owquantities. Figure 2 (a)-(d) show the in uences of per-tinent parameters which are present in the momentumequation of the considered problem. Velocity pro�lehas accelerated behavior for rising values of curvatureparameter k (see Figure 2(a)). The e�ects of thecompliant wall parameters on velocity pro�le, �2, aredrawn in Figure 2(b). Also, disturbance is observed inthe symmetry of velocity pro�le, �2, about the centerline of the channel, and symmetry retains its positionfor larger curvature parameter k. It is observed thatlarger values of E1, E2, and E3 enhance the velocity. Itis also important to know that an increase in E1 and E2enhances velocity inside the channel, while it decreaseswhen damping parameter, E3, increases. Figure 2(c)explores that velocity pro�le, �2, augments in non-Newtonian uids (�1 6= 0) when matched with theNewtonian liquid (�1 = 0). The behavior of Hartmannumber on velocity pro�le, �2, is plotted in Figure 2(d).It reveals that the velocity decreases in larger radialmagnetic �eld parameters, i.e. the magnetic force hasa resistive role in the ow.

Figure 3(a)-(e) indicate the variation in tem-perature �, for the pertinent physical parameters.Figure 3(a) illustrates that the temperature decreasesin the case of curvature parameter, k. Temperatureincreases in the case of di�erent values of compliantwall parameters, E1, E2, and E3 (see Figure 3(b)). Itis observed that the temperature is enhanced insidethe channel for larger wall elastance parameter, E1,or mass characterizing parameter, E2. Note thatthe temperature drops within the channel for greaterdamping parameter, E3. Figure 3(c) depicts thattemperature pro�le � increases in di�erent values of


Figure 2. Variation in �2 for di�erent parameters when = 0:7, x = �0:2, t = 0:2: (a) �1 = 0:2, Ha = 1:0, E1 = 0:2,E2 = 0:1, E3 = 0:1, (b) �1 = 0:2, Ha = 1:0, k = 2:5, (c) k = 2:5, Ha = 1:0, E1 = 0:2, E2 = 0:1, E3 = 0:1, and (d) �1 = 0:2,k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1.

Figure 3. Variation in temperature � for di�erent sundry parameters when, = 0:7, x = �0:2, t = 0:2, Pr = 2:0, Sr = 0:5,Sc = 0:5, Ha = 1:0: (a) �1 = 0:2, Br = 0:5, E1 = 0:2, E2 = 0:1, E3 = 0:1, Du = 0:5, (b) �1 = 0:2, Br = 0:5, k = 2:5, Du= 0:5, (c) k = 2:5, Br = 0:5, E1 = 0:2, E2 = 0:1, E3 = 0:1, Du = 0:5, (d) �1 = 0:2, k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1,Du = 0:5, and (e) �1 = 0:2, Br = 0:5, k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1.

Je�rey uid parameter (�1, i.e. the relaxation factoris the only cause of increasing the temperature of uidin the vicinity of the upper wall of the curved channel,Figure 3(d) shows that the temperature rises in termsof Brinkman number Br. Physically, it means that theincreasing Br indicates larger amount of energy loss,i.e. maximum heat is produced because of its resistanceagainst the shear stress in ow �eld, which enhances the

temperature of the material. The temperature attainsits maximum value towards the upper wall of thecurved channel. It is also found that the temperatureenhances upon increasing the values of Dufour numberDu (see Figure 3(e)). It reveals that the temperatureattains its maximum value at the center of the curvedchannel.

The e�ects of pertinent sundry variables on con-


centration, �, can be seen through Figures 4(a)-(e).Concentration, �, shows the decreasing behavior byincreasing curvature, k (see Figure 4(a)). Figure 4(b)characterizes that the concentration pro�le decreasesin non-Newtonian uid when correlated with the New-tonian liquid. It means that the retardation time ofthe Je�rey uid parameter retards concentration, �,of the uid near the lower wall of the curved channel.Figure 4(c) indicates uid concentration, �, for partic-ular values of elasticity parameters. Surprisingly, �,

becomes negative in few values of parameters. It isrealistic when nutrients are dispersed away from theblood vessels towards the nearby tissues. We have seenthat � reduces upon enhancing wall elastance, E1, andwall mass characterizing, E2. However, concentration,�, is an increasing quantity of E3. Figure 4(d) and (e)present that the concentration decreases in distinctvalues of Soret number, Sr, and Schmidt number, Sc.

Figure 5(a)-(e) indicate the uctuation in heattransfer coe�cient, Z. Obviously, the behavior of

Figure 4. Variation in concentration � for di�erent pertinent parameters when = 0:7, x = �0:2, t = 0:2, Pr = 2:0, Br= 0:5, Du = 0:5, Ha = 1:0: (a) �1 = 0:2, Sr = 0:5, E1 = 0:2, E2 = 0:1, E3 = 0:1, Sc = 0:5, (b) Sr = 0:5, k = 2:5, E1 = 0:2,E2 = 0:1, E3 = 0:1, Sc = 0:5, (c) �1 = 0:2, k = 2:5, Sr = 0:5, Sc = 0:5, (d) �1 = 0:2, k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1,Sc = 0:5, and (e) �1 = 0:2, Sr = 0:5, k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1.

