haar wavelet solution of the mhd jeffery hamel flow and heat transfer in eyring powell fluid

AIP ADVANCES 6, 115102 2016Haar wavelet solution of the MHD Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid Najeeb Alam Khan,1, aFaqiha Sultan,2Amber Shaikh,1Asmat Ara,3 and

Powell fluid

Najeeb Alam Khan, Faqiha Sultan, Amber Shaikh, Asmat Ara, and Qammar Rubbab

Citation: AIP Advances 6, 115102 (2016); doi: 10.1063/1.4967212

View online: http://dx.doi.org/10.1063/1.4967212

View Table of Contents: http://aip.scitation.org/toc/adv/6/11

Published by the American Institute of Physics

AIP ADVANCES 6, 115102 (2016)

Haar wavelet solution of the MHD Jeffery-Hamel flow

and heat transfer in Eyring-Powell fluid

Najeeb Alam Khan,1, aFaqiha Sultan,2Amber Shaikh,1Asmat Ara,3

and Qammar Rubbab4

1Department of Mathematics, University of Karachi, Karachi 75270, Pakistan

2Department of Sciences and Humanities, National University of Computer and Emerging

Sciences, Karachi 75030, Pakistan

3Mohammad Ali Jinnah University, Karachi 75400, Pakistan

4Department of Mathematics, Air University, Multan Campus, Pakistan

(Received 8 August 2016; accepted 24 October 2016; published online 2 November 2016)

This study deals with the numerical investigation of Jeffery-Hamel flow and heat trans-fer in Eyring-Powell fluid in the presence of an outer magnetic field by using Haar wavelet method Jeffery-Hamel flows occur in various practical situations involving flow between two non-parallel walls Applications of such fluids in biological and industrial sciences brought a great concern to the investigation of flow characteristics

in converging and diverging channels A suitable similarity transformation is applied

to transform the nonlinear coupled partial differential equations (PDEs) into nonlin-ear coupled ordinary differential equations (ODEs), which govern the momentum and heat transfer properties of the fluid Due to the high nonlinearity of resulting cou-pled ODEs, the exact solution is unlikely Thus, the solution is approximated using a numerical scheme based on Haar wavelets and the results are verified by comparing with 4thorder Runge-Kutta results © 2016 Author(s) All article content, except where

otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4967212]

I INTRODUCTION

Flows through convergent-divergent channels gained importance in early nineteenth century after the revolutionary works of Refs.1and2 In modern eras, Jeffery-Hamel flows have various applica-tions in fluid mechanics, aerospace, civil, bio-mechanical, mechanical, chemical, and environmental engineering along with exploring the rivers and canals Practical applications of these types of flows include flow through rivers, canals, and different biological flows such as flow through arteries and venous blood vessels Since the pioneer work of Jeffery and Hamel, many researchers have been investigating the applications of this fluid, and they have reported the variation in flow characteristics

by changing the angle between the two channel.3 5Jeffery-Hamel flows are considered with various other flow conditions and the effects of many other fluid characteristics have been investigated.6 investigated the thermal radiation effects on the conventional Jeffery-Hamel flow caused by a point source or sink in convergent/divergent channels with stretching or shrinking walls of the stationary channel.7explored the effects of magnetic field applied transversely on Jeffery-Hamel flow using Cu-water nanofluid in the middle of two nonparallel plane walls.8 obtained the similarity solutions for the flow of Jeffery-Hamel fluid and described its relation to flow in a converging-diverging channel

A number of scientific problems are inherently nonlinear such as the flow models in fluid mechan-ics including Jeffery-Hamel flows and as a result of this nonlinear dynamic system of the models, the exact solutions are improbable So, various numerical techniques have been presented to solve these systems.9obtained the approximate homotopy analysis solution for the Jeffery-Hamel flows.10 presented an improved homotopy analysis solution of the nonlinear equations of Jeffery-Hamel

a Corresponding author E-Mail: njbalam@yahoo.com Tel.: +923333012008

2158-3226/2016/6(11)/115102/13 6, 115102-1 © Author(s) 2016

fluid flow.11have proposed another new technique for MHD Jeffery-Hamel flow, named spectral-homotopy analysis method.12used the Adomian decomposition method to analytically investigate the Jeffery-Hamel flow with nanoparticles and high magnetic field.13introduced the optimal homo-topy asymptotic solution of the problem Recently, a massive attention has been directed towards the artificial intelligence techniques to numerically solve the nonlinear differential equations.14have recently introduced a numerical treatment based on stochastic algorithms for the nonlinear MHD Jeffery-Hamel problems

