Consequence of nanofluid on peristaltic transport of a hyperbolic tangent fluid model in the occurrence of apt (tending) magnetic field

Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

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Consequence of nanofluid on peristaltic transport of a hyperbolic tangent fluid model in the occurrence of apt (tending) magnetic field Safia Akram a,n, S. Nadeem b a b

Department of Basic Sciences, MCS, National University of Sciences and Technology, Rawalpindi 46000, 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 18 July 2013 Received in revised form 18 January 2014 Available online 2 February 2014

In the current study, sway of nanofluid on peristaltic transport of a hyperbolic tangent fluid model in the incidence of tending magnetic field has been argued. The governing equations of a nanofluid are first modeled and then simplified under lubrication approach. The coupled nonlinear equations of temperature and nano particle volume fraction are solved analytically using a homotopy perturbation technique. The analytical solution of the stream function and pressure gradient are carried out using perturbation technique. The graphical results of the problem under discussion are also being brought under consideration to see the behavior of various physical parameters. & 2014 Elsevier B.V. All rights reserved.

Keywords: Peristaltic flow Nano fluid particles Hyperbolic tangent fluid model Apt magnetic field Asymmetric channel

1. Introduction Peristaltic transports are a vigorous research area because of their spacious range applications in physiology and industry. Such flows occurs in urine transport from kidney to bladder, swallowing food through the esophagus, mixing of food and chyme movement in the intestine, circulation of blood in small blood vessels and blood pumps in heart lung machines. To recognize peristaltic action numerous theoretical and experimental studies have been accomplished [1–10]. In recent years, the study of MHD flow problems has achieved significant interest because of its wide-ranging engineering and medical applications [11–14]. An effect of inclined magnetic field on magneto fluid flow through porous medium between two inclined wavy porous plates was explored in [15]. Recently, Nadeem and Akram [16] have discussed the inclined magnetic field in viscous peristaltic phenomena in presence of heat and mass transfer, where an exact solution of reduced equations has been carried out. The study of nano fluids is another important area which has recently attracted the attention of many researchers. Since the pioneering work done by Choi [17], various aspects of nanofluid have been discussed. Masuda et al. [18] have examined that the effective thermal conductivity of nano fluids is expected to enhance the heat transfer as compared to conventional heat transfer. Some recent

studies of nano fluid due to stretching sheet and peristaltic motion are given in Refs. [19–24]. In this paper we have discussed the influence of nanofluid on peristaltic transport of a hyperbolic tangent fluid model under the effects of inclined magnetic field. The paper is arranged as: The mathematical formulation of the present problem is given in Section 2. In Section 3, the analytical solution of the proposed problem is computed with the help of homotopy perturbation and regular perturbation technique. The graphical results of the present problem are defined in Section 4.

2. Mathematical formulation We consider the peristaltic transport of an incompressible nonNewtonian fluid (hyperbolic tangent model) in a two dimensional channel of width d1 þd2, under the effects of apt magnetic field. The channel asymmetry is produced due to different amplitudes and phases of the peristaltic waves. Heat transfer along with nano particle phenomena has been taken into description. The lower wall of the channel is sustained at temperature T1 and nano particle volume fraction C1 while the upper wall has temperature T0 and nano particle volume fraction C0 The geometry of the wall surface is defined as Y ¼ H 1 ¼ d1 þ a1 cos

n

Corresponding author. E-mail addresses: safi[email protected], drsafi[email protected] (S. Akram). 0304-8853/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2014.01.052


2π

λ


  2π ðX  ctÞ ; Y ¼ H 2 ¼  d2  b1 cos ðX  ctÞ þ ϕ ;

λ

ð1Þ where a1 and b1 are the amplitudes of the waves, λ is the wave length, d1 þ d2 is the width of the channel, c is the velocity of

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S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

propagation, t is the time and X is the direction of wave propagation, the phase difference ϕ varies in the range 0 r ϕ r π ; ϕ ¼ 0 corresponds to symmetric channel with waves out of phase and ϕ ¼ π the waves are in phase, and further a1 ; b1 ; d1 ; d2 and ϕ satisfies the condition a21 þb1 þ 2a1 b1 cos ϕ r ðd1 þ d2 Þ2 :

d1 d1 Γ c _ γ_ d1 ; γ¼ S ; S ¼ S ; We ¼ ; M¼ d1 η0 c xy yy η0 c yy c

Pr ¼

ν τDT ðT 1  T 0 Þ τDB ðC 1  C 0 Þ ρgαd21 ðT 1  T 0 Þ ; ; NT ¼ ; Nb ¼ ; Gr ¼ T 0ν υ η0 c α

Br ¼

ρg αd21 ðC 1  C 0 Þ υ ; Le ¼ : η0 c DB

2

The governing equations for an incompressible nanofluid under the effect of inclined magnetic field are given by [16,22] ∂U ∂V þ ¼ 0; ∂X ∂Y   ∂U ∂U ∂U ∂P ∂ ∂ þU þV ¼ þ ðSXX Þ þ ðSXY Þ ρf ∂t ∂X ∂Y ∂X ∂X ∂Y

ð2Þ

Re δðΨ y Ψ xy  Ψ x Ψ yy Þ ¼ 

∂V ∂V ∂V þU þV ∂t ∂X ∂Y

 ¼



∂T ∂T ∂T þU þV ∂t ∂X ∂Y

∂C ∂C ∂C þU þV ∂t ∂X ∂Y



¼α



∂2 T





þ þ τ DB ∂X 2 ∂Y 2 "    #) ∂T 2 ∂T 2 þ ;  ∂X ∂Y



 ¼ DB

∂ C 2

∂X 2

þ

∂ C 2

∂Y 2





∂C ∂T ∂C ∂T þ ∂X ∂X ∂Y ∂Y

3



DT T0

S ¼  ½½η1 þðη0 þ η1 Þ tan hðΓ γ_ Þn γ_ ;

1 2 2 ðθyy þ δ θxx Þ þ Nb ðδ θx Φx þ θy Φy Þ Pr 2 þN T ðδ ðθx Þ2 þðθy Þ2 Þ;

Re δðΨ y θx  Ψ x θy Þ ¼

Re δ LeðΨ y Φx  Ψ x Φy Þ ¼ ðΦyy þ δ Φxx Þ þ δ 2

ð6Þ

ð7Þ

in which S is the extra stress tensor, η1 is the infinite shear rate viscosity, η0 is the zero shear rate viscosity, Γ is the time constant, n is the power law index and γ_ is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi ffi u u1 1 t ð8Þ ∑ ∑ γ_ γ_ ¼ γ_ ¼ Π; 2 i j ij ji 2 where

Π ¼ tracðgradV þ ðgradVÞT Þ2 here Π is the second invariant strain tensor. We consider the constitution Eq. (7), the case for which η1 ¼ 0 and Γ γ_ o1: The component of extra stress tensor therefore, can be written as S ¼  η0 ½ðΓ γ_ Þn  γ_ ¼  η0 ½ð1 þ Γ γ_ 1Þn γ_ ¼  η0 ½1 þ nðΓ γ_ 1Þγ_ :

ð9Þ

The coordinate frames are related by the following transformation x ¼ X  ct; y ¼ Y; u ¼ U  c; v ¼ V; and pðxÞ ¼ PðX; tÞ:

ð10Þ

Defining the following non-dimensional quantities 2

y u v d1 d2 d p ct H1 x ¼ ; y ¼ ; u ¼ ; v ¼ ; δ ¼ ; d ¼ ; p ¼ 1 ; t ¼ ; h1 ¼ ; d1 c c d1 d1 λ λ η0 cλ λ h2 ¼

ð13Þ



ð5Þ

  2  DT ∂ T ∂2 T þ þ ; T0 ∂X 2 ∂Y 2

∂p ∂ 2 ∂ þδ ðSyx Þ þ δ ðSyy Þ ∂y ∂x ∂y

þ M 2 δ sin ΘððΨ y þ 1Þ cos Θ þ δΨ x sin ΘÞ;

ð4Þ

where U, V are the velocities in X and Y directions in fixed frame, ρf is density of fluid base, P is the pressure, ν is the kinematic viscosity, is the temperature, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, τ ¼ ððρcÞp =ðρcÞf Þ is the ratio of the effective heat capacity of the nanoparticle material and heat capacity of the fluid with ρ being the density, c is the volumetric volume expansion coefficient and ρp is the density of the particles. The constitutive equation for hyperbolic tangent fluid is given by [7]

x

ð12Þ Re δ ð  Ψ y Ψ xx þ Ψ x Ψ xy Þ ¼ 

∂P ∂ ∂ þ ðSYX Þ þ ðSYY Þ ∂Y ∂X ∂Y

∂2 T

∂p ∂ ∂ þ δ ðSxx Þ þ ðSxy Þ þ Gr θ þ Br Φ ∂x ∂x ∂y

ð3Þ

þ sB20 sin ΘðU cos Θ  V sin ΘÞ; 

ð11Þ

 M 2 cos ΘððΨ y þ1Þ cos Θ þ δΨ x sin ΘÞ;

þ ρg αðT  T 0 Þ þ ρg αðC C 0 Þ; 

s K' ; B d ; α¼ υ 0 1 cp ρ

Using Eqs. (10) and (11) the resulting equations in terms of stream function Ψ ðdropping the bars; u ¼ ð∂Ψ =∂yÞ; v ¼  δð∂Ψ =∂xÞÞ can be written as;

 sB20 cos ΘðU cos Θ  V sin ΘÞ

ρf

rffiffiffiffiffi

Sxy ¼

H2 a1 b1 cd1 Ψ T T0 λ ; Ψ¼ Sxx ; ; a ¼ ; b ¼ ; Re ¼ ; θ¼ ; S ¼ d2 d1 d1 v cd1 T 1  T 0 xx η0 c

2 NT

Nb

θxx þ

NT θyy ; Nb

ð14Þ ð15Þ

where ∂2 Ψ ; Sxx ¼ 2½1 þnðWeγ_  1Þ ∂x∂y  2  2 ∂ Ψ 2∂ Ψ Sxy ¼ ½1 þ nðWeγ_  1Þ ; δ 2 2 ∂y ∂x ∂2 Ψ ; Syy ¼  2δ½1 þ nðWeγ_  1Þ ∂x∂y "  2 2  2  2 2 #1=2 2 2 ∂ Ψ ∂ Ψ 2∂ Ψ 2 ∂ Ψ γ_ ¼ 2δ2 þ  δ þ 2 δ ; ∂x∂y ∂x∂y ∂y2 ∂x2

ð16Þ

The corresponding boundary conditions in terms of stream function are defined as q at y ¼ h1 ¼ 1 þ a cos 2π x; 2 q Ψ ¼  at y ¼ h2 ¼  d  b cos ð2π x þ ϕÞ; 2 ∂Ψ ¼  1 at y ¼ h1 and y ¼ h2 ; ∂y

Ψ¼

ð17Þ

θ ¼ 0 on y ¼ h1 ; θ ¼ 1 on y ¼ h2 ;

ð18Þ

Φ ¼ 0 on y ¼ h1 ; Φ ¼ 1 on y ¼ h2 :

ð19Þ

where q is the flux in the wave frame, a; b; ϕ and d satisfy the relation a2 þ b þ 2ab cos ϕ rð1 þdÞ2 : 2

Under the assumption of long wave length Reynolds number, Eqs. (12)–(16) become
   2  ∂p ∂ ∂2 Ψ ∂ Ψ 1 þn We 2  1 0¼  þ ∂x ∂y ∂y ∂y2  M 2 cos 2 ΘðΨ y þ1Þ þGr θ þ Br Φ;

δ⪡1 and low

ð20Þ

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

∂p ; ∂y

185

2a02 a0 a01 2 a02 3 y 2 y y  y m41 m1 2m21 3m21 !

ð21Þ

M 2 cos 2 ½Θ B 

 2 1 ∂2 θ ∂θ ∂Φ ∂θ þ N þ N ¼ 0; T b Pr ∂y2 ∂y ∂y ∂y

ð22Þ

þB1 m1 cosh ½m1 y þ A1 m1 sinh ½m1 y þ 1

∂2 Φ N T ∂ 2 θ þ ¼ 0: ∂y2 Nb ∂y2

ð23Þ

þWeð2n 

ð24Þ

A1 m31 sinh ½m1 y  M 2 ð cos ½ΘÞ2 ðB2 þ2b8 y þ3b9 y2

0¼ 

2a02 a0 a01 a02  2  2 y  2 y2 þA1 m21 cosh ½m1 y m41 m1 m1 m1 ! a01 2a02 þB1 m21 sinh ½m1 y  2  2 y þB1 m31 cosh ½m1 y m1 m1 !

Elimination of pressure from Eqs. (20) and (21), yields 2

∂ ∂y2




  2  ∂2 Ψ ∂ Ψ ∂2 Ψ ∂θ ∂Φ ¼ 0;  M 2 cos 2 Θ 2 þ Gr þ Br 1 þ n We 2  1 ∂y ∂y ∂y ∂y2 ∂y

þ4b10 y3 þ ðb1 þA4 m1 b4 m1 þ 2b2 y þ b5 m1 yþ 3b3 y2 þ b6 m1 y2

3. Solution of the problem

þb7 m1 y3 Þcosh ½m1 y þ2b12 m1 cosh ½2m1 y  ðb5 þ A3 m1 þ b0 m1 þ 2b6 yþ b1 m1 y  3b7 y2 þ b2 m1 y2

3.1. Homotopy perturbation method

þb3 m1 y3 Þsinh ½m1 y þ 2b11 m1 sinh ½2m1 yÞ  þð1 nÞð6b9 þ 24b10 y þ 6b3 þ 6b6 m1 þ3b1 m21 þ A4 m31

Since Eqs. (22) and (23) are coupled nonlinear equation so the exact solution of Eqs. (22) and (23) seems to be impossible. The homotopy perturbation technique is applied to calculate the solution of the coupled nonlinear equations. For our convenience we have taken L ¼ ð∂2 =∂y2 Þ as the linear operator. We define the initial guess as

θ10 ¼

h1  y h1  y ; Φ10 ¼ : h1  h2 h1 h2

þb4 m31 þ 18b7 m1 y þ 6b2 m21 yþ b5 m31 y þ 9b3 m21 y2 þb6 m31 y2 þb7 m31 y3 cosh ½m1 y þ 8b12 m31 cosh ½2m1 y  þ 6b7 þ 6b2 m1 þ 3b5 m21 þ A3 m31 þ b0 m31 þ 18b3 m1 y þ 6b6 m21 y þb1 m31 y þ 9b7 m21 y2 þ b2 m31 y2 þ b3 m31 y3 sinh ½m1 y

ð25Þ

þ8b11 m31 sinh ½2m1 yÞÞ

Using the similar procedure of homotopy perturbation method as done in [25–27], the solution of Eqs. (22) and (23) is straight forward written as

θ¼

þGr

h1 y Pr þ ðN T þ N b Þð  y2 þ ðh1 þ h2 Þy  h1 h2 Þ h1  h2 2ðh1  h2 Þ2

þ

Pr2 ðN T þ N b Þð2N T þ N b Þ 2ðh1  h2 Þ

3 3

3



þ

h1 h ðh1 þ h2 Þ Þ; left: 1 2 3

3

Φ¼

3

3

ðh1 þ h2 Þ2 h1 ðh2  h1 Þ h1 ðh1 þ h2 Þ2 yþ þ 3ðh1  h2 Þ 2 2

h1  y PrN T ðNT þN b Þ 2 þ ðy  ðh1 þ h2 Þy þ h1 h2 Þ: h1  h2 2Nb ðh1 h2 Þ2

ð27Þ

3

Δp ¼



3

2

ð26Þ

3

Pr2 ðN T þ N b Þð2N T þ N b Þ

y3 y2 ðh  h1 Þ þ ðh1 þ h2 Þ  2 y 3 2 3ðh1  h2 Þ 2ðh1  h2 Þ !! 3 3 3 2 ðh1 þh2 Þ2 h1 ðh2 h1 Þ h1 ðh1 þ h2 Þ2 h1 h1 ðh1 þ h2 Þ  yþ þ þ 3ðh1 h2 Þ 2 2 2 3 ! h1  y PrN T ðN T þ N b Þ 2 þ ðy  ðh1 þh2 Þy þh1 h2 Þ ; ð30Þ þBr h1  h2 2N b ðh1  h2 Þ2 þ

y3 y2 ðh  h1 Þ þ ðh1 þh2 Þ right:  2 y 3ðh1 h2 Þ 3 2



h1  y Pr þ ðN T þ Nb Þð  y2 þ ðh1 þ h2 Þy  h1 h2 Þ h1  h2 2ðh1  h2 Þ2

Z

1

0



 dp j ; dx y ¼ 0

ð31Þ

where the constants appearing in Eqs. (29) and (30) are defined in Appendix.

Eqs. (20) and (24) are highly non-linear equations, so the exact solutions are looking difficult, Therefore, we apply regular perturbation technique. Now we expand ψ, p and q as;

0.5

Ψ ¼ Ψ 0 þWeðΨ 1 Þ; p ¼ p0 þWeðp1 Þ;

ð28Þ

q ¼ q0 þ Weðq1 Þ: 0

u

Substituting Eq. (28) into Eqs. (17), (20) and (24), then solving the resulting zeroth and first order systems, we arrive at

Ψ ¼ A þ By þ A1 cosh ½m1 y þ B1 sinh ½m1 y 

a02 a0 þ m41 2m21

!

y2 

a01 3 a02 4 y  y WeðA2 þB2 y þA3 cosh ½m1 y 6m21 12m21

-0.5

M = 1.0 M = 1.5

þA4 sinh ½m1 y þ b0 cosh ½m1 y þ b1 y cosh ½m1 yb2 y2 cosh ½m1 y

M = 2.0

þb3 y3 cosh ½m1 y þ b4 sinh ½m1 y þ b5 y sinh ½m1 y

M = 2.5

þb6 y2 sinh ½m1 yb7 y3 sinh ½m1 y þ b8 y2 þ b9 y3 þ b10 y4 þb11 cosh ½2m1 y þ b12 sinh ½2m1 yÞ; dp a01 2a02 ¼ ð1  nÞ  2  2 þ B1 m31 cosh ½m1 y þ A1 m31 sinh ½m1 y dx m1 m1

ð29Þ !

-1 -1

-0.5

0

0.5

1

1.5

y Fig. 1. Velocity profile for different values of M for fixed a ¼ 0:7; b ¼ 0:7; d ¼ 1; x ¼ 0; We ¼ 0:02; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; Q ¼ 2; Θ ¼ ðπ=8Þ; n ¼ 2; ϕ ¼ ðπ=2Þ:

186

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

4. Numerical results and discussion

3 2 1 0 Δp

The objective of this section is to study the graphical consequences of the current problem. The expression for pressure rise and pressure gradient is calculated using a mathematics software because the definition of pressure rise involves integration of dp/dx which is not solvable analytically. In order to see the behavior of velocity profile for different parameters of interest Figs. 1–3 are prepared. Fig. 1 shows the velocity profile for different values of Hartmann number M. It is observed from Fig. 1 that with an increase in M the velocity profile decreases in the magnitude at the boundary and the amplitude of the velocity profile reduce at the center of the channel. This is for the reason that the magnetic field acts in the transverse direction to the flow and magnetic force resists the flow. Moreover the velocity profiles are parabolic in nature. Fig. 2 shows the velocity profile for different values of the inclination of the magnetic field Θ. It is observed that the behavior of Θ on velocity profile is quite opposite as compare to those of Hartmann number M. Moreover, it is also observed that at the center of the channel the amplitude of the

-1 M = 1.0 -2

M = 1.5 M = 2.0

-3

M = 2.5 -4 -1

-0.5

0

0.5

1

1.5

2

2.5

3

Q Fig. 4. Variation of pressure rise Δp with volume flow rate Q for different values of M and fixed values of a ¼ 0:7; b ¼ 0:7; We ¼ 0:02; Θ ¼ ðπ=6Þ; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; d ¼ 2; n ¼ 2; ϕ ¼ ðπ=2Þ:

-0.1 -0.2 -0.3

3

-0.4

n = 0.0 2

n = 0.2

u

-0.5

n = 0.6

-0.6 -0.7

Θ = 0.0

-0.8

Θ = π/4

-0.9

Θ = π/3

-1 -1

-0.5

0

0.5

Δp

1

n = 0.8

0

-1 1

1.5

-2

y Fig. 2. Velocity profile for different values of Θ for fixed a ¼ 0:7; b ¼ 0:7; d ¼ 1; x ¼ 0; We ¼ 0:01; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; Q ¼ 1; M ¼ 3; n ¼ 2; ϕ ¼ ðπ=2Þ:

-3 -1

-0.5

0

0.5

1

1.5

2

2.5

3

Q 40

Fig. 5. Variation of pressure rise Δp with volume flow rate Q for different values of n and fixed values of a ¼ 0:7; b ¼ 0:7; We ¼ 0:02; Θ ¼ ðπ=6Þ; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; d ¼ 2; M ¼ 1; ϕ ¼ ðπ=2Þ:

d = 0.6

30

d = 0.7 20

d = 0.8

Δp

10

d = 0.9

0 -10 -20 -30 -40 -50 -1

-0.5

0

0.5

1

1.5

2

2.5

3

Q Fig. 3. Variation of pressure rise Δp with volume flow rate Q for different values of d and fixed values of a ¼ 0:7; b ¼ 0:7; We ¼ 0:02; Θ ¼ ðπ=6Þ; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; M ¼ 1; n ¼ 2; ϕ ¼ ðπ=2Þ:

velocity profile increases with an increase in Θ. Figs. 3–6 show the variation of pressure rise with volume flow rate Q for different values of width of the channel d, Hartmann number M, Power law index n and inclination angle of magnetic field Θ. It is depicted from Fig. 3 that in the augmented pumping region ðΔp o 0; Q 4 0Þ the pressure rise increases with an increase in the width of the channel d, while in the peristaltic pumping ðΔp 4 0; Q 4 0Þ region the pressure rise decreases. Fig. 4 shows the behavior of pressure rise with volume flow rate Q for different values of M. It is observed from Fig. 4 that in the retrograde pumping ðΔp 40; Q o 0Þ region the pressure rise increases with an increase in M, whereas in peristaltic ðΔp 4 0; Q 4 0Þ and augmented pumping ðΔp o0; Q 4 0Þ regions the behavior is quite opposite, here the pressure rise decreases with an increase in M. Fig. 5 is plotted to see the effects of pressure with volume flow rate Q for different values of power law index n. It is depicted from Fig. 5 that the pressure rise decreases in the retrograde pumping ðΔp 40; Q o 0Þ region, while in Peristaltic

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

0.75

1 Θ = π/9

0.7

Θ = π/8 Θ = π/7

0.65

Θ = π/6

0.9

Pr = 0.2

0.8

Pr = 0.4

0.7

Pr = 0.6

0.55

0.5

0.5

0.4

0.45

0.3

0.4

0.2

0.35 -1

0.1 -0.5

0

0.5

1

1.5

2

2.5

3

0 -2

Q Fig. 6. Variation of pressure rise Δp with volume flow rate Q for different values of Θ and fixed values of a ¼ 0:7; b ¼ 0:7; We ¼ 0:02; n ¼ 2; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Gr ¼ 0:8; Br ¼ 0:9; d ¼ 2; M ¼ 1; ϕ ¼ ðπ=2Þ:

-1.5

-1

-0.5

0

0.5

1

1.5

y Fig. 9. Temperature profile for different values of Pr for fixed values of a ¼ 0:7; b ¼ 1:2; N T ¼ 0:9; N b ¼ 1:0; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

1

1

0.9

0.9

N = 0.2

0.8

0.8

N = 0.4

0.7

N = 0.6

0.7

Φ

0.5

T T

N = 0.8

N = 0.2

0.4

0.3

N = 0.4

0.3

0.2

N = 0.6

0.2

0.1

N = 0.8

0.1

T T T T

-1.5

-1

-0.5

0

0.5

1

0 -2

1.5

T

0.5

0.4

0 -2

T

0.6

0.6 θ

Pr = 0.8

0.6 θ

Δp

0.6

-1.5

-1

-0.5

y

0

0.5

1

Fig. 10. Concentration profile for different values of N T for fixed values of a ¼ 0:7; b ¼ 1:2; Pr ¼ 1; N b ¼ 0:6; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

1

0.9

0.9

Pr = 0.2

0.8

0.8

Pr = 0.4

0.7

0.7

Pr = 0.6

0.6

0.6

Pr = 0.8

Φ

1

0.5 0.4

0.5 0.4

N = 0.1 b

0.3

1.5

y

Fig. 7. Temperature profile for different values of N T for fixed values of a ¼ 0:7; b ¼ 1:2; N b ¼ 0:6; Pr ¼ 1; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

θ

187

0.3

N = 0.3 b

0.2

N = 0.5

0.2

0.1

N = 0.7

0.1

0 -2

b b

-1.5

-1

-0.5

0

0.5

1

1.5

y Fig. 8. Temperature profile for different values of N b for fixed values of a ¼ 0:7; b ¼ 1:2; N T ¼ 0:9; Pr ¼ 1:2; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

y Fig. 11. Concentration profile for different values of Pr for fixed values of a ¼ 0:7; b ¼ 1:2; N b ¼ 1; N T ¼ 0:9; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

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S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

1 N = 0.1

0.8

b

N = 0.3 b

0.6

N = 0.5 b

0.4

N = 0.7 b

Φ

0.2 0 -0.2 -0.4 -0.6 -0.8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

y Fig. 12. Concentration profile for different values of N b for fixed values of a ¼ 0:7; b ¼ 1:2; Pr ¼ 1; N T ¼ 0:9; d ¼ 2; x ¼ 0, ϕ ¼ ðπ=2Þ:

ðΔp 4 0; Q 4 0Þ and augmented ðΔp o0; Q 4 0Þ pumping regions the pressure rise increases with an increase in power law index n. Fig. 6 shows the variation of pressure rise for different values of inclination angle of magnetic field Θ. It is observed that the behavior of the pressure rise is same throughout all the regions. Figs. 7–9 are displayed to analysis the influence of temperature profile on NT ; N b and Pr: It is explored from Figs. 7–9 that the temperature profile increases with an increase in N T ; N b and Pr. This is physically valid because these parameters show a direct relationship with temperature. To examine the effects of concentration profile on N T ; N b and Pr; Figs. 10–12 are prepared. It is illustrated from figures that the concentration profile decreases with an increase in NT and Pr and increases with an increase in N b . Stream lines for different values of Gr; λ1 and NT are shown in Figs. 13–15. It is depicted from Figs. 13 and 14 that the size of the trapping bolus increases in both upper and lower half of the channel with an increase in Q and We: It is observed form Fig. 15 that the size of the trapping bolus decreases with an increase in Hartmann number M:

Fig. 13. Stream lines for different values of Q ; ðaÞ for Q ¼ 1:9, ðbÞ for Q ¼ 1:96. The other parameters are a ¼ 0:7; b ¼ 0:7; d ¼ 1; ϕ ¼ 0:1; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; We ¼ 0:009; Br ¼ 0:9; Gr ¼ 0:8; M ¼ 1; n ¼ 2; Θ ¼ ðπ=6Þ:

Fig. 14. Stream lines for different values of We; ðaÞ for We ¼ 0, ðbÞ for We ¼ 0:009: The other parameters are a ¼ 0:7; b ¼ 0:7; d ¼ 1; ϕ ¼ 0:1; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Q ¼ 1:9; Br ¼ 0:9; Gr ¼ 0:8; M ¼ 1; n ¼ 2; Θ ¼ ðπ=6Þ:

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

189

Fig. 15. Stream lines for different values of M, ðaÞ for M ¼ 1:2, ðbÞ for M ¼ 1:3: The other parameters are a ¼ 0:7; b ¼ 0:7; d ¼ 1; ϕ ¼ 0:1; N T ¼ 0:9; N b ¼ 0:5; Pr ¼ 2; Q ¼ 1:9; Br ¼ 0:9; Gr ¼ 0:8; We ¼ 0:009; n ¼ 2; Θ ¼ ðπ=6Þ:

5. Concluding remarks In the present study we discuss the effects of nanofluid on peristaltic transport of a hyperbolic tangent fluid model in the presence of inclined magnetic field. The governing equations of a nanofluid are first modeled and then simplified using long wave length approximation. The coupled nonlinear equations of temperature and nano particle volume fraction are solved analytically. The results are discussed through graphs. The main finding can be summarized as follows: (1) It is observed the velocity profile decreases in the magnitude at the boundary and the amplitude of the velocity profile reduce at the center of the channel with an increase in M: (2) It is observed that at the center of the channel the amplitude of the velocity profile increases with an increase in Θ. (3) The temperature profile increases with an increase in N T ; N b and Pr. This is physically valid because these parameters show a direct relationship with temperature. (4) The concentration profile decreases with an increase in N T and Pr and increases with an increase in N b . (5) The size of the trapping bolus increases in both upper and lower half of the channel with an increase in Q and We: (6) The size of the trapping bolus decreases with an increase in Hartmann number M:

 

ðGr þ Br Þ PrðN T þ N b Þ þ ðn  1Þðh1  h2 Þ 2ðn  1Þðh1  h2 Þ2 GrPrð2NT þN b Þ 3ðh1  h2 Þ2

þ

3

3

ðh2  h1 Þ 

 Br N T ðh1 þ h2 Þ ; Nb

a01 ¼ 

;

12a202 n 12a01 a02 n ; a05 ¼ ; 4 ðn  1Þm1 ðn  1Þm41

n   2na02 A1 m21 ð  8a02 B1 m1  2a01 A1 m21 Þ; a07 ¼ ; a06 ¼ n1 n1

n  ð2A21 þ 2B21 Þm61 ; a08 ¼ n1

n  a09 ¼ ð  8a02 B1  4a01 A1 m1  2a0 B1 m21 Þ; n1

n  ð  8a02 A1 m1  2a01 B1 m21 Þ; a10 ¼ n1

n   2na02 B1 m21 ; a12 ¼ a11 ¼ ð2A21 m61 þ 2B21 m61 Þ; n 1 n1

n  8nA1 B1 m61 ð  8a02 A1  4a01 B1 m1  2a0 A1 m21 Þ; a14 ¼ ; a13 ¼ n1 ðn  1Þ a04 ¼

b0 ¼  b2 ¼  b3 ¼

49a07 17a10 5a13 17a11 5a06 a09 þ  ; b1 ¼  þ ; 8m51 4m51 8m61 4m41 4m41 2m31

5a07 a10 þ ; 4m41 4m31

a11 49a11 17a06 5a09 17a07 5a10 a13 ; b4 ¼  þ  ; b5 ¼  þ ; 8m51 4m51 6m31 8m61 4m41 4m41 2m31

5a11 a06 a07 a08 a12 a05 a03 þ ; b7 ¼ ; b8 ¼  þ   ; 4m41 4m31 6m31 4m21 4m21 m41 2m21 a04 a05 ; b10 ¼  ; b9 ¼  6m21 12m21

Gr PrðNT þN b Þ ðn  1Þðh1  h2 Þ

2

 Gr ðh1 þ h2 Þ

Gr Prð2NT þN b Þðh1 þ h2 Þ2 2ðh1 h2 Þ

b11 ¼

ða08 þ a12 Þ a14 ðk0  k1  k2 Þ k4  k5 ; b12 ¼ ; A¼ ; B¼ ; k3 m1 k3 24m41 24m41

 cosech ½ð1=2Þðh1  h2 Þm1 ðk6  k7 þ k8 Þ ; 2k3 cosech ½ð1=2Þðh1  h2 Þm1 ðk9 þ k10 k11 Þ B1 ¼ ; 2k3 k12  k13 þ k14 þk15 þk16 þk17 þk18 þ k19 þ k20 þ k21 þ k22 þ k23 þ k24 ; A2 ¼ k25 A1 ¼

þ

Br PrN T ðN T þ N b Þ N b ðn  1Þðh1  h2 Þ2

a02 ¼ 

!

b6 ¼ 

Appendix a0 ¼ 

n  8a2 2a2 4a0 a02 02 a03 ¼ þ 01 þ n1 m61 m41 m41

Gr Pr2 ðN T þ N b Þð2N T þ N b Þðh1 þh2 Þ 2ðh1 h2 Þ3 ðn  1Þ ;

Gr Pr2 ðN T þ N b Þð2N T þ N b Þ 2ðn  1Þðh1  h2 Þ3

ðk26 þ k27 þk28 þk29 þk30 Þsinh ½ð1=2Þðh1  h2 Þ ; k31 k37 sinh ½2h2 m1  þ k36 cosh ½h1 m1  ; A3 ¼ 2m1 k31

B2 ¼

;

190

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

A4 ¼

ðk38 þ k39 þ k40 þ k41 þ k42 Þ ; 2m1 k31

2

3

2

þ 3a01 ðh1 þ h2 ÞÞm21

2 2 þ h1 h2 ð6a0 þ 2a01 ðh1 þ h2 Þ þa02 ðh1 þ h1 h2 þ h2 ÞÞm41 Þ

k1 ¼ 6ðh1 þ h2 Þm61 q cosh ½ð1=2Þðh1 h2 Þm1 

2

 ðh1  h2 Þ m1 ð12a02 þ ð6a0 þ 3a01 ðh1 þ h2 Þ 4

2

3

3

4

k2 ¼ m1 ð12a02 ðh1  4h1 h2 þh2 Þ þ a02 ðh1  4h1 h2 4h1 h2 þ h2 Þm21 2

þ2ðh1  4h1 h2 þ h2 Þð3a0 þ a01 ðh1 þ h2 ÞÞm21 þ12ðh1 þ h2 Þm41 Þsinh

½ð1=2Þðh1  h2 Þm1 ;

k3 ¼ 12m51 ððh1  h2 Þm1 cosh ½ð1=2Þðh1  h2 Þm1  2 sinh ½ð1=2Þðh1  h2 Þm1 Þ; 3

3

k4 ¼ ð2a01 h1 m31  2a01 h2 m31 þ 6a0 ðh1  h2 Þðh1 þ h2 Þm31 2

2

þa02 ðh1  h2 Þðh1 þ h2 Þm1 ð12 þ ðh1 þ h2 Þm21 Þ þ12m51 qÞcosh

½ð1=2Þðh1  h2 Þm1 ; 2

2

k5 ¼ ð2ð12a02 ðh1 þ h2 Þ þ ð6a0 ðh1 þ h2 Þ þ 3a01 ðh1 þ h2 Þ 3

3

2

3 2 3 4 þ 2b7 h2 m1 þ 2b1 h1 m21 þ 2b2 h1 m21 þ 2b3 h1 m21  2b1 h1 h2 m21 2 3  2b2 h1 h2 m21  2b3 h1 h2 m21 Þsinh ½h1 m1 ; 2 k21 ¼ ð  2b12 m1 þ 4b11 h1 m1 Þsinh ½2h1 m1  þ ðb12 m1 þ b11 h1 m21 Þsinh ½ðh1 3h2 Þm1 ; 2 k22 ¼ ðb1 þ 2b2 h2 þ 3b3 h2 þ b5 h1 m1 þ 2b6 h1 h2 m1 2 þ 3b7 h1 h2 m1 Þsinh ½ðh1  2h2 Þm1  2 3 2 þ ð  4b8 h1  6b9 h1  8b10 h1 þ 4b8 h2 þ6b9 h2 3 2 3 þ 8b10 h2 þ 2b8 h1 h2 m21 þ 2b9 h1 h2 m21 4 2 2 3 2 þ 2b10 h1 h2 m1  2b8 h1 h2 m1  2b9 h1 h2 m21 4 2  2b10 h1 h2 m1 Þsinh ½ðh1  h2 Þm1 ;

3

þ2a02 ðh1 þ h2 ÞÞm21  12m41 Þsinh ½ð1=2Þðh1  h2 Þm1 ; k6 ¼  ðh1  h2 Þ m1 ð12a02 ðh1  h2 Þ þ ðh1  h2 Þð6a0 þ 2a01 ðh1 þ 2h2 Þ 2

2

þa02 ðh1 þ 2h1 h2 þ 3h2 ÞÞm21 þ 12m41 Þcosh ½h1 m1  12m51 q

2

þ 2b5 h2 m1  2b6 h1 h2 m1 þ 2b6 h2 m1  3b7 h1 h2 m1

2

þ 2a02 ðh1 þ h1 h2 þ h2 ÞÞm21 cosh ½ð1=2Þðh1  h2 Þm1 ; 2

½h2 m1 ;

k20 ¼ ðb1 þ 2b2 h2 þ 3b3 h2  b5 h1 m1 þ 2b6 h1 m1 þ 4b7 h1 m1

2

2

3

2 3 4 þ 2b5 h2 m21 þ 2b6 h2 m21 þ 2b7 h2 m21 Þcosh

k19 ¼ ð  2b11 m1 þ 4b12 h2 m21 Þcosh ½2h2 m1   3m1 ð  2b11 þ b12 ðh1 þh2 Þm1 Þcosh ½ðh1 þ h2 Þm1 ;

cosh ½ð1=2Þðh1  h2 Þm1 ;

2

2

þ 4b3 h2 m1  2b5 h1 h2 m21  2b6 h1 h2 m21  2b7 h1 h2 m21

k0 ¼  ðh1  h2 Þð  24a02  2ð6a0 þ 2a02 ðh1 h2 Þ

2

2

 b1 h2 m1  2b2 h1 h2 m1  3b3 h1 h2 m1 þ2b2 h2 m1

cosh ½h1 m1 ;

k7 ¼ ðh1  h2 Þm1 ð12a02 ðh1  h2 Þ 2

þðh1  h2 Þð6a0 þ h1 ð4a01 þ 3a02 h1 Þ2ða01 þ a02 h1 Þh2 þ a02 h2 Þm21 12m41 Þcosh ½h2 m1  þ 12m51 q cosh ½h2 m1 ; k8 ¼ 2ðh1  h2 Þð12a02 þ ð6a0 þ 3a01 ðh1 þ h2 Þ 2

2

2

2

þ2a02 ðh1 þ h1 h2 þ h2 ÞÞm21 Þðsinh ½h1 m1   sinh ½h2 m1 Þ; k9 ¼ 2ðh1  h2 Þð12a02 þ ð6a0 þ 3a01 ðh1 þ h2 Þ þ2a02 ðh1 þ h1 h2 þ h2 ÞÞm21 Þðcosh ½h1 m1   cosh ½h2 m1 Þ; k10 ¼  ðh1  h2 Þm1 ð12a02 ðh1  h2 Þ þ ðh1  h2 Þð6a0 þ 2a01 ðh1 þ 2h2 Þ 2

2

þ a02 ðh1 þ2h1 h2 þ 3h2 ÞÞm21 þ 12m41 Þsinh ½h1 m1 

2

 b12 m1  b11 h2 m21 Þsinh ½ð2h1  h2 Þm1 ; 2 2 3 k24 ¼ ðb1 þ 2b2 h1 þ 3b3 h1 þ 2b5 h1 m1 þ 2b6 h1 m1 þ2b7 h1 m1 2 2  b5 h2 m1  2b6 h1 h2 m1  3b7 h1 h2 m1 þ2b6 h2 m1 3 2 3 2 2 þ 4b7 h2 m1  2b1 h1 h2 m1  2b2 h1 h2 m1  2b3 h1 h2 m21 2 2 3 2 4 2 þ 2b1 h2 m1 þ 2b2 h2 m1 þ 2b3 h2 m1 Þsinh ½h2 m1  þ ð  2b12 m1 þ4b11 h2 m21 Þsinh ½2h2 m1  þ ð  3m1 ð  2b12 þb11 ðh1 þ h2 Þm1 Þsinh ½ðh1 þ h2 Þm1 ; k25 ¼ 2m1 ð2  2 cosh ½ðh1 h2 Þm1  þ ðh1  h2 Þm1 sinh ½ðh1  h2 Þm1 ; k26 ¼ b11 m1 cosh ½ð1=2Þðh1  5h2 Þm1   ðb5 þ h2 ð2b6 þ 3b7 h2 ÞÞcosh ½ð1=2Þðh1  3h2 Þm1 ; 2

2

k27 ¼ ð  2h1 ðb8 þh1 ðb9 þ b10 h1 ÞÞm1 þ 2b8 h2 m1 3

 12m51 q sinh ½h1 m1  þ m1 ðh1 h2 Þ; k11 ¼ ð 12a02 ðh1  h2 Þ ðh1  h2 Þð6a0 þ h1 ð4a01 þ 3a02 h1 Þ 2

þ 2ða01 þa02 h1 Þh2 þ a02 h2 Þm21 þ 12m41 Þsinh ½h2 m1 

4

þ 2b9 h2 m1 þ 2b10 h2 m1 Þcosh ½ð1=2Þðh1  h2 Þ m1  þ ðb5 þ h1 ð2b6 þ 3b7 h1 Þcosh ½ð1=2Þð3h1  h2 Þm1  þ b11 m1 cosh ½ð1=2Þð5h1  h2 Þm1 ; k28 ¼ ðh1  h2 Þð2b6 þ 3b7 ðh1 þ h2 Þ þ2ðb1 þ b2 ðh1 þh2 Þ 2

þ 12m51 q sinh ½h2 m1 ; 2 2 3 3 4 4 k12 ¼ 2ðb8 ðh1 þ h2 Þ þ 2b9 ðh1 þh2 Þ þ 3b10 ðh1 þ h2 ÞÞm1

þ ðb5 þ h1 ð b1 þ 2h1 ðb2 þ 2b3 h1 ÞÞm1 2

2

þ 2h1 ðh1  h2 Þðb5 þ b6 ðh1 þ h2 Þ þ b7 ðh1 þ h1 h2 þ h2 ÞÞm21 þ h2 ð2b6 þ 3b7 h2 þ 2b1 2b2 h1 þ 2b2 h2 2

 3b3 h1 h2 þ 2b3 h2 Þm1 ÞÞcosh ½h1 m1 ; k13 ¼ 2m1 ðb11  2b12 h1 m1 Þcosh ½2h1 m1 ; k14 ¼ ð b11 m1  b12 h1 m21 Þcosh ½ðh1  3h2 Þm1  þ ð  ðb5 þ h2 ð2b6 þ 3b7 h2 ÞÞ  b1 h1 m1 2

 2b2 h1 h2 m1  3b3 h1 h2 m1 Þcosh ½ðh1  2h2 Þm1 ;

2

þ b3 ðh1 þ h1 h2 þ h2 ÞÞm1 Þ cosh ½ð1=2Þðh1 þ h2 Þm1   3b11 m1 cosh ½ð1=2Þð3h1 þ h2 Þm1  þ 3b11 m1 cosh ½ð1=2Þðh1 þ 3h2 Þm1  þ b12 m1 sinh ½ð1=2Þðh1  5h2 Þm1 ; k29 ¼ ðb1 þ h2 ð2b2 þ 3b3 h2 ÞÞsinh ½ð1=2Þðh1  3h2 Þm1  2

 b5 ðcosh ½ð2h1  h2 Þm1   cosh ½h2 m1 Þ; 2 2 3 k18 ¼ ð2b6 h1 þ 3b7 h1 þ 2b1 h1 m1 þ 2b2 h1 m1 þ 2b3 h1 m1

2

þ ð4b8 ðh1 þh2 Þ þ6b9 ðh1 þ h2 Þ 3

3

þ 8b10 ðh1 þh2 ÞÞsinh ½ð1=2Þðh1  h2 Þm1  þ ðb1 þ h1 ð2b2 þ 3b3 h1 ÞÞsinh ½ð1=2Þð3h1  h2 Þm1  þ b12 m1 sinh ½ð1=2Þð5h1  h2 Þm1 ; k30 ¼ ðh1  h2 Þð2b2 þ 3b3 ðh1 þ h2 Þ þ2ðb5 þ b6 ðh1 þh2 Þ 2

2 3 4 2 3 k15 ¼ ð2b8 h1 þ 2b9 h1 þ 2b10 h1  8b8 h1 h2  6b9 h1 h2  8b10 h1 h2 2 2 3 3 2 2 þ 2b8 h2  6b9 h1 h2 þ2b9 h2  8b10 h1 h2 þ 2b8 h2  6b9 h1 h2 3 3 4 þ 2b9 h2  8b10 h1 h2 þ 2b10 h2 Þm1 cosh ½ðh1 h2 Þm1 ; 2 k16 ¼ ð 2b6 h1  3b7 h1 b1 h2 m1  2b2 h1 h2 m1 2  3b3 h1 h2 m1 Þcosh ½ð2h1 h2 Þm1 ; k17 ¼ ð b11 m1  b12 h2 m21 Þcosh ½ð3h1  h2 Þm1 

2

k23 ¼ ð  b1  2b2 h1  3b3 h1  b5 h2 m1  2b6 h1 h2 3b7 h1 h2 m1

2

þ b7 ðh1 þ h1 h2 þ h2 ÞÞm1 Þsinh ½ð1=2Þðh1 þ h2 Þm1   3b12 m1 sinh ½ð1=2Þð3h1 þ h2 Þm1  þ 3b12 m1 sinh ½ð1=2Þðh1 þ 3h2 Þm1 ; k31 ¼ 2  2 cosh ½ðh1  h2 Þm1  þðh1  h2 Þm1 sinh ½ðh1  h2 Þm1 ; 2

2

k32 ¼ 2b5  2b6 ðh1  h2 Þ 3b7 ðh1 þ h2 Þ  2ð2b0 þ b1 ðh1 þ h2 Þ 2 2 3 3 þ b2 ðh1 þ h2 Þ þ b3 ðh1 þ h2 ÞÞm1 þ b11 m1 ðcosh

½3h1 m1 

 3 cosh ½h2 m1 Þ þ b11 m1 cosh ½3h2 m1  3

3

þ ð8b10 h1 þ4b8 ðh1  h2 Þ 8b10 h2 þ 6b9 ðh1  h2 Þðh1 þ h2 Þ  3b12 m1 Þsinh ½h1 m1  þ cosh ½2h2 m1 ðb5 þ h2 ð2b6 þ 3b7 h2 Þ

S. Akram, S. Nadeem / Journal of Magnetism and Magnetic Materials 358-359 (2014) 183–191

 4b12 m1 sinh ½h1 m1 Þ þ ðb1 þ h1 ð2b2 þ 3b3 h1 ÞÞsinh ½2h1 m1 ;

þ 2b11 ðh1 h2 Þm1 Þsinh ½h1 m1 Þsinh ½2h2 m1   b11 m1 sinh ½3h2 m1 ;

k33 ¼ 2 cosh ½h2 m1 ð  ðh1  h2 Þ2 ðb8 þ b9 ð2h1 þh2 Þ 2

191

2

2

þ b10 ð3h1 þ 2h1 h2 þ h2 ÞÞm1  ðb1 þ 2b2 h2 þ 3b3 h2 2

þ ðh1  h2 Þ ðb6 þb7 ð2h1 þ h2 Þm1

References

þ ðb0 þ h1 ðb1 þ h1 ðb2 þ b3 h1 ÞÞÞðh1  h2 Þm21 Þsinh ½h1 m1  þ m1 ðb12 þ 2b11 ðh2  h1 Þm1 Þsinh ½2h1 m1 Þ; 3

k34 ¼ b12 m1 sinh ½3h1 m1   ð8b10 h1 þ 4b8 ðh1  h2 Þ 3  8b10 h2 þ 6b9 ðh1  h2 Þðh1 þ h2 Þ þ3b12 m1 2 2 þ 2ð2b5 þ 2b6 ðh1 þh2 Þ þ3b7 ðh1 þ h2 Þ þð2b0 þ b1 ðh1 þ h2 Þ 2 2 3 3 þ b2 ðh1 þh2 Þ þ b3 ðh1 þ h2 ÞÞm1

þ 4b11 m1 cosh ½h1 m1 Þsinh ½h1 m1 Þsinh ½h2 m1  2

þ cosh ½2h1 m1 ðb5 þ 2b6 h1 þ 3b7 h1 þ 2m1 ðb11 þ 2b12 ðh2  h1 Þm1 Þcosh ½h2 m1  4b12 m1 sinh ½h2 m1 Þ; k35 ¼  ð3b11 þ 2ðh1  h2 Þ2 ðb8 þ h1 ðb9 þ b10 h1 Þ 2

þ 2ðb9 þb10 h1 Þh2 þ 3b10 h2 ÞÞm1  2m1 ð 2b0  b1 ðh1 þ h2 Þ 2

2

þ b3 ðh1 2h2 Þð2h1  h2 Þðh1 þh2 Þ þ b2 ðh1  4h1 h2 þ h2 Þ þ ðh1  h2 Þ2 ðb5 þb6 ðh1 þ h2 Þ 2

2

þ b7 ðh1 þh1 h2 þ h2 ÞÞm1 Þcosh ½h2 m1 ; k36 ¼ ðk35 þ 2m1 ðb11 þ 2b12 ðh1  h2 Þm1 Þcosh ½2h2 m1  2

 2ðb1 þ2b2 h1 þ 3b3 h1 þðh1 h2 Þ2 ðb6 þ b7 ðh1 þ2h2 ÞÞm1  ðh1  h2 Þðb0 þ h2 ðb1 þh2 ðb2 þ b3 h2 ÞÞÞm21 Þsinh ½h2 m1  þ 2m1 ðb12 þ2b11 ðh1  h2 Þm1 Þsinh ½2h2 m1 Þ þ b12 m1 sinh ½3h2 m1 ; k37 ¼ k32 þ k33 þ k34 þ ðb1 þ h2 ð2b2 þ 3b3 h2 Þ  4b11 m1 sinh ½h1 m1 Þ; 2

2

k38 ¼  2b1  2b2 ðh1 þ h2 Þ  3b3 ðh1 þ h2 Þ  2ð2b4 þ b5 ðh1 þ h2 Þ 2

2

3

3

þ b6 ðh1 þh2 Þ þ b7 ðh1 þ h2 ÞÞm1  b12 m1 cosh ½3h1 m1   b12 m1 cosh ½3h2 m1  þ ð  3b11 þ 2ðh1  h2 Þ2 ðb8 þ h1 ðb9 þb10 h1 Þ 2

þ 2ðb9 þb10 h1 Þh2 þ 3b10 h2 ÞÞm1 sinh ½h1 m1  2

þ cosh ½2h2 m1 ð  b1  2b2 h2  3b3 h2  2m1 ðb11 þ2b12 ðh1  h2 Þm1 Þsinh ½h1 m1 ; 3

3

k39 ¼ cosh ½h2 m1 ð8b10 h1 þ 4b8 ðh1  h2 Þ  8b10 h2 2

þ 6b9 ðh1  h2 Þðh1 þh2 Þ  3b12 m1 þ 2ðb5 þ 2b6 h1 þ 3b7 h1 þ ðh1  h2 Þ2 ðb2 þb3 ðh1 þ 2h2 ÞÞm1  ðh1  h2 Þðb4 þ h2 ðb5 þh2 ðb6 þ b7 h2 ÞÞÞm21 Þsinh ½h1 m1 Þ  ðb5 þ h1 ð2b6 þ 3b7 h1 ÞÞsinh ½2h1 m1   b11 m1 sinh ½3h1 m1 ; k40 ¼ m1 ð  3b11 þ2ðh1  h2 Þ2 ðb8 þ b9 ð2h1 þ h2 Þ 2

2

þ b10 ð3h1 þ 2h1 h2 þ h2 ÞÞ  2ð2b4 þ b5 ðh1 þ h2 Þ 2

2

 b7 ðÞð2h1  h2 Þðh1 þ h2 Þ  b6 ðh1  4h1 h2 þ h2 Þ 2

2

 ðh1  h2 Þ2 ðb1 þb2 ðh1 þ h2 Þ þ b3 ðh1 þ h1 h2 þ h2 ÞÞm1 Þsinh ½h1 m1   2ðb12 þ 2b11 ðh2  h1 Þm1 Þsinh ½2h1 m1 Þsinh ½h2 m1  2

þ cosh ½2h1 m1 ð  b1  2b2 h1  3b3 h1 þ 4b12 m1 cosh ½h2 m1   2m1 ðb11 þ2b12 ðh2  h1 Þm1 Þsinh ½h2 m1 ; k41 ¼ cosh ½h1 m1 ð  2h1 ð2b8 þ h1 ð3b9 þ 4b10 h1 ÞÞ þ 4b8 h2 2

3

þ 6b9 h2 þ8b10 h2  3b12 m1 þ 4b12 m1 cosh ½2h2 m1  3

þ 2ðb5 þ2b6 h2 þ 3b7 h2 þðh1 h2 Þ2 ðb2 þ b3 ð2h1 þ h2 ÞÞm1 þ ðb4 þ h1 ðb5 þ h1 ðb6 þ b7 h1 ÞÞÞðh2  h1 Þm21 Þsinh ½h2 m1  2

2

þ 2 cosh ½h2 m1 ð2b1 þ 2b2 ðh1 þ h2 Þ þ 3b3 ðh1 þ h2 Þ 2

2

3

3

þ ð2b4 þb5 ðh1 þ h2 Þ þ b6 ðh1 þ h2 Þ þ b7 ðh1 þ h2 ÞÞm1 þ 4b11 m1 ðsinh ½h1 m1  þ sinh ½h2 m1 ÞÞÞ; 2

k42 ¼ ð  b5  2b6 h2  3b7 h2  2m1 ðb12

[1] T.K. Mitra, S.N. Prasad, On the influence of wall properties and Poiseuille flow in peristalsis, J. Biomech. 6 (1973) 681–693. [2] T.D. Brown, T.K. Hung, Computational and experimental investigations of two dimensional non linear peristaltic flows, J. Fluid Mech. 83 (1977) 249–272. [3] S. Nadeem Safia Akram, Influence of induced magnetic field and heat transfer on the peristaltic motion of a Jeffrey fluid in an asymmetric channel: closed form solutions, J. Magn. Magn. Mater. 328 (2013) 11–20. [4] S. Nadeem Safia Akram, Simulation of heat and mass transfer on peristaltic flow of hyperbolic tangent fluid in an asymmetric channel, Int. J. Numer. Methods Fluids 70 (2012) 1475–1493. [5] N.T. Eldabe, M.F. El-Sayed, A.Y. Ghaly, H.M. Sayed, Peristaltically induced transport of a MHD biviscosity fluid in a non-uniform tube, Physica A 383 (2007) 253–266. [6] R.S. Agarwal, C. Dhanapal, Numerical solution to the flow of a micropolar fluid between porous walls of different permeability, Int. J. Eng. Sci. 25 (1987) 325–336. [7] Safia Akram, S. Nadeem, M. Hanif, Numerical and analytical treatment on peristaltic flow of Williamson fluid in the occurrence of induced magnetic field, J. Magn. Magn. Mater. 346 (2013) 142–151. [8] S. Nadeem, Safia Akram, Noreen Sher Akbar, Simulation of heat and chemical reactions on peristaltic flow of a Williamson fluid in an inclined asymmetric channel, Iran. J. Chem. Chem. Eng. 32 (2013) 93–107. [9] Safia Akram, S. Nadeem, Changhoon Lee, Influence of lateral walls on Peristaltic flow of a Jeffrey fluid in a rectangular duct with partial slip, Int. J. Appl. Math. 14 (2012) 449–463. [10] Safia Akram, Abdul Ghafoor, S. Nadeem, Mixed convective heat and Mass transfer on a peristaltic flow of a non-Newtonian fluid in a vertical asymmetric channel, Heat Transfer Asian Res. 41 (2012) 613–633. [11] Mehdi Dehghan, Rezvan Salehi, A meshfree weak-strong (MWS) form method for the unsteady magnetohydrodynamic (MHD) flow in pipe with arbitrary wall conductivity, Comput. Mech. 52 (2013) 1445–1462. [12] Fatemeh Shakeri, Mehdi Dehghan, A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl. Numer. Math. 61 (2011) 1–23. [13] Mehdi Dehghan, Davoud Mirzaei, Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math. 59 (2009) 1043–1058. [14] Mehdi Dehghan, Davoud Mirzaei, Meshless local boundary integral equation (LBIE)method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput. Phys. Commun. 180 (2009) 1458–1466. [15] E.F. Elshehawey, M.E. Elsayed Elbarbary, S. Nasser Elgazery, Effects of inclined magnetic field on magneto fluid flow through a porous medium between two inclined wavy porous plates (numerical study), Appl. Math. Comput. 135 (2003) 85–103. [16] S. Nadeem, Safia Akram, Influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric or asymmetric channel, Asia Pac. J. Chem. Eng. 7 (2012) 33–44. [17] S.U.S. Choi, Enhancing Thermal Conductivity of Fluid with Nanoparticles Developments and Applications of Non-Newtonian Flow, ASME FED 66 (231) 99–105. [18] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu. Bussei 7 (1993) 227–233. [19] K.B. Anoop, T. Sundararajan, Sarit K. Das, Effect of particle size on the convective heat transfer in nanofluid in the developing region, Int. J. Heat Mass Transfer 52 (2009) 2189–2195. [20] N. Bachok, A. Ishak, I. Pop, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Therm. Sci. 49 (2010) 1663–1668. [21] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2008) 240–250. [22] D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Transfer 51 (2008) 2967–2979. [23] Safia Akram, Effects of nanofluid on peristaltic flow of a Carreau fluid model in an inclined magnetic field, Heat Transfer Asian Res. [24] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [25] D.D. Ganji, A. Sadighi, Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) 411–418. [26] D.D. Ganji, The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. A 355 (2006) 337–341. [27] D.D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transfer 33 (2006) 391–400.