Figure 5. Variation in heat transfer coe�cient Z for di�erent parameters when = 0:7, t = 0:2, Pr = 2:0, Sr = 0:5, Sc= 0:5, Ha = 1:0: (a) �1 = 0:2, Br = 0:5, E1 = 0:2, E2 = 0:1, E3 = 0:1, Du = 0:5, (b) Br = 0:5, k = 2:5, Du = 0:5,E1 = 0:2, E2 = 0:1, E3 = 0:1, (c) �1 = 0:2, k = 2:5, Br = 0:5, Du = 0:5, (d) �1 = 0:2, k = 2:5, E1 = 0:2, E2 = 0:1,E3 = 0:1, Du = 0:5, and (e) �1 = 0:2, Br = 0:5, k = 2:5, E1 = 0:2, E2 = 0:1, E3 = 0:1.


Z is oscillatory. Figure 5(a) depicts that the heattransfer coe�cient decays when curvature parameterk enhances. Figure 5(b) portrays that an increase in�1 causes a decrease in the coe�cient of heat transfer.In Figure 5(c), it is observed that compliant wallparameters, E1, E2, and E3, decrease, Z. Variationin Z for larger viscous dissipation e�ects on (i.e., ofBrinkman number) Br is depicted in Figure 5(d). It isrevealed that Z decreases by Br. Figure 5(e) illustratesthat Z is also the decreasing function of Du number.

Figures 6-8 show the variation in streamlinesfor di�erent physical parameters. Figure 6(a)-(c) areplotted for variation of curvature parameter, k, instream function, . It depicts that, in larger curvatureparameter, k, reduces the size and circulation oftrapped bolus in upper half of the channel, whereasthe size of trapped bolus decreases and the number ofcirculation increases in lower half of channel. Figure7(a)-(c) re ect that the number of circulation and sizeof the trapped bolus reduce in case of larger radialmagnetic parameter Ha. Figure 8(a)-(d) show that the

size of trapped bolus decreases when compliant walls'parameters E1, E2, and E3 increase.

5. Concluding remarks

Dufour and Soret e�ects on peristalsis subject to radialmagnetic �eld are analyzed. The main �ndings arepointed out as follows:

� Velocity in inviscid uid is more than the viscousmaterial;

� The e�ects of E1 and E2 on temperature are oppo-site to that of E3;

� There is an enhancement of temperature forBrinkman and Dufour numbers;

� As expected, the heat transfer coe�cient has oscil-latory characteristics;

� Circulation of trapped bolus decreases, whereas thesize of the bolus increases in larger Ha;

Figure 6. In uence of curvature parameter k on streamlines when = 0:7, t = 0:2, Ha= 3:0, �1 = 0:2, E1 = 0:2, E2 = 0:1,and E3 = 0:1.

Figure 7. In uence of Hartman number, Ha, on streamlines when = 0:7, t = 0:2, �1 = 0:2, k = 2:5, E1 = 0:2, E2 = 0:1,E3 = 0:1.


Figure 8. In uence of compliant wall properties, E1, E2, E3, on streamlines when = 0:7, t = 0:2, �1 = 0:2, k = 2:5,Ha = 3:0.

� The e�ects of E1 and E3 on trapped bolus are quiteopposite to that of E2.

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Biographies

Tasawar Hayat received his PhD degree in AppliedMathematics from Quaid-i-Azam University, Islam-abad, Pakistan, in 1999. Since then, he has beenin teaching and research at Quaid-i-Azam University,Islamabad, Pakistan, where he joined as a lecturer inDepartment of Mathematics, 1998, and has been pro-moted to a Professor. Besides, he has been appointedas Distinguished National Professor by the HigherCommission of Pakistan. He has published extensivelyin Newtonian and non-Newtonian uid mechanics. Hehas received a number of national and internationalawards, and is well known internationally.

Shahid Farooq received his PhD degree in AppliedMathematics from Quaid-i-Azam University, Islam-abad, Pakistan, in 2018. His works mainly addressthe peristaltic ows of Newtonian and non-Newtonian


materials. He published interesting articles in interna-tional Journals of high repute.

Bashir Ahmad received his PhD in 1995 from Quaid-i-Azam University, Islamabad, Pakistan. His researchinterest includes nonlinear ordinary/integro-di�erentialequations, fractional di�erential equations, stabilityanalysis, and wave motion. He has published exten-

sively. He has been declared as the best researcherof King Abdulaziz University in 2009. At present,he has 18 h-index. He is the reviewer of severalinternational journals. He is the assistant managingeditor of Bulletin Mathematical Sciences and associateeditor of Advances in Di�erence Equations. He hasalready completed 11 research projects successfully. Healso has supervised several research students.

Documents

The impact of compliant walls on magneto hydrodynamics