Generally, fluids used in chemical engineering processes and polymer processing are non-Newtonian in nature Realizing the fact that these fluids possess many industrial applications, Powell and Eyring presented such a fluid model in 1944, acknowledged as Eyring-Powell fluid model15that gives several benefits beyond the other non-Newtonian fluids Although this model is more complex mathematically, additional consideration is dedicated to it owing to its distinctive benefits above the Power-law model Mainly because it is acquired using the kinetic theory of liquid instead of the empirical relation, unlike power-law models Additionally, it truly exhibits Newtonian character when shear rate is high and low Considering the significance of this fluid, vital researches have been presented to study the impact of several fluid conditions Some of the recent studies include: a research

on the effects of double diffusion on the flow of Eyring-Powell fluid over a cone conducted by,16 investigation of MHD flow, heat and mass transfer in radiating Eyring-Powell fluid with suspended nanoparticles over a stretching sheet in three dimensions by,17the study the flow of nano-Eyring-Powell fluid and heat and mass transfer in a two-layer channel by,18moreover, another research is dedicated to examine the effect of slip conditions and wall properties on MHD peristaltic flow and heat/mass transfer in Eyring-Powell fluid by.19

The analysis of the MHD Jeffery-Hamel flow of Eyring-Powell fluid with heat transfer has not been considered yet This paper aims to fill this gap and presents an investigation on the effects

of magnetic field on Hamel flow and heat transfer in Eyring-Powell fluid The Jeffery-Hamel problem occurs in various flow phenomena of fluid flow between two no-parallel walls The mathematical model of Jeffery-Hamel problems defines a nonlinear dynamic system, which does not possess an exact solution To overcome this issue, a numerical method based on Haar wavelets is used to find the numerical solution of the system of complicated nonlinear equations governing the flow and heat transfer characteristics Moreover, a shooting technique via NDSolve command in Mathematica 10 is also used to solve and verify the solution The results obtained

by these two methods are presented through graphs and tables and discussed in detail for the variation of some important parameters Comparison of these two methods is made by compar-ing the numerical values of the physical quantities, for the sake of the validity of the obtained results

II MATHEMATICAL MODEL

The extra-stress tensor for Eyring-Powell fluid is defined as:

Γ= µ + β ˙γ1 sinh−1 1

dγ˙

! !

where µ represents the shear viscosity, β and d are the characteristics of the Eyring-Powell model,

d has the dimension of (time)−1, ˙γ =q1

2trA2

1, and A1= ∇V + (∇V) t is the kinematical tensor The second order approximation of sinh−11dγ˙

d1γ −˙ 1 6

1

dγ˙3 withγd˙55<<<1, discussed in detail by,20

is used to get the momentum equation Eq (1) now takes the form:

Γ= µ + βd1

!

A1 − γ˙2

The shears of Eyring-Powell fluid in polar coordinates are derived by using the theory of rate process Thus, Eq (2) gives:

Γrr= µ + βd1 ! 2∂ u ∂ r − γ˙2

6 βd3

2∂ u

115102-3 Khan et al. AIP Advances 6, 115102 (2016)

Γr θ= µ + βd1

! 1

r

∂u

∂θ +

∂v

∂r −

v r

!

− γ˙2

6 βd3

1

r

∂u

∂θ +

∂v

∂r −

v r

!

(4)

Γθ θ= 2 µ + βd1

! 1

r

∂v

∂θ +

u r

!

− γ˙2

6 βd3

1

r

∂v

∂θ +

u r

!

