Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel

Journal of King Saud University – Engineering Sciences (2016) xxx, xxx–xxx

King Saud University

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ORIGINAL ARTICLES

Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel M. Kothandapani a,*, V. Pushparaj b, J. Prakash c a Department of Mathematics, University College of Engineering Arni (A Constituent College of Anna University Chennai), Arni 632 326, Tamil Nadu, India b Department of Mathematics, C. Abdul Hakeem College of Engineering and Technology, Melvisharam 632 509, Tamil Nadu, India c Department of Mathematics, Arulmigu Meenakshi Amman College of Engineering, Vadamavandal 604 410, Tamil Nadu, India

Received 31 July 2014; accepted 31 December 2015

KEYWORDS Peristalsis; Fourth-grade fluid; Tapered asymmetric channel; Hartmann number

Abstract The problem of peristaltic transport of an incompressible non-Newtonian fluid in a tapered asymmetric channel is debated under long-wavelength and low-Reynolds number assumptions. The fluid is considered to be fourth order and electrically conducting by a transverse magnetic field. The tapered asymmetry in the flow is induced by taking peristaltic wave imposed on the non uniform boundary walls to possess different amplitudes and phase. Series solutions for stream function, axial velocity and pressure gradient are given using regular perturbation technique, when Deborah number is small. Numerical computations have been performed for the pressure rise per wavelength. Influences of different physical parameters entering into the problem have also been discussed in detail. Ó 2016 Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Peristaltic transport is a form of material transport induced by a progressive wave of contraction or expansion along the length of a distensible tube, mixing and transporting the fluid in the direction of the wave propagation. This kind of phenomenon is termed as peristalsis. It plays an indispensable role in transporting many physiological fluids in the body under * Corresponding author. Tel./fax: +91 4173 244400. E-mail address: [email protected] (M. Kothandapani). Peer review under responsibility of King Saud University.

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various situations as urine transport form kidney to bladder, the movement of chyme in the gastrointestinal tracts, 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 the fluids without internal moving parts. For example, the blood pump in the heart- lung machine and peristaltic transport of noxious fluid in nuclear industry. The mechanism of peristaltic transport has attracted the attention of many investigators since its investigation by Latham (1966), a number of analytical, numerical and experimental studies (Burns and Pareks, 1967; Shapiro et al., 1969; Fung and Yih, 1968; Shapiro et al., 1969; Takabatake and Ayukawa, 1982; Akram and Nadeem, 2013; Mekheimer and El Kot, 2008;

http://dx.doi.org/10.1016/j.jksues.2015.12.009 1018-3639 Ó 2016 Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

2 Mekheimer and Al-Arabi, 2003; Mekheimer, 2003; Nadeem and Akbar, 2010; Kothandapani et al., 2015a) of peristaltic flow for different fluids have been reported under various conditions with reference to physiological and mechanical situations. Most of these investigations are confined to the peristaltic flow only in a symmetric channel or tube. Recently, physiologists observed that the intra-uterine fluid flow due to myometrial contractions is peristaltic type motion and the myometrial contractions may occur in both symmetric and asymmetric directions. Eytan and Elad (1999) have established a mathematical model of wall induced peristaltic fluid flow in a two-dimensional channel with wave strains having a phase difference, moving independently on the upper and lower walls to stimulate intra-uterine fluid motion in a sagittal cross-section of the uterus. In another work, Eytan et al. (1999) pointed out that the width of the sagittal cross-section of the uterine cavity increases toward the fundus and the cavity is not fully occluded during the contractions. Pozrikidis (1987) has extensively studied the streamline pattern and mean flow rate due to different amplitudes and phases of the wall deformation. Mishra and Ramachandra Rao (2003) have discussed the peristaltic motion of viscous fluid in a two-dimensional asymmetric channel under assumptions of long-wavelength and low-Reynolds number in a wave frame of reference. Followed by the above works exhaustive studies have been made by many authors in the domain (Kothandapani and Srinivas, 2008a,b; Siddiqui and Schwarz, 1993; Hayat et al., 2003; El Hakeem et al., 2006; Hayat and Ali, 2006;Srinivas and Kothandapani, 2008). Moreover, Srinivas and Pushparaj (2007) have investigated the peristaltic pumping of MHD gravity flow of a viscous incompressible fluid in a two-dimensional asymmetric channel having electrically insulated walls. It is well-known that the flow phenomena of nonNewtonian fluids arise in various Engineering, industrial and technological applications. In view of different physical structure and behavior of non-Newtonian fluids, there is no single mathematical expression which describes all the characteristics of non-Newtonian fluids. Hence, several mathematical models of non-Newtonian fluids such as micropolar fluid, power-law fluids, viscoelastic fluids etc. were developed by researchers. Moreover, the formulation of a constitutive equation of nonNewtonian fluids is greatly higher-order nonlinear and complex in nature. Among the differential-type non-Newtonian fluid models, the most generalized model is fourth grade fluid model which represents most of the non-Newtonian flow properties at one time. One of the important features of fourth grade fluid is capable to exhibit normal stress differences in simple shear flows, leading to characteristic phenomena such as rod-climbing or die-swell. The main aim of this work is to analyze a tapered asymmetric wall-induced peristaltic motion of a fourth grade fluid with the presence of a uniform magnetic field. Mention in this contest to some recent interesting analytical, experimental and review studies pertaining to nonNewtonian fluids which may give valuable insights into their behaviour (Haynes, 1960; Joseph, 1980; Srivastava et al., 1983; Renardy, 1988; Vajravelu et al., 2013; Akram et al., 2013;Nadeem et al., 2014a,b; Riaz et al., 2014a,b; Umavathi and Mohite, 2014; Oahimire and Olajuwon, 2014; Kothandapani and Prakash, 2015b). Further, Haroun (2007) has also analyzed the problem of non-linear peristaltic flow of a fourth-grade fluid in an inclined channel. Moreover, Ali

M. Kothandapani et al. and Hayat (2007) have developed a mathematical model for the flow of an incompressible Carreau fluid in an asymmetric channel using a perturbation technique for a small Weissenberg number. Radhakrishnamacharya and Radhakrishna Murty (1993) discussed the problem of Heat transfer to peristaltic transport in a Non-uniform channel. Vajravelu et al. (2012) have developed a peristaltic transport of a Williamson fluid in asymmetric channels with permeable walls. They have reported a perturbation analysis with a Weissenberg number and elucidated that the pressure gradient decreases with increasing Weissenberg number. More recently, Akram (2014) has scrutinized the effects of nanofluid on peristaltic flow of a Carreau fluid model in the presence of an inclined channel and magnetic field. The effects of permeable walls and magnetic field on the peristaltic flow of a Carreau fluid in a tapered asymmetric channel have been studied by Kothandapani et al. (2015b). The mathematical observations for the peristaltic flow of a Williamson fluid model in a cross-section of a rectangular duct having compliant walls was reported by Ellahi et al. (2013). The effects of magnetohydrodynamics on the peristaltic flow of Jeffrey fluid in a rectangular duct under the constraints of low Reynold number and long wavelength have been discussed by Bhatti et al. (2014). Khan et al. (2014) have analyzed the peristaltic motion of Oldroyd fluid in an inclined asymmetric channel by regular perturbation method. Akram et al. (2014) studied the peristaltic flow of a couple stress fluid in a non-uniform rectangular duct. Akbar et al. (2015) considered the peristaltic flow in asymmetric channel with permeable wall. In their study, the influences of induced magnetic field, heat generation, heat flux water and CNTs nanofluid have been taken into account. The influence of walls attributes on the peristaltic transport in a three dimensional rectangular channel has been analyzed by Riaz et al. (2014). Kothandapani and Prakash (2015a) have studied the influences of inclined magnetic field, heat source, thermal radiation, and chemical reactions on peristaltic flow of a Newtonian nanofluid in a vertical generalized channel. They have also noted that the temperature profile increases with an increase of the non-uniform parameter, while it decreases when the thermal radiation parameter is increased. Akram and Nadeem (2014) have discussed the influence of nanofluid on peristaltic transport of a hyperbolic tangent fluid model under the effects of inclined magnetic field. Further, it is worthwhile to mention here that the intrauterine fluid flow in a sagittal cross - section of the uterus discloses a narrow channel enclosed by two fairly parallel walls with wave trains having different amplitudes and phase difference (Eytan et al., 2001). With the aid of sufficient literature support, a theoretical investigation on the peristaltic motion of a fourth-grade fluid in a tapered channel or non-uniform asymmetric channel is carried out. To the best of our familiarity, so far no attempt has been made to examine the MHD Peristaltic transport of the fourth-grade in the tapered asymmetric channel which may help to imitate intra-uterine fluid motion in a sagittal cross-section of the uterus. The governing equations are solved under the long-wavelength and low-Reynolds number approximations. The stream function and the pressure gradient could be expanded in perturbation series in a small material parameter. The effects of emerging parameters on the axial velocity and pressure drop could be studied and the phenomenon of trapping is also discussable.

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

Effect of magnetic field on peristaltic flow

3

2. Mathematical formulation

The governing equations of the continuity and momentum equations for two-dimensional case are

Let us consider the MHD flow of an incompressible and electrically conducting fourth order-fluid through a twodimensional tapered asymmetric channel. We assume that infinite wave train traveling with velocity c along the non uniform walls. We choose a rectangular coordinate system for the channel with X along the direction of wave propagation and parallel to the centerline and Y transverse to it. The wall of the tapered asymmetric channel are given Fig. 1 by the equations  
  2p
0

H2 X; t ¼ d þ m X þ a2 cos X  ct . . . upper wall; k  
  2p
H1 X; t
¼ d  m0 X  a1 cos X  ct
þ / . . . lower wall; k

@U @U þ ¼ 0; @X @Y   @U @U
@U @P @ @
q þV þ S þ S ¼ þU @ t
@X @Y @X @X XX @Y XY

ð1Þ where a1 ; a2 are the amplitudes of the waves, k is the wave length, 2d is the width of the channel at the inlet, m0 ðm0  1Þ is the non-uniform parameter, the phase difference / varies in the range 0 6 / 6 p; / ¼ 0 represents to symmetric channel in which waves are out of phase and when / ¼ p the waves are in phase, and further a1 ; b1 ; d and / satisfies the condition a21 þ a21 þ 2a1 a1 cos / 6 ð2dÞ2 : ð2Þ
 The Cauchy / and extra S stress tensors (Haynes, 1960) are given by T ¼ P I þ S;

ð3Þ


 S ¼ lA1 þ a1 A2 þ a2 A21 þ b1 A3 þ b2 A1 A2 þ A2 A1

 þ b3 trA21 A1 þ c1 A4 þ c2 A1 A3 þ A3 A1 þ c3 A22


 þ c4 A21 A2 þ A2 A21 þ c5 trA2 A2 þ c6 trA2 A21   þ c7 trA3 þ c8 trA2 A1 A1 ;

ð4Þ

where l is the constant viscosity and a1 ; a2 ; b1  b3 and c1  c8
 are the material constants. The Rivilin – Ericksen tensors An are given by A1 ¼ ðgradVÞ þ ðgrad VÞT ; An ¼

dAn1 þ An1 ðgrad VÞ þ ðgradVÞT An1 ; dt

Y

n > 1:

ð5Þ

H2

c a2

λ

0

d

φ

d a1

B

0

Fig. 1

H1 A physical sketch of the problem.

 rB20 U;

ð6Þ

ð7Þ


 @V @ V

@ V
@P @ @
þU ð8Þ SYX þ SYY ; q þV ¼ þ @ t
@X @Y @Y @X @Y
 in which q is the fluid density, U; V are velocity components
 in the direction of the laboratory frame X; Y ; r is the electrical conductivity and B0 is the constant magnetic field. We employ the following dimensionless variables in the governing equations of motion h
1 X Y ct
U V
; y ¼ ; t ¼ ; u ¼ ; v ¼ ; h1 ¼ ; k d k c dc d h
2 d2 P a1 a2 ; a ¼ ;b ¼ ; h2 ¼ ; p ¼ ckl d d d d Sij ¼ Sij ðXÞ; ðfor i; j ¼ 1; 2; 3Þ: l x¼

ð9Þ

The stream function wðx; yÞ is defined by u¼

@w ; @y

v ¼ d

@w : @x

ð10Þ

Eq. (6) is satisfied automatically and Eqs. (7) and (8) become   @w @ @w @ @w @p @ @  ¼ þ d Sxx þ Sxy dR @y @x @x @y @y @x @x @y @w ; ð11Þ  M2 @y d3 R

  @w @ @w @ @w @p @ @  ¼  þ d2 Sxy þ d Syy ; @y @x @x @y @x @y @x @y ð12Þ

where p is the pressure, l is the viscosity and S is the extra stress. In the above, the wave number d, the Hartmann number M, the Reynolds number R and the material coefficients are defined respectively as rffiffiffi 2pd1 r qcd1 ; M¼ ; d¼ B0 d1 ; R ¼ l k l k1 ¼

a1 c a2 c b c2 b c2 ; k2 ¼ ; n1 ¼ 1 2 ; n2 ¼ 2 2 ; ld1 ld1 ld1 ld1

n3 ¼

b3 c2 c c3 c c3 c c3 ; g1 ¼ 1 3 ; g2 ¼ 2 3 ; g3 ¼ 3 3 ; 2 ld1 ld1 ld1 ld1

g4 ¼

c4 c3 c c3 c c3 ; g5 ¼ 5 3 ; g6 ¼ 6 3 ; 3 ld1 ld1 ld1

g7 ¼

c7 c3 c c3 ; g8 ¼ 8 3 : 3 ld1 ld1

X

ð13Þ

We obtain, under the long wavelength and low Reynolds number approximations, the following expressions

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

4 

M. Kothandapani et al. @p @ @w þ Syy  M2 ¼ 0; @x @y @y

@p ¼ 0; @y Sxy ¼

 2 3 @2w @ w þ 2C ; @y2 @y2

ð14Þ ð15Þ

2 @ 4 w0 2 @ w0  M ¼ 0; @y4 @y2

ð21Þ

ð16Þ

@p0 @ 3 w0 @w ¼  M2 0 ; @x @y3 @y

ð22Þ

in which Cð¼ n2 þ n3 Þ is the Deborah number, w the stream function, q the fluid density, l the constant viscosity and continuity equation is automatically satisfied. Further, Eq. (15) indicates p is independent of y. From Eqs. (14) and (15), we have @
 @ w Sxy  M2 2 ¼ 0: @y @y2 2

3.1. For the system of order zero

2

w0 ¼

F0 @w0 ; ¼ 0; 2 @y

F0 @w w0 ¼  ; 0 ¼ 0; 2 @y

at y ¼ h1 :

3.2. For the system of order one

F @w ; ¼ 0; at y ¼ h2 ; 2 @y F @w ¼ 0; at y ¼ h1 : w¼ ; 2 @y

w¼

ð18Þ

We note that h1 ðx; tÞ and h2 ðx; tÞ represent the dimensionless form of the surfaces of the peristaltic walls h2 ¼ 1 þ mx þ b sinð2pðx  tÞÞ and h1 ¼ 1  mx  a sinð2pðx  tÞ þ /Þ:

ð19Þ

@ 4 w1 @ 2 w1 @2  M2 ¼ 2 2 4 2 @y @y @y

w1 ¼

F0 @w1 ; ¼ 0; 2 @y

3. Method of solution It is clear that the resulting equation of motion Eq. (17) is nonlinear. It seems to be impossible to obtain the general solution

" 3 # @ 2 w0 ; @y2

"  2 3 # @p1 @ @ 2 w1 @ w0 @w ¼  M2 1 ; þ2 @x @y @y2 @y2 @y

w1 ¼ 

þ

ð23Þ

ð17Þ

The appropriate boundary conditions in dimensionless form are

w0 ¼ 

at y ¼ h2 ;

F0 @w1 ; ¼ 0; 2 @y

ð26Þ at y ¼ h1 :

3.3. Solution for system of order Cð0Þ

4sinh2 ðMðh1  h2 Þ=2Þ  Mðh1  h2 Þ sinhðMðh1  h2 ÞÞ

;

ð27Þ

@p0 F0 M3 sinhðMðh1  h2 ÞÞ : ¼ @x 4sinh2 ðMðh1  h2 Þ=2Þ  Mðh1  h2 Þ sinhðMðh1  h2 ÞÞ ð28Þ 3.4. Solution for system of order Cð 1Þ w1 ¼ A1 þ A2 y þ A3 coshðMyÞ þ A4 sinhðMyÞ  M4
coshð3MyÞðA35 þ 3A5 A26 Þ 16
 M4 sinhð3MyÞðB3 þ 3A6 A25 Þ þ 16 3M5 ðA35  A5 A26 Þy sinhðMyÞ  4 3M5 ðA36  A25 A6 Þy coshðMyÞ ; þ 4 

w ¼ w0 þ Cw1 þ    F ¼ F0 þ CF1 þ   

ð25Þ

at y ¼ h2 ;

F0 ðM=2Þðh1 þ h2  2yÞ sinhðMðh1  h2 ÞÞ 4sinh2 ðMðh1  h2 Þ=2Þ  Mðh1  h2 Þ sinhðMðh1  h2 ÞÞ

  F0 2 sinh2 ðMh1 =2Þ  sinh2 ðMh2 =2Þ coshðMyÞ  ðsinhðMh1 Þ  sinhðMh2 ÞÞ sinhðMyÞ

in closed form for arbitrary values of all parameters arising in this non-linear equation. We are interested in the case of fluids that are shear thinning but exhibit weak normal stresses. We note that all the terms involving the Reynolds number R and material coefficients k1 ; k2 ; n1 ; g1 ; g2 ; g3 ; g4 ; g5 ; g6 ; g7 ; g8 disappear automatically. We seek the solution of the problem as a power series expansion in small parameter C ¼ ðn2 þ n3 Þ. For perturbation solution, we expand w; F and p as

ð24Þ

ð20Þ

p ¼ p0 þ Cp1 þ    Substitution of above equations into Eqs. (14) and (17) and boundary conditions Eq. (18), and collection of terms with respect to like powers of C yields the following systems:

ð29Þ

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

Effect of magnetic field on peristaltic flow

5

@p1 ¼ A3 M3 sinhðMh2 Þ þ A4 M3 coshðMh2 Þ þ A9 M2 ; @x where A5 ¼


 2F0 sinh2 ðMh1 =2Þ  sinh2 ðMh2 =2Þ

4sinh ðMðh1  h2 Þ=2Þ  Mðh1  h2 Þ sinhðMðh1  h2 ÞÞ 2

ð30Þ

;

The non-dimensional expression for the average rise in pressure Dp is given as follows: Z 1Z 1 @p Dp ¼ dxdt: ð33Þ 0 0 @x

A6 ¼ 

F0 ðsinhðMh1 Þ  sinhðMh2 ÞÞ ; 4sinh2 ðMðh1  h2 Þ=2Þ  Mðh1  h2 Þ sinhðMðh1  h2 ÞÞ

4. Results and discussion

A7 ¼ 

F1 M4 ðA3 þ 3AB2 Þ coshð3Mh2 Þ  16 2

In order to discuss the results quantitatively, we assume the instantaneous volume rate of the flow Fðx; tÞ, periodic in ðx  tÞ, (Kothandapani and Prakash, 2015a; Kothandapani et al., 2015c; Srivastava et al., 1983) as



M4 ðB3 þ 3BA2 Þ sinhð3Mh2 Þ M5 3ðAB2  A3 Þh2 sinhðMh2 Þ þ 16 4



M5 3ðA2 B  B3 Þh2 coshðMh2 Þ ; 4

Fðx; tÞ ¼ h þ a sin ½2pðx  tÞ þ / þ b sin 2pðx  tÞ;

F1 M4 ðA3 þ 3AB2 Þ coshð3Mh1 Þ M4 ðB3 þ 3BA2 Þ sinhð3Mh1 Þ  A8 ¼  16 16 2 M5 3ðAB2  A3 Þh1 sinhðMh1 Þ M5 3ðA2 B  B3 Þh1 coshðMh1 Þ  ; 4 4

 3M5 B3 þ 3A2 B coshð3Mh2 Þ 3M5 A3 þ 3B2 A sinhð3Mh2 Þ  A9 ¼  16 16 þ



3ðA2 B  B3 Þ coshðMh2 Þ 3ðAB2  A3 Þ sinhðMh2 Þ þ 4 4

Mh2 3ðAB2  A3 Þ coshðMh2 Þ Mh2 3ðA2 B  B3 Þ sinhðMh2 Þ  ; 4 4

 3M5 B3 þ 3A2 B coshð3Mh1 Þ 3M5 A3 þ 3B2 A sinhð3Mh1 Þ A10 ¼   16 16 þ



3ðA2 B  B3 Þ coshðMh1 Þ 3ðAB2  A3 Þ sinhðMh1 Þ þ 4 4

þ

Mh1 3ðAB2  A3 Þ coshðMh1 Þ Mh1 3ðA2 B  B3 Þ sinhðMh1 Þ  ; 4 4

A11 ¼ MðsinhðMh1 Þ  sinhðMh2 ÞÞ; A12 ¼ MðcoshðMh2 Þ  coshðMh2 ÞÞ; A13 ¼ coshðMh2 Þ  coshðMh1 Þ þ Mðh1  h2 ÞsinhðMh2 Þ; A14 ¼ sinhðMh2 Þ  sinhðMh1 Þ þ Mðh1  h2 Þ coshðMh2 Þ; A4 ¼

k11 ðA10  A9 Þ  A11 A8 þ A11 A7  A11 A9 ðh2  h1 Þ ; A12 k11  A11 k12

A4 ¼

A13 ðA10  A9 Þ  A11 A8 þ A11 A7  A11 A9 ðh2  h1 Þ ; A12 A13  A11 A14

A3 ¼

A10  A9  A4 A12 ; A11

A2 ¼ A3 M sinhðMh2 Þ  A4 M coshðMh2 Þ  A9 ; A1 ¼ A2 h2  A3 coshðMh2 Þ  A4 sinhðMh2 Þ  A7 :

Solving the resulting zeroth and first order systems and then invoking F ¼ F0 þ CF1 :

ð31Þ

Summarizing the perturbation solutions up to first order for w; dp=dx and Dp as w ¼ w0 þ Cw1 ;

dp dp0 dp þC 1; ¼ dx dx dx

Dp ¼ Dp0 þ CDp1 :

ð32Þ

Using F0 ¼ F  oðCF1 Þ and then neglecting the terms greater than oðCÞ the results given by Eq. (32) can be explicitly calculated.

ð34Þ

in which h is the time-average of the flow over one period of the wave and Z h2 udy: ð35Þ F¼ h1

To study the behavior of solutions, numerical calculations for several values of material parameter ðCÞ, Hartmann number ðMÞ, the phase difference ð/Þ, the non uniform parameter of the channel (m) and the amplitudes of upper and lower walls ða&bÞ have been carried out. The effect of these parameters on Dp, the integral appearing in Eq. (46) has been evaluated numerically using mathematica and the results are presented graphically. Fig. 2 shows the variation of Dp against time mean flow rate h. The whole region is considered into five parts: (i) Peristaltic pumping region where Dp > 0 and h > 0. (ii) augmented pumping region where Dp < 0 and h > 0. (iii) When Dp > 0 and h < 0, then it is a retrograde pumping region. (iv) There is a co-pumping region where Dp < 0 and h < 0. (v) Dp ¼ 0 corresponds to the free pumping region. Fig. 2(a) elucidates the variation of average rise in pressure with mean flow rate h for different values of Hartmann number (M). Due to the influence of magnetic parameter, the relation between h and Dp is a non-linear in retrograde and peristaltic pumping. The pumping rate found to increase with an increase in magnetic parameter. Fig. 2(b) shows the variation of Dp against dimensionless flow rate h for various values of Deborah number C. It indicates in respect of Newtonian fluid ðC ¼ 0Þ, the relation between average rise in pressure and mean flow rate is linear in character whereas for non-Newtonian fluid, it is non-linear and further peristaltic pumping increases with increases C. It has been observed from Fig. 2(c) that average rise in pressure increases with the increase in amplitude of the upper wall (b). Fig. 2(d) is plotted to see the effect of average rise in pressure Dp against dimensionless flow rate h for various values of nonuniform parameter (m). One could perceive that with an increase in non-uniform parameter m, the pumping decreases in the regions of peristaltic pumping, retrograde pumping and while in the co-pumping region the pumping found to increase. Fig. 2(f) reveals the average rise in pressure decreases for all the regions of study with an increase in phase difference. Influences of geometric parameters on the velocity distribution have been illustrated in Fig. 3. The change in values of m on the axial velocity u is shown in Fig. 3(a). It is interesting to note that an increase in m causes an increase in the

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

6

M. Kothandapani et al.

5

15 10 5

a = 0.5, b = 0.6, m = 0.1,

φ = π /4, Γ = 0.05.

-0.5

(a) 0

0.5

θ

-0.5

0

θ

0.5

1

m=0 m = 0.1 m = 0.2 m = 0.3

2 1

1

Δp

Δp

1

b=0 b = 0.2 b = 0.4 b = 0.6

2

0 φ = π /4, M = 1, Γ = 0.05.

(c) -0.5

0 -1

a = 0.2, m = 0.3,

-1

(b)

0 -1 3

3

-1

a = 0.6, b = 0.7, m = 0.1,

φ = π /2, M = 0.2.

0

-5 -1

Γ=0 Γ = 0.1 Γ = 0.2 Γ = 0.3

20

Δp

Δp

25

M = 0.2 M=1 M=2 M=3

10

0

0.5

θ

a = 0.4, b = 0.3, M = 0.5, φ = π /2, Γ = 0.1.

-2 -1

1

-0.5

6

0

θ

0.5

1

φ=0 φ = π/4 φ = π/3 φ = π/2

4

Δp

(d)

2 0 -2 -1

Fig. 2

a = 0.5, b = 0.4, m = 0.2, M = 2, Γ = 0.05.

-0.5

(e) 0

θ

0.5

1

Variation of average rise in pressure Dp versus h.

magnitude of u at the boundaries. However, at the center of the channel the magnitude of u gets decreased. A similar behavior is seen for the case of Hartmann parameter and it is projected in Fig. 3(b). From Fig. 3(c) it appears that the velocity profile traces a parabolic path. It increases with an increase of C in the upper wall and converse in behavior is observed at the lower part of channel. Fig. 3(d) shows that with an increase in mean volume flow rate h, the axial velocity increases. The influence of the amplitude of the upper wall b on the velocity is depicted in Fig. 3(e) for a fixed value of other parameters. It could be observed that an increase in the amplitude of upper wall b increases the magnitude of the velocity in the upstream. The axial velocity for the phase angle / is shown in Fig. 3(f). It has been observed that an increase in / values causes as decrease in the magnitude of axial velocity u at the lower wall.

The phenomenon of trapping is another interesting topic in peristaltic transport. The formation of an internally circulating bolus of fluid through closed streamlines is called trapping and this trapped bolus is pushed ahead along with the peristaltic wave. The streamlines for different a are shown in Fig. 4. It has been noticed that the bolus increases in the lower and upper of the tapered channel with increasing a. The impact of non-uniform parameter m on trapping is shown in Fig. 5. It is examined that the size of the bolus increases with an increase in m. The streamlines for the different values of Hartmann number M are plotted in Fig. 6 for the fixed values of all other parameters. One could observe that the volume of the bolus decreases with increasing M. Fig. 7 illustrates the variation of mean flow rate h on trapping. The size of the trapping bolus appears to be increasing with an increase in h and it is complementary when / is increased as shown Fig. 8.

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

Effect of magnetic field on peristaltic flow

7

1

0

M = 0.1 M=1 M=2 M=3

0.5

y

y

0.5

-0.5

1

m=0 m = 0.1 m = 0.2 m = 0.3

0 a = 0.4, b = 0.3, φ = π /3, θ = 1.8, M = 0.5, Γ = 0.2, x = 0.4, t = 0.2.

-0.5

-1

-1

(a)

0

0.2

0.4

0.6

a = 0.3, b = 0.4, m = 0.2, φ = π /4, θ = 1.6, Γ = 0.1, x = 0.4, t = 0.2.

0.8

u

1

1.2

(b)

0

1.4

0.2

0.4

0.6

u

0.8

1

1.2

1.5 1 1

Γ=0 Γ = 0.1 Γ = 0.2 Γ = 0.3

0

y

y

0.5

0

a = 0.6, b = 0.5, m = 0.1, φ = π /3, θ = 1.4, M = 0.2, x = 0.4, t = 0.2.

-0.5

θ = 1.4 θ = 1.6 θ = 1.8 θ=2

0.5

a = 0.2, b = 0.1, m = 0.3, φ = π /2, M = 1, Γ = 0.2, x = 0.4, t = 0.2.

-0.5

-1

(c)

u

1

-0.5

0.8

1

0

-0.7

-0.8

-0.

4

0

0. 4

0. 7

-1.5

0. 2

0. 8

0. 6

-0.7

0.7

0.7

0.9

0. 6

-0 .6

-2

1

1.5

x Fig. 4

9 -0.

0.7

0.5

1

4 0.

0

0.8

0

0.9

0

-2

0

-1

0. 7

0. 9

-0. 9 8

2 -0.

-0. 4

-0.5

4 0.

0. 6

0. 8

0. 8

-0.

0.5 0

0. 4

-0.5

1

-0.4

y

0

0.6

-0 .7

-0.7

6

u

(b)

1.5

-0 .6

-0.

0.4

Axial velocity profile uðyÞ.

-0. 9 -0 .8

.8 -0

0.2

0. 6

-0. 2

-0. 4 0

0.5

1.4

-0.6

-0.7

1.2

(f)

-0.8

0.6

u

-0.9

1

1

0. 9

0.4

2

1.5

u

0.8

a = 0.4, m = 0.3, φ = π /4, θ = 1.5, M = 2, Γ = 0.1, x = 0.4, t = 0.2.

-1

(e) 0.2

0.6

b=0 b = 0.2 b = 0.4 b = 0.6

0

(a)

-1.5

0.4

0.5

Fig. 3

-1

0.2

1

a = 0.3, b = 0.2, m = 0.2, θ = 1.4, M = 2, Γ = 0.1, x = 0.4, t = 0.2.

0

y

0

1.5

-1

2

(d)

-1 1.2

y

y -0.5

0.8

φ=0 φ = π/2 φ = 2π/3 φ=π

0.5 0

0.6

0 0. 7 . 6

1

0.4

-0 . 6 -0.7

0.2

0.8

0

2

2.5

0

0.5

1

1.5

2

2.5

x

Streamlines for b ¼ 0:3; m ¼ 0:2; / ¼ p=2; h ¼ 1:5; M ¼ 1; C ¼ 0:1; t ¼ 0:5, (a) a ¼ 0, (b) a ¼ 0:3.

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

8

M. Kothandapani et al.

0.6

-1

0. 4

-1.5

0. 7

0. 7

0

0

0.4 0. 6

7 0.

-1 -1.5

-2

-0.9

-0 .9

0 -0.5

0. 9

0. 8

0. 7 0. 8 0.9

-0.5

0. 4

-0.4 -0. 2

0. 4

0. 7

0

0 0.4

y

0

7 -0. 4 -0. -0.2

0.5

0. 8

-0.4

0. 9

-0 .9

8 -0.

-0. 9

-0. 4

1

.7 -0

-0.6

-0. 7

0.4

0.5

1

1.5

2

2.5

0

0.5

1

1.5

x

0

0.4

0. 8

-1.5

0.2

0.5

-2

1

1.5

2

0.2

0.5

1

-0. 7

-0 .7

1.5

2

2.5

x

x

y

-0.7

-1.1

6 -0.

0

1 1.

-2 2.5

0

0.5

0. 7 0.8 0.9

0. 6

1 1. 1

-1.5

1

0.8

0. 7 0. 9

0. 6

-0. 7

-0. -1 9

0. 6

0 1 .9

0. 8

0. 0. 6 7

Fig. 7

2

0

0.6

1.5

x

-0. 7

1

-0.6

0.5

8

0

-0.

1

-2

0

2

0. 2

0. 6

0. 2

0

-1

0. 7 0.9

9 -0.

-0. 6

-0.

-0.5

0.6

-1.5

0.8

.9 -0

0

0

-1

-0.5

0.5 0

0. 2

0

1

-0.

-0. 6

-1

0

-0.4

.1 -1 -0. 8

0.5

7 -0.

-1

9 -0. 6 -0.

-0.7

1

1.5

-0. 7

1.5

(b)

2

-0.6

.8 -0

0 -0. 4

-0.8

(a)

-0. 6

Streamlines for a ¼ 0:2; b ¼ 0:1; / ¼ p=4; h ¼ 1:6; m ¼ 0:2; C ¼ 0:3; t ¼ 0:5, (a) M ¼ 0:1, (b) M ¼ 3.

Fig. 6

-1

0. 7 0. 8

(b) 0

2.5

0. 4

0.8

-1.5

(a) 0

0. 4 0. 7

-1

0. 9

0. 9

-1

0.7 0.8

0

0

-0.5

0. 4

0.6

0. 7 0. 8

0. 7

-0.5

-0. 4

-0.4

0

-

0.7

0

0.5

-0. 9

-0.8

-0 .9

-0.7 -0.8

-0. 9

-0. 4

0. 9

0

-0.6

-0.7

8

0. 9

-0. 4

y

0.5

-0.

1

.8 -0

-0.8

1

-0.4

1.5

-0.7

1.5

2

2.5

2

-0.2

-0 .9

2

-2

2

x

Streamlines for a ¼ 0:2; b ¼ 0:1; / ¼ p=3; h ¼ 1:6; M ¼ 0:5; C ¼ 0:2; t ¼ 0:5, (a) m ¼ 0, (b) m ¼ 0:3.

Fig. 5

y

7 0.

-2

0

y

0.8 0. 9

0. 8

y

1.5

-0.4

-0.8

-0.8 -0.7

0.5

-0 .8

8 -0.

-0. 4

-0. 8 -0.7

1.5 1

(b)

2

-0. 7

(a)

2

1

1.5

2

2.5

x

Streamlines for a ¼ 0:4; b ¼ 0:2; / ¼ p=3; M ¼ 2:m ¼ 0:3; C ¼ 0:2; t ¼ 0:5, (a) h ¼ 1:6, (b) h ¼ 1:7.

Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009

Effect of magnetic field on peristaltic flow

-0.9 -0. 7

-0 .4

0. 6

0. 9

0. 6

0. 6

y 0.6

-0. 8 -0 .6

-0.

-1

1

0. 0. 8 7

4 0.

1 0. 9

7 0.

-1

.6 -0

0

0

-1.5

-2

-0.8

-0 .2

0 -0.5

0. 7

-0.2

-1.5

-0.2

0. 4

0. 4 0.6 0.8 0.9

0. 8

7 0.

-1

0. 6

-0 .7

0.5

0

0. 2

1

-0.6

-0.2

0 4 0.

-0. 8 -0.6 -0.4

-0.9

0

-0.8

1.5 -0. 6

-0.6 -0.7 -0.4

0.5

-0.2

.9 -0 .7 -0

-1

1

9

1.5

-0.5

(b)

2

0.9 0. 8

(a)

2

y

9

-2

0

0.5

1

1.5

2

2.5

0

x Fig. 8

0.5

1

1.5

2

2.5

x

Streamlines for a ¼ 0:3; b ¼ 0:4; h ¼ 1:5; M ¼ 1:5:m ¼ 0:1; C ¼ 0:1; t ¼ 0:5, (a) / ¼ 0, (b) / ¼ p=2.

5. Conclusion

References

A mathematical model has been developed to study the peristaltic transport of a fourth-grade fluid in the tapered asymmetric channel under the influence of magnetic field. The channel asymmetry is produced by choosing the peristaltic wave train on the non-uniform walls to have different amplitudes and phase. A long-wavelength and low-Reynolds number approximations are adopted. A regular perturbation method is employed to obtain the expression for stream function, axial velocity and pressure gradient. Numerical study has been conducted for the average rise in pressure over a wavelength. The effects of material parameter ðCÞ, Hartmann number ðMÞ, wave amplitudes, channel width and phase angle on the pressure rise, axial velocity and stream lines are also investigated in detail. The following observations are made in the present study.

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 The average rise in pressure increases with increase of M; C and b while it decreases by increasing m and /.  The average rise in pressure per wavelength in respect of fourth-grade fluid is higher as compared with that of Newtonian fluid.  The axial velocity decreases with increase of M and m.  The velocity profile increases with an increasing C in the upper part of the channel. However, the converse behavior is noticed in the case of lower part of the tapered asymmetric channel.  The size of trapped bolus decreases with an increase in Hartmann number.  Finally, the results of Haroun (2007) in the absence inclined parameter can be captured as a special case of our analysis by taking M ¼ 0 and m ¼ 0. Acknowledgement The authors are highly thankful to the Editor-in-Chief and the anonymous reviewers for their valuable suggestions which have been immensely helped to bring this work in the present form.

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Please cite this article in press as: Kothandapani, M. et al., Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. Journal of King Saud University – Engineering Sciences (2016), http://dx.doi.org/10.1016/j.jksues.2015.12.009