Numerical and analytical treatment on peristaltic flow of Williamson fluid in the occurrence of induced magnetic field

Journal of Magnetism and Magnetic Materials 346 (2013) 142–151

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Numerical and analytical treatment on peristaltic flow of Williamson fluid in the occurrence of induced magnetic field Safia Akram a,n, S. Nadeem b, M. Hanif a a b

Department of Basic Sciences, Military College of Signals, National University of Sciences and Technology, Pakistan Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

art ic l e i nf o

a b s t r a c t

Article history: Received 6 January 2013 Received in revised form 27 June 2013 Accepted 5 July 2013 Available online 20 July 2013

In this paper the effects of induced magnetic field on the peristaltic transport of a Williamson fluid model in an asymmetric channel has been investigated. The problem is simplified by using long wave length and low Reynolds number approximations. The perturbation and numerical solutions have been presented. The expressions for pressure rise, pressure gradient, stream function, magnetic force function, current density distribution have been computed. The results of pertinent parameters have been discussed graphically. The trapping phenomena for different wave forms have been also discussed. & 2013 Elsevier B.V. All rights reserved.

Keywords: Williamson fluid model Peristaltic motion Induced magnetic field Perturbation method Numerical Solution Different wave form

1. Introduction Since the pioneering work done by Latham [1], considerable attention has been given to the study of peristaltic flows of both Newtonian and non-Newtonian fluids with different flow geometries because of their importance in many engineering and Biomedical applications. In biological systems it is involved in urine transport from kidney to bladder, swallowing food through esophagus, chyme motion in the gastrointestinal tract, vasomotion of small blood vessels and movement of spermatozoa in the human reproductive tract. There are many engineering processes as well in which peristaltic pumps are used to handle a wide range of fluids particularly in chemical and pharmaceutical industries. It is used in sanitary fluid transport, blood pumps in heart lungs machine and transport of corrosive fluids where the contact of the fluid with the machinery parts are prohibited. Peristaltic flows of Newtonian and non-Newtonian fluids with different physical geometries and wave shapes have been studied by number of authors. To mention a few, Nadeem and Akbar [2] have examined the effects of temperature dependent viscosity on peristaltic flow of a Jeffrey six constant fluid in a non-uniform tube. Lozano and Sen [3] have highlighted the stream lines patterns and their local and global bifurcation in a two dimensional planner and axisymmetric peristaltic flow of a Newtonian fluid. They [3] discussed

n

Corresponding author. E-mail address: safi[email protected] (S. Akram).

0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.07.014

three bifurcation regions and verify their results with the experimental data. The peristaltic flow of a couple stress fluid in an annulus have been studied by Mekheimer and Abd-elmaboud [4]. Nadeem and Akram [5–8] have discussed the peristaltic flows of Newtonian and non-Newtonian fluid in symmetric and asymmetric channels. In few other papers, Nadeem and Akbar [9–12] have discussed the peristaltic flows in cylindrical geometry with different wave forms. A new numerical solution for MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method has been examined by Ebaid [13]. Some more useful papers on this subject are cited in the Refs. [14–25] In view of amount of work done on peristaltic flows it becomes interesting to investigate the effects of induced magnetic field on the peristaltic flow of Williamson fluid model in symmetric and asymmetric channel. The modeled nonlinear equations of Williamson model for two dimensional peristaltic flow are simplified using the well-known long wave length and low Reynolds number approximations. The reduced equations are then solved analytically and numerically. A comparison of both the solutions is also given. The expressions for pressure rise, velocity, induced magnetic field function and stream lines are discussed through graphs for different physical parameters Tables 1 and 2. 2. Mathematical formulation Let us consider the peristaltic flow of an incompressible, electrically conducting non-Newtonian fluid (Williamson fluid)

S. Akram et al. / Journal of Magnetism and Magnetic Materials 346 (2013) 142–151

Nomenclature U,V U,v ρ p s M Re

velocity components in the X and Y directions in fixed frame velocity components in the x and y directions in wave frame constant velocity pressure electrical conductivity Hartmann number Reynolds number

in a two dimensional channel of width d1+d2. The flow is generated by sinusoidal wave trains propagating with constant speed c along the channel walls. We choose a rectangular coordinate system for the channel with X along the center line of the channel and Y is transverse to it. An external transverse uniform constant magnetic field H0, induced magnetic field H (hX(X,Y,t),H0+hY(X,Y,t),0) and the total magnetic field H+(hX(X,Y,t), H0+hY(X,Y,t),0) are taken into account. A schematic diagram of the geometry of the problem under consideration is shown in Fig. (a). The channel walls are considered to be non-conductive and the geometry of the wall surface is defined as
 2π Y ¼ H 1 ¼ d1 þ a1 Cos ðXct Þ ; λ
 2π Y ¼ H 2 ¼ d2 b1 Cos ðXct Þ þ ϕ ; ð1Þ λ where a1 and b1 are the amplitudes of the waves, λ is the wave length, d1+d2 is the width of the channel, c is the velocity of propagation, t is the time and X is the direction of wave propagation, the phase difference ϕ varies in the range 0≤ϕ≤π, ϕ¼ 0 corresponds

Φ a1;b1 λ c Q δ ψ S1 Rm Φ τ

143

amplitude ratio amplitude of waves wave length velocity of propagation volume flow rate long wave length stream function Strommer's number (magnetic force number) magnetic Reynolds numbers magnetic force function extra stress tensor

to symmetric channel with waves out of phase and ϕ¼π, the waves are in phase, further a1,b1,d1,d2 and ϕ satisfy the condition 2

2

a21 þ b1 þ 2a1 b1 cos ϕ≤ðd1 þ d2 Þ : The equations governing the flow are given by (i). Maxwell's equation ∇H ¼ 0; ∇E ¼ 0;

ð2Þ

∇∧H ¼ J; withJ ¼ rfEþμe ðV∧HÞg;

ð3Þ

∇∧E ¼ μe

∂H ⋅ ∂t

ð4Þ

(ii). The continuity equation ∇V ¼ 0:

ð5Þ

(iii). The equation of motion       ∂V 1  þ 2 þ ðV⋅∇Þ V ¼ divðpIþτÞ∇ μe H μe Hþ ⋅∇ Hþ ; ρ ∂t 2 ð6Þ

Table 1 Shows the comparison of Numerical and Perturbation solution. y

Perturbation solution

Numerical solution

 1.5  1.2  0.9  0.6  0.3 0 0.3 0.6 0.9 1.2 1.5

1  1.75134  2.32132  2.71565  2.93986  2.99925  2.89899  2.6441  2.23945  1.68984 1

1  1.726386988  2.273336808  2.649944075  2.863689314  2.920718863  2.826056927  2.583769801  2.197095457  1.668547535 1

in which the extra stress tensor τ for Williamson fluid in defined by [5] h τ ¼ μ0 ð1Γ γ_ Þ1  γ_ ¼ μ0 ½ð1 þ Γ γ_ Þ γ_ ⋅ ð7Þ With the help of Eqs. (2)–(4), we obtain the induction equation as follows:   1 ∂Hþ ¼ ∇∧ V∧Hþ þ ∇2 Hþ ; ξ ∂t where ξ ¼ sμ1 is the magnetic diffusively. e

Table 2 Shows the comparison of Numerical and perturbation solution. Y

Perturbation solution

Numerical solution

 1.5  1.2  0.9  0.6  0.3 0 0.3 0.6 0.9 1.2 1.5

1  1.73181  2.28729  2.67189  2.89086  2.94927  2.85202  2.60387  2.20946  1.67335 1

1  1.726386988  2.273336808  2.649944075  2.863689314  2.920718863  2.826056927  2.583769801  2.197095457  1.668547535 1 Fig. (a). Geometry of the problem.

ð8Þ

144

S. Akram et al. / Journal of Magnetism and Magnetic Materials 346 (2013) 142–151

Introducing a wave frame (x,y) moving with velocity c away from the fixed frame (X,Y) by the transformation x ¼ Xct; y ¼ Y; u ¼ Uc; v ¼ V: Defining 2

x y u v d1 d2 d p ; y¼ ; d¼ ; u¼ ; v¼ ; δ¼ ; p¼ 1 ; λ d1 c c μ0 cλ λ d1 ct H1 t ¼ ; h1 ¼ ; λ d1 H2 a1 b1 cd1 Ψ Φ ; Ψ¼ h2 ¼ ; a¼ ; b¼ ; Re ¼ ; Φ¼ ; cd1 H 0 d1 d2 d1 d1 ν  þ 2 μ H 1 pm ¼ p þ Reδ e 2 ; 2 ρc rffiffiffiffiffi H 0 μe λ d1 ; τ ¼ τxx ; τxy ¼ τxy ; Rm ¼ sμe d1 c; S1 ¼ c ρ xx μ0 c μ0 c

in which δ, Re, We, S1 Rm, and E represent the wave, Reynolds, Weissenberg numbers, Strommer's (magnetic force number), magnetic Reynolds numbers and electric field respectively. Under the assumptions of long wavelength δo o1 and low Reynolds number, neglecting the terms of order δ and higher, Eqs. (10)–(13) take the form

x¼

τyy ¼

d1 γd1 τyy ; γ ¼ : μ0 c c

∂P ∂ ¼ ∂x ∂y

Reδ Ψ x Ψ xy Ψ y Ψ xx



∂2 ∂y2

1 ðΦyy þ δ2 Φxx Þ ¼ E; Rm

ð12Þ

ð17Þ

Combining Eqs. (16) and (17), we obtain ∂2 ∂y2


   ∂ 2 Ψ ∂2 Ψ ∂2 Ψ 1 þ We 2 ReS21 Rm ð 2 Þ ¼ 0; ∂y ∂y2 ∂y -0.5

-1 Numerical Solution Perturbation Solution

-1.5

u

ð11Þ

ð16Þ


   ∂ 2 Ψ ∂2 Ψ 1 þ We 2 þ ReS21 Φyyy ¼ 0⋅ ∂y ∂y2

ð10Þ

Reδ2 S21 Φyy ReS21 δ3 ðΦy Φxx Φx Φxy Þ;

∂Ψ Þ⋅ ∂y

Elimination of pressure from Eqs. (14) and (15), yields

∂p ∂   ∂ τyx þ δ ðτyy Þ ¼  m þ δ2 ∂x ∂y ∂y

Ψ yδðΨ y Φx Ψ x Φy Þ þ

ð15Þ

Φyy ¼ Rm ðE

Using the above non-dimensional quantities, the equations which govern the MHD flow in terms of the stream function Ψ(x,y) and magnetic-force function Φ(x,y) (dropping the bars and using ∂Ψ ∂Φ ∂Φ u ¼ ∂Ψ ∂y ; v ¼ δ ∂x ; hx ¼ ∂y ; hy ¼ δ ∂x Þ are

 3

ð14Þ

∂P ¼ 0; ∂y

ð9Þ

∂p ∂ ∂   τxy ReδðΨ y Ψ xy Ψ x Ψ yy Þ ¼  m þ δ ðτxx Þ þ ∂x ∂y ∂x  2 2  þReS1 Φyy þ ReS1 δ Φy Φxy Φx Φyy ;


   ∂ 2 Ψ ∂2 Ψ 1 þ We 2 þ ReS21 Φyy ; ∂y ∂y2

where -2

∂2 Ψ ; τxx ¼ 2½1 þ We_γ  ∂x ∂y  2 2  ∂ Ψ 2∂ Ψ τxy ¼ ½1 þ We_γ  δ ; ∂y2 ∂x2

-2.5

∂2 Ψ ; ∂x∂y "  2 2  2  2 2 #1=2 2 2 ∂ Ψ ∂ Ψ 2∂ Ψ 2 ∂ Ψ γ_ ¼ 2δ2 þ δ þ 2δ ; ∂x∂y ∂x∂y ∂y2 ∂x2

τyy ¼ 2δ½1 þ We_γ 

-3 -1.5

-1

-0.5

0

0.5

1

1.5

y

ð13Þ

Fig. 2. Comparison of numerical and perturbation solution for fixed values of a¼ 0.5, b¼ 1.2, d ¼1.5, φ ¼ 2π ; We¼ 0.03, x¼ 0.

-0.5

2

-1

1.5 Numerical Solution

1

Perturbation Solution

M = 1.0

-2

y

u

-1.5

M = 2.0 M = 3.0

0

-2.5

-0.5

-3

-3.5 -1.5

0.5

-1

-0.5

0

0.5

1

1.5

y Fig. 1. Comparison of numerical and perturbation solution for fixed values of a¼ 0.5, b¼ 1.2, d ¼ 1.5, φ ¼ 2π ; We ¼0.03, x ¼0.

-1 -1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

u Fig. 3. Velocity profile for different values of M at a¼ 0.7, b¼ 0.7, d ¼1, x ¼0, φ ¼ 2π , We ¼0.03, Q ¼0.5.

S. Akram et al. / Journal of Magnetism and Magnetic Materials 346 (2013) 142–151




∂2 ∂y2

1 þ We

∂2 Ψ ∂y2



 ∂2 Ψ ∂2 Ψ M 2 2 ¼ 0; 2 ∂y ∂y

ð18Þ

3. Methods of solution Since Eq. (18) is highly non linear, therefore, the exact solution of Eq. (18) seems to be impossible. Adopting the similar procedure as done by Nadeem and Akram [5], the solution of Eq. (18) straight

where M 2 ¼ ReS21 Rm is the Hartmann number. The dimensionless mean flow Q is defined as Q ¼ F þ 1 þ d;

ð19Þ

60

in which Z

h1 ðxÞ

F¼

h2 ðxÞ

∂Ψ dy ¼ Ψ ðh1 ðxÞh2 ðxÞÞ; ∂y

a = 0.1 a = 0.3 a = 0.5 a = 0.7

40

ð20Þ

20

h1 ðxÞ ¼ 1 þ a cos 2πx; h2 ðxÞ ¼ db cos ð2πx þ ϕÞ⋅

ð21Þ

Δp

where

The boundary conditions in terms of stream function Ψ are defined as Ψ¼

145

0 -20 -40

F ∂Ψ ; ¼ 1at y ¼ h1 ðxÞ; 2 ∂y

-60

F ∂Ψ ¼ 1 at y ¼ h2 ðxÞ; Ψ ¼ ; 2 ∂y

ð22Þ

Φ ¼ 0 at y ¼ h1 ðxÞ and y ¼ h2 ðxÞ:

-80 -1

0

1

2

ð23Þ

Fig. 6. Variation of pressure rise with volume flow rate Q for different values of a at b¼ 0.3, d ¼0.2, φ ¼ 2π ; M ¼0.5, We¼ 0.03, E¼ 2.

2

40 d = 0.1 d = 0.3 d = 0.5 d = 0.7

1.5 20 1 0

0.5

Q = 0.2

0

Q = 0.4

Δp

Q = 0.0

y

3

Q

-20 -40

-0.5 -60 -1 -1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-80 -1

u Fig. 4. Velocity profile for different values of Q at a¼ 0.7, b ¼ 0.7, d ¼ 1, x¼ 0, φ ¼ We¼ 0.03, M¼ 2.

π 2,

0

1 Q

2

Fig. 7. Variation of pressure rise with volume flow rate Q for different values of d at b¼ 0.3, a¼ 0.5, φ ¼ 2π ; M ¼ 0.5, We¼0.03, and E¼ 2.

80

2

φ=0

60 1.5

0.5 0

φ =π/6

40

φ =π/4 φ =π/2

20

We = 0.0

Δp

y

1

3

We = 0.1

0 -20

We = 0.2

-40 -60

-0.5

-80 -1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

u Fig. 5. Velocity profile for different values of We at a ¼0.7, b ¼ 0.7, d ¼1, x¼ 0, φ ¼ 2π , Q¼ 2, M ¼ 2.

-100 -1

0

1

2

3

Q Fig. 8. Variation of pressure rise with volume flow rate Q for different values of ϕ at b¼ 0.3, d ¼0.2, a ¼ 0.5, M¼ 0.5, We¼ 0.03, and E¼ 2.

146

S. Akram et al. / Journal of Magnetism and Magnetic Materials 346 (2013) 142–151

1



ðð2ð2ð3h1 þ 3h2 12ððh1 þ h2 ÞMcosh 12 ðh1 h2 ÞM þ 2sinh 12ðh1 h2 ÞM Þ2

forward can be written as Ψ¼

1



6FMðh1 þ h2 2yÞcosh 2ðh1 h2 ÞM 



þ 12 ðh1 h2 ÞMcosh 12 ðh1 h2 ÞM 2sinh 12ðh1 h2 ÞM Þ

þF 2 M 2 We þ ðh1 h2 Þð2F þ h1 h2 ÞM 2 We6yÞ

0

150 M = 1.0

-5

M = 2.0

100

-10

M = 4.0

-15

Δp

dp/dx

50

M = 3.0

0

-20 -25

φ φ φ φ

-30

-50

-35 -100 -1

0

1

2

-40

3

0

0.2

0.4

0.8

Fig. 9. Variation of pressure rise with volume flow rate Q for different values of M at b¼ 0.3, d ¼0.2, φ ¼ 8π ; a¼ 0.5, We ¼0.03, and E ¼2.

Fig. 12. Variation of dp/dx with x for different values of ϕ at b ¼0.3, a¼ 0.5, d ¼0.2, M¼ 1, We¼ 0.03, E¼ 2, and Q ¼1.0.

10

30 E = 0.0

20

E = 5.0 E = 10.0

0

E = 15.0

0

dp/dx

10

-10

-10

-20

-20

M = 0.5 M = 1.0

-30

-30

M = 1.5

-40

M = 2.0

-50 -1

0

1

2

-40

3

0

0.2

Fig. 10. Variation of pressure rise with volume flow rate Q for different values of E at b¼ 0.3, d ¼0.2, φ ¼ 2π ; M ¼0.5, We¼ 0.03, and a¼ 0.5.

0.6

0.8

1

Fig. 13. Variation of dp/dx with x for different values of M at b¼ 0.3, a¼ 0.5, φ ¼ 2π ; d¼ 0.2, We¼ 0.03, E¼ 2, and Q¼ 1.0.

0

0

M = 2.0

-5

-2

M = 3.0

-10

M = 4.0

-4

Φ

-15

dp/dx

0.4

x

Q

-20 -25

d = 0.1 d = 0.2 d = 0.3 d = 0.4

-30 -35 -40

1

x

Q

Δp

0.6

=0 = π/6 = π/4 = π/2

0

0.2

0.4

0.6

0.8

-6 -8 -10

1

x Fig. 11. Variation of dp/dx with x for different values of d at b¼ 0.3, a¼ 0.5, φ ¼ 2π ; M ¼0.5, We¼ 0.03, E¼ 2, and Q ¼1.0.

-12 -2

-1

0

1

2

y Fig. 14. Variation of magnetic force function Φ with y for different values of M at a¼ 0.7, b¼ 1.2, d ¼ 2, x¼ 0, Q ¼4, φ ¼ 2π ; We¼ 0.02, E¼ 2, and Rm ¼ 4.

S. Akram et al. / Journal of Magnetism and Magnetic Materials 346 (2013) 142–151

147

0

ð6h1 6h2 þ F 2 M 2 We 2

þðh1 h2 Þð2F þ h1 h2 ÞM We þ 12yÞcosh½ðh1 h2 ÞM

Rm = 2.0

2

þ2ðF þ h1 h2 Þð3 þ ðF þ h1 h2 ÞM WeÞcosh½ðh1 yÞM

Rm = 3.0

þ2ðF þ h1 þ h2 Þð3 þ ðF þ h1 h2 ÞM 2 WeÞ

Rm = 4.0

-5

Φ

cosh½ðh2 yÞMMððF þ h1 h2 Þ2 MWecosh½Mðh1 þ h2 2yÞ þ3ðh1 h2 Þ ðh1 þ h2 2yÞsinh½ðh1 h2 ÞMÞ þ 3ðh1 h2 ÞðF þ h1 h2 ÞMsinh½Mðh1 yÞ

þ3ðh1 h2 ÞðF þ h1 h2 ÞMsinh½Mðh2 yÞÞÞÞ⋅

ð24Þ

-10

The axial pressure gradient is obtained from Eq. (14) and is defined as



ðM 2 ðMðF þ ðh1 þ h2 ÞEÞcosh 12 ðh1 h2 ÞM þ 2ð1 þ EÞsinh 12ðh1 h2 ÞM ÞÞ dp



¼ ⋅ dx ðh1 h2 ÞMcosh 12 ðh1 h2 ÞM 2sinh 12ðh1 h2 ÞM

-15 -2

ð25Þ With the help of Eq. (24) the solution of Eq. (16) is straight forward written as 2

3

11.5

M = 1.0

11

M = 2.0

10.5

M = 3.0

10

ð26Þ

9.5

The expression for axial induced magnetic field can be obtained with the help of hx ¼ ∂Φ ∂y ; which are as follows

9

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 hx ðx; yÞ ¼

ðh1 þ h2 2yÞ3ðh1 h2 Þð4ð1 þ EÞ þ ðh1 h2 ÞM 2 ðF þ ðh1 h2 ÞEÞÞðh1 þ h2 2yÞcosh½ðh1 h2 ÞM þ2ðh1 h2 ÞðF þ h1 h2 Þ2 M 2 Wecosh½Mðh1 þ h2 2yÞ 2

4ðh1 h2 ÞðF þ h1 h2 Þð3 þ ðF þ h1 h2 ÞM WeÞcosh½Mðh1 yÞ 4ðh1 h2 ÞðF þ h1 h2 Þð3 þ ðF þ h1 h2 ÞM 2 WeÞcosh½Mðh2 yÞ þ6MðF 2 We þ ðh1 h2 Þ2 ðh1 þ h2 þ 2h1 E þ 2h2 E þ We2y4EyÞ Fðh1 h2 Þðh1 þ h2 2ðWe þ yÞÞÞsinh½ðh1 h2 ÞM6ðh1 h2 Þ2 ðF þ h1 h2 ÞMsinh½Mðh1 yÞ6ðh1 h2 Þ2 ðF þ h1 h2 ÞMsinh½Mðh2 yÞÞÞ



ð12ðh1 h2 Þððh1 h2 ÞMcosh 12 ðh1 h2 ÞM 2sinh 12ðh1 h2 ÞM Þ2 Þ

2

12

ð12ðh1 h2 ÞMððh1 h2 ÞM cosh½ð1=2Þðh1 h2 ÞM2sinh½ð1=2Þðh1 h2 ÞMÞ2 Þ

ðRm ð3ðh1 h2 Þð4ð1 þ EÞ þ ðh1 h2 ÞM 2 ðF þ ðh1 h2 ÞEÞÞ

1

Fig. 16. Variation of magnetic force function Φ with y for different values of Rm at a¼ 0.7, b ¼1.2, d ¼ 2, x ¼0, Q ¼2, We¼ 0.02, E¼ 2, φ ¼ 2π ; and M ¼2.

7 6 7 6 ðF þ h1 h2 ÞM cosh½Mðh2 yÞ þ 3ððh1 h2 ÞMð2h22 þ h1 ð2h1 7 6 7 6 7 6 h2 ð4 þ ðh1 h2 Þ2 M 2 ÞEÞFðh1 h2 Þð2 þ M 2 ðh1 yÞðh2 yÞÞ 7 6 6 ðh þ h Þð4 þ ð4 þ ðh h Þ2 M 2 ÞEÞy þ ð4 þ ð4 þ ðh h Þ2 M 2 ÞEÞy2 Þ 7 1 2 1 2 1 2 7 6 7 6 7 6 þðh1 h2 ÞMð2h2 Mð2h12 þ 4h1 h2 E þ h13 h2 M 2 E2h12 h22 M 2 E 7 6 7 6 2 7 6 þh1 h3 M 2 EFðh1 h2 Þð2 þ M 2 ðh1 yÞðh2 yÞÞðh1 þ h2 Þð4 þ ð4þ 7 6 7 6 2 2 2 2 2 7 6 ðh1 h2 Þ M ÞEÞy þ ð4 þ ð4 þ ðh1 h2 Þ M ÞEÞy Þ cosh½ðh1 h2 ÞM 7 6 7 6 ðF 2 M 2 Weðh1þ h2 2yÞ þ ðh1 h2 Þ2ð4 þ M 2 ð2h1 h2þ 4h1 h2 E þ h1 We 7 6 7 6 2 7 6 þh2 We2ðh1 þ h2 þ 2ðh1 þ h2 ÞE þ WeÞy þ 2ð1 þ 2EÞy ÞÞ þ 2Fðh1 h2 Þ 7 6 6 ð2 þ M ðh ðWe þ yÞ þ h ðh þ We þ yÞyð2We þ yÞÞÞÞ sinh½ðh h ÞM 7 2 2 1 2 1 2 7 6 7 6 7 6 ðh1 h2 ÞðF þ h1 h2 Þ2 M 2 We sinh½Mðh1 þ h2 2yÞ þ 4ðh1 h2 Þ 7 6 7 6 ðF þ h h Þð3 þ ðF þ h h ÞM 2 WeÞsinh½Mðh yÞ þ 4ðh h Þ 1 2 1 2 1 1 2 5 4 2 ðF þ h1 h2 Þð3 þ ðF þ h1 h2 ÞM WeÞsinh½Mðh2 yÞÞÞ

2

0

y

Jz

Φ¼

ðRm ð6ðh1 h2 Þ2 ðF þ h1 h2 ÞM cosh½Mðh1 yÞ þ 6ðh1 h2 Þ2

-1

3

8.5

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

8 7.5 -2

-1

0

1

2

y Fig. 17. Variation of current density distribution Jz with y for different values of M at a ¼0.7, b¼ 1.2, d ¼ 2, φ ¼ 2π ; E ¼2, Q ¼2, Rm ¼ 4, We¼0.01, x ¼0.

⋅

ð27Þ

0

12 Q = 0.0

Q = 0.0 Q = 0.5

-5

Q = 0.5

11.5

Q = 1.0

Q = 1.0

Jz

Φ

11

-10

10.5

-15

-20 -2

10

-1

0

1

2

y Fig. 15. Variation of magnetic force function Φ with y for different values of Q at a¼ 0.7, b ¼1.2, d ¼2, x ¼0, M¼ 2, We¼ 0.02, φ ¼ 2π ; E ¼2, and Rm ¼ 4.

9.5 -2

-1

0

1

2

y Fig. 18. Variation of current density distribution Jz with y for different values of Q at a¼ 0.7, b ¼1.2, d ¼ 2, φ ¼ 2π ; E¼ 2, M ¼4, Rm ¼4, We¼ 0.01, and x¼ 0.

148

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Fig. 19. Stream lines for different values of d. (a) and (b) for d ¼ 1.7, (c) and (d) for d ¼1.68, (a) and (c) for ϕ ¼0, (b) and (d) for φ ¼ 2π : The other parameters are a ¼0.5, b¼ 0.7, M ¼2.1, Q¼ 2.8, and We¼ 0.02.

Fig. 20. Stream lines for different values of M. (a) and (b) for M ¼2.6, (c) and (d) for M ¼ 2.4, (a) and (c) for ϕ¼ 0, (b) and (d) for φ ¼ 2π : The other parameters are a¼ 0.5, b¼ 0.7, d ¼1.5, Q¼ 2.8, and We ¼0.02.

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2. Multisinusoidal wave

Also, the current density distribution takes the following form 2

ðRm ð3ð4ð1 þ EÞ þ ðh1 h2 ÞM 2 ðF þ ðh1 h2 ÞEÞÞ þ 3ð4ð1 þ EÞ

3

7 6 7 6 þðh1 h2 ÞM 2 ðF þ ðh1 h2 ÞEÞÞcosh½ðh1 h2 ÞM þ 3ðh1 h2 Þ 7 6 7 6 ðF þ h1 h2 ÞM 2 cosh½Mðh1 yÞ þ Mð3ðh1 h2 ÞðF þ h1 h2 Þ 7 6 7 6 7 6 Mcosh½Mðh2 yÞ þ 2ð3ðFh1 þ h2 2h1 E þ 2h2 EÞsinh½ðh1 h2 ÞM 7 6 6 þðF þ h h Þð3sinh½Mðh yÞ þ 3sinh½Mðh yÞ þ ðF þ h h ÞM 2 We 7 1 2 1 2 1 2 5 4 ðsinh½Mðh1 þ h2 2yÞ þ sinh½Mðh1 yÞ þ sinh½Mðh2 yÞÞÞÞÞÞÞ ⋅ J z ðx; yÞ ¼



6ððh1 h2 ÞMcosh 12 ðh1 h2 ÞM 2sinh 12ðh1 h2 ÞM Þ2 Þ

ð28Þ 4. Numerical solution To get the numerical solution of the velocity profile we solved the Eq. (14) with the relevant boundary conditions given in Eq. (22) by employing shooting method. The comparative study is also made to see the validity of the results. Case a (Q¼  4.5) Case b (Q¼  4.4)

h1 ðxÞ ¼ 1 þ a sin 2nπx; h2 ðxÞ ¼ db sin ð2nπx þ ϕÞ: 3. Triangular wave " # 8 1 ð1Þmþ1 sin ð 2π ð 2m1 Þ x Þ ; h1 ðxÞ ¼ 1 þ a 3 ∑ π m ¼ 1 ð2m1Þ2 "

# 8 1 ð1Þmþ1 h2 ðxÞ ¼ db 3 ∑ sin ð2π ð2m1Þ x þ ϕÞ : π m ¼ 1 ð2m1Þ2 4. Trapezoidal wave " # 32 1 sin 8π ð2m1Þ h1 ðxÞ ¼ 1 þ a 2 ∑ sin ð 2π ð 2m1 Þ x Þ ; π m ¼ 1 ð2m1Þ2 " h2 ðxÞ ¼ db

5. Expressions for different wave shape The non- dimensional expressions for three considered wave form are given as 1. Sinusoidal wave h1 ðxÞ ¼ 1 þ a sin 2πx; h2 ðxÞ ¼ db sin ð2πx þ ϕÞ:

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# 32 1 sin 8π ð2m1Þ ∑ sin ð 2π ð 2m1 Þ x þ ϕ Þ : π 2 m ¼ 1 ð2m1Þ2

5. Square wave " # 4 1 ð1Þmþ1 ∑ cos ð2ð2m1Þ πxÞ ; h1 ðxÞ ¼ 1 þ a π m ¼ 1 ð2m1Þ " h2 ðxÞ ¼ db

# 4 1 ð1Þmþ1 ∑ cos ð2ð2m1Þ πx þ ϕÞ ; π m ¼ 1 ð2m1Þ

Fig. 21. Stream lines for different values of Q. (a) and (b) for Q ¼3, (c) and (d) for Q ¼ 3.8, (a) and (c) for ϕ¼ 0, (b) and (d) for φ ¼ 2π : The other parameters are a¼ 0.5, b ¼0.7, d ¼1.5, M¼ 2.1, and We¼ 0.02.

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6. Graphical results and discussion In this section graphical results of the problem under discussion are discussed. The expression for pressure rise is calculated numerically using mathematics software Mathematica. The validity of the results is discussed through the comparison of perturbation and numerical solution. Figs. 1 and 2 show the comparison between perturbation and numerical solution. It is observed from Figs. 1 and 2 that for large values of volume flow rate Q, the perturbation and numerical solutions are almost identical (see Fig. 1), while for small values of volume flow rate Q there is a

difference between the perturbation and numerical solutions (see Fig. 2). The velocity profile for different values of M, Q and We are shown in Figs. 3–5. It is observed from Fig. 3 that with an increase in M, the amplitude of the velocity decreases in the center and near the channel wall the velocity increases. From Fig. 4 it is observed that the magnitude of the velocity profile decreases with an increase in volume flow rate Q. It is observed from Fig. 5 that the velocity profile increases in the upper half of the channel and decreases in the lower half of the channel with an increase in Weissenberg number We. The graphical results of pressure rise for different values of amplitude of wave a, width of the channel d,

Fig. 22. Stream lines for different wave forms. (a) Sinusoidal wave (b) Multisinusoidal wave (c) Triangular wave (d) Trapezoidal wave and (e) Square wave. The parameters are a ¼0.5, b ¼ 0.7, d ¼1.8, M ¼ 2.1, Q¼ 3.4, We ¼0.02, and ϕ ¼0.

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amplitude ratio ϕ, Hartmann number M and electric field E are shown in Figs. 6–10. It is observed from Fig. 6 that the pressure rise increases in the retrograde (Δp 40,Qo 0) and peristaltic pumping (Δp 40,Q40) regions, while in the copumping region (Δp o0,Q40) the behavior is quite opposite. Here the pressure rise decreases with an increase in a. From Figs. 7 and 8 it is observed that the pressure rise decreases in the retrograde and peristaltic pumping regions with an increase in d and ϕ, while in copumping region the pressure rise increases with an increase in d and ϕ. It is observed from Fig. 9 that with an increase in M the pressure rise increases in peristaltic and retrograde pumping regions. It is observed that the behavior of pressure rise in all the regions is same with an increase in E (see Fig. 10 ). Here pressure rise increases with an increase in E. The graphical results of the pressure gradient for different values of d, ϕ and M are shown in Figs. 11–13. The graphical results of the magnetic force function Φ for different values of M, Q and Rm are shown in Figs. 14–16. It is observed that the magnitude of Φ increases with an increase in M and Rm while it decreases with an increase in Q. The graphical results of the current density distribution Jz for different values of M and Q are shown in Figs. 17 and 18. It is observed from Fig. 17 that the current density distribution Jz increases at the center of the channel with an increase in M. The current density distribution Jz decreases with an increase in Q (See Fig. 18). Another interesting phenomena in peristaltic motion is trapping. It is basically the formulation of an internally circulating bolus of fluid by closed stream lines. The trapped bolus pushed a head along a peristaltic waves. Figs. 19–21 illustrate, the stream lines for different values of d, M and Q for both symmetric and asymmetric channel. It is observed that the size of the trapping bolus increases with an increase in d. It is also observed that for a symmetric channel the trapping bolus is symmetric about the center line of the channel (see panels (a) and (c)), while in case of asymmetric channel the bolus tends to shift towards left side of the channel due to phase angle (see panels (b) and (d)). It is observed from Figs. 20 and 21 that the size of the trapping bolus decreases with an increase in M and Q. Fig. 22 shows the stream lines for five different wave forms. 7. Concluding remarks The effects of induced magnetic field on the peristaltic transport of a Williamson fluid model in an asymmetric channel is studied. The problem is simplified using lubrication approach. Both analytical and numerical solutions is caluclated for the present problem. The expressions for pressure rise, pressure gradient, stream function, magnetic force function, current density distribution are obtained analytically. The main findings can be summarized as follows. 1. The amplitude of the velocity decreases in the center and near the channel wall the velocity increases with an increase in M, while the magnitude of the velocity profile decreases with an increase in volume flow rate Q. 2. The velocity profile increases in the upper half of the channel and decreases in the lower half of the channel with an increase in Weissenberg number We. 3. The magnitude value of magnetic force function increases with an increase in M and Rm while it decreases with an increase in Q. 4. The current density distribution Jz increases at the centre of the channel with an increase in M, while current density distribution Jz decreases with an increase in Q The size of the trapping bolus increases with an increase in d. It is also observed that for a symmetric channel the

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