(5) where,

˙

γ2= 2 ∂u ∂r

!2 + ∂v ∂r +1

r

∂u

∂θ −

v r

!2 + 2 1

r

∂v

∂θ +

u r

!2

(6)

The scalar momentum equations in r and θ-directions are:

ρ ∂ u

∂t + u

∂u

∂r +

v r

∂u

∂θ −

v2 r

!

= −∂p ∂r +1

r

∂(r Γ rr)

∂r +1

r

∂Γrθ

∂θ −

Γθθ

ρ ∂ v

∂t + u

∂v

∂r +

v r

∂v

∂θ +

uv r

!

= −1

r

∂ p

∂θ +

∂Γrθ

∂r +

1

r

∂Γθθ

∂θ +2

Γrθ

where p is the hydrostatic pressure.

A Problem statement

Consider a steady, unidirectional, incompressible flow of Eyring-Powell fluid in polar coordinates

(r, θ), the angle between two rigid non-parallel plane walls at the intersection is 2α as sketched in

Fig.1 The rigid walls are known to be convergent or divergent for α < 0 and α > 0, respectively The flow is produced in such a way that streamlines of the flow are created by the family of straight lines passing through a point of source or sink Therefore, the velocity is the function of polar angle θ as

it varies from line to line The electric and magnetic fields obey the Ohm’s law J = σ(E + V × B), where J is the Joule current, E is the electric field, V = (u, v, 0) is the velocity vector with components

in the radial and tangential direction, respectively, and B = (0, Bθ, 0) is the external magnetic field acting vertically downward to the top wall The velocity is assumed to be along the radial direction

only and depends on r and θ such that V = (u (r, θ), 0, 0) A purely symmetric radial flow is assumed

so the tangential velocity v = 0 and to match this simple dependence on r, the magnetic field is chosen such that Bθ=B0

r 21A constant temperature T wis maintained along the walls By considering all the above assumptions, the continuity, momentum, and heat transfer equations can be presented in in polar coordinates as:

FIG 1 Physical model of the MHD Jeffery-Hamel flow.

r

∂

ρ u ∂u

∂r =−

∂ p

∂ r +

∂Γrr

∂r +

1

r

∂Γrθ

∂θ +

Γrr− Γθθ

r −σ B2

0

u

0= −1

r

∂ p

∂ θ +

∂ Γr θ

∂ r +

1

r

∂ Γθ θ

∂ θ +2

Γr θ

ρ C P u ∂T

∂r = k

∂2T

∂r2 +1

r

∂T

∂r +

1

r2

∂2T

∂θ2

! + 2 ∂ u ∂ r Γrr+2 u

r Γθθ+2

r

∂u

∂θΓr θ (12) Subject to the boundary conditions:

∂u (r, θ)

∂θ = 0, ∂T∂θ =0, u (r, θ) = Umax at the centerline of the channel

u (r, θ)= 0, T = T w at the walls of the channel

Where k is the thermal conductivity, C Pis the specific heat constant, σ is the electrical conductivity,

and Umaxis the velocity at the centerline of the channel where θ= 0

B Similarity transformation

The similarity transformation for Jeffery-Hamel flow is obtained from Eq (9) as:

Introducing:

f(η)=f(θ)

fmax, g (η)= T

Using the above similarity transformation in Eqs (10) - (12) and removing the pressure term from Eqs (10) and (11), the PDEs are now transformed into ODEs as:

(1+ E P)f000+ 4α2f0 + 2 Re α f f0

−Ha α2f0−3 E Pλ

α2

1

2f

02f000+ f0f002

!

−λEP

72f2f0+ 2f03+ 32f f0f00+ 2f2f000 = 0 (15)

g00+ Pr Ec 2 (1+ E P)f02+ 4 α2f2−8 E Pλ

f02+ 2 α2f2f2−E Pλ

α2 f04

!

= 0 (16) And the boundary conditions now become:

f(0)= 1, f0

(0)= 0, f (1) = 0, g0

(0)= 0, g (1) = 1 (17)

In Eqs (15) and (16), E P= 1

µ β d is the dimensionless Eyring-Powell parameter, λ= U2

max

3r2d2 is the local non-Newtonian parameter, Re=fmax α

ν =Umaxrα

ν is the local Reynolds number, Ha=σ Bµ2 is the Hartmann number, Pr=µ C p

k is the Prandtl number, and Ec=U2

max

T w C p is the Eckert number

Some significant physical quantities are of prime importance in fluid mechanics and have great influence on fluid flow and heat transfer properties of the fluid The skin friction coefficient based on the wall shear stress and the Nusselt number based on the heat flux have the utmost importance in engineering processes These quantities are defined as:



C f , N u = Γw

ρ f2 max

, r q w

k T w

! θ=α

(18) Where Γw= Γr θ=µ + 1

β d − 6 β d13γ˙2 1

r ∂ u

∂ θ +∂ v

∂ r−v r



is the wall shear stress of Eyring-Powell

fluid and q w=−k

r

∂T

∂θ is the surface heat flux Using these definitions in Eq (18), C f and N uare obtained as:



C f , N u = 1

r2Re (1+ E P ) − E Pλ 2f (1)2+1

2f

0 (1)2

! !

f0(1) , g

0 (1) α

!

(19)

115102-5 Khan et al. AIP Advances 6, 115102 (2016)

III HAAR WAVELETS

Haar wavelet is the firstly recognized simplest wavelet, offered by Ref.22 in 1910, used as

a primary example for orthonormal wavelet transform with compact support In wavelet analy-sis, the functions are expanded with respect to the basis functions, in terms of wavelets generated from dilations and translations of a fixed function known as mother wavelet The structure of Haar

wavelet family is based on multi resolution analysis (MRA) For Z ∈ [0, 1], the i th Haar wavelet is defined as:23

h i (Z)=









1 Z ∈ [α1, α2)

−1 Z ∈[α2,α3)

0 elsewhere

(20)

where, α1=K

m, α2=K+0.5

m , and α3=K+1

m

In which, m= 2j (j = 0, 1, , J) is the level of wavelet and K = 0, 1, , m − 1 is the translational parameter In levels of wavelet, J is the maximum level of resolution and i = K + m + 1 is the formula for calculating index i which has the maximal value i= 2 ˜M= 2J+1 In case of minimal values i.e.

m = 1 and K = 0, the starting value of i is 2 For i = 1, the function is defined as:

h1(Z)=( 1 Z ∈ [0, 1) ,

Now, the collocation points by which discretized form of Haar function can be acquired, is defined as:

Z j=j − 0.5

2 ˜M j= 1, 2, , 2 ˜M (22) The following definitions of integration will also be used

P i,1 (Z)=

Z

 0

P i,v+1(Z)=

Z

 0

Eqs (20) and (23) yield:

P i,1 (Z)=





Z - α1 Z ∈[α1, α2)

α3- Z Z ∈[α2, α3)

0 elsewhere

(25)

Integrating the above equation gives:

P i,2 (Z)=

















1

2(Z − α1)2 Z ∈[α1, α2) 1

4m2 −1

2(α3−Z)2 Z ∈[α2, α3) , 1

4m2 Z ∈[α3, 1) ,

(26)

P i,3 (Z)=

Z

 0

P i,2 (Z)=















1

6(Z − α1)3 Z ∈[α1, α2) , 1

4m2(Z − α2)+1

6(α3−Z)3 Z ∈[α2, α3) , 1

4m2 (Z − α2) Z ∈[α3, 1) ,

(27)

A Employment of Haar wavelet collocation method

Consider the system of Eqs (15) and (16) satisfying the boundary conditions in Eq (17) Assume that the highest derivative term of the formulated system of equations for each dependent variable can be expressed in terms of Haar wavelet series while the values of lower derivatives and dependent function can be obtained through the integration of Haar wavelet series in a particular way Substitution

of these values in Eqs.(15) and (16), will generate a system of 4M nonlinear algebraic equations that

can be solved with the help of a computational software, Mathematica 10

1 For momentum equation

Assume that the highest derivative term of momentum equation can be written as:

f000(η)=

2 ˜M

X

i=1

Integrating Eq (28) from 0 to η gives:

f00(η) − f00(0)=

2 ˜M

X

i=1

Integrating Eq (29) from 0 to η yields:

f0(η) − f0(0) − η f00(0)=

2 ˜M

X

i=1

Since f0(0)= 0, Eq (30) becomes:

f0(η) − η f00(0)=

2 ˜M

X

i=1

Now integrating Eq (31) from 0 to 1 provides:

f (1) − f (0) − η2

2 f

00 (0)

1

0

=

2 ˜M

X

i=1

a i C i,3(η) (32) where,

C i,3=

1

 0

Using the boundary conditions from Eq (17), Eq (32) becomes:

f00(0)= −2 *

,

2 ˜M

X

i=1

ai C i,3(η)+ 1+/

-(34)

Now, inserting the value of f00(0) in Eqs (29) and (31) yield:

f00(η)=

2 ˜M

X

i=1

ai P i,1(η) − 2 *.

,

2 ˜M

X

i=1

ai C i,3(η)+ 1+/

-(35)

f0(η)=

2 ˜M

X

i=1

ai P i,2(η) − 2η *

,

2 ˜M

X

i=1

ai C i,3(η)+ 1+/

-(36) Integrating Eq (36) from 0 to η and substituting the boundary condition provide:

f(η)= 1 +

2 ˜M

X

i=1

ai P i,3(η) − η2 *

2 ˜M

X

i=1

ai C i,3(η)+ 1+/ (37)

115102-7 Khan et al. AIP Advances 6, 115102 (2016)

FIG 2 The effect of α on velocity profiles in convergent α < 0 and divergent α > 0 channels keeping Re = 20.

2 For energy equation

Consider the highest derivative term of energy equation that can be expressed as:

g00(η)=

2 ˜M

X

i=1

Integrating Eq (38) from 0 to η and substituting the boundary condition g0(0)= 0 yield:

g0(η)=

2 ˜M

X

i=1

Now integrating Eq (39) from 0 to 1 and substituting the boundary condition g (1)= 1 provide:

g(0)= 1 −

2 ˜M

X

i=1

FIG 3 The effect of α on temperature profiles in convergent α < 0 and divergent α > 0 channels keeping Re = 20.

FIG 4 The effect of Eyring-Powell parameter E Pon velocity profiles in a divergent channel keeping α = 2.5 and Re = 20.

where,

C i,2=

1

 0

Finally, integrating Eq (39) from 0 to η and substituting the value of g (0) provide:

g(η)= 1 +

2 ˜M

X

i=1

bi P i,2(η) −

2 ˜M

X

i=1

IV RESULTS AND DISCUSSION

With the purpose of having a clear insight of the flow and heat transfer properties of the fluid, the system of governing equations is solved by Haar wavelet and 4thorder Runge-Kutta method by

FIG 5 The effect of Eyring-Powell parameter E P on temperature profiles in a divergent channel keeping α = 2.5 and

Re = 20.

115102-9 Khan et al. AIP Advances 6, 115102 (2016)

FIG 6 The effect of Hartman number Ha on velocity profiles in a divergent channel keeping α= 2.5 and Re = 20.

using computational software The influence of the magnetic field and other fluid parameters has been investigated on flow and heat transfer in a wedge shaped region The effects of pertinent parameters have been observed through Figs 2-12and TablesI–II The results obtained by Haar wavelet are validated, by comparison of numerical values of the physical quantities, with the values obtained by NDSolve For the purpose of conciseness, the constant values of the parameters are assumed to be

E P = 0.2, λ = 0.1, Ha = 5, Pr = 6.2, Ec = 0.5, and Re = 20.

The overall structure and the effect of angle α on the fluid velocity and temperature distribution

of Eyring-Powell fluid flow over a wedge shaped region is presented in Figs 2-3 Fig.2 depicts the velocity profile of the flow in convergent/divergent channel In a convergent channel i.e α < 0 and for adequately high Reynolds number, the velocity profile remains almost constant over a big portion and rapidly declines to zero near the walls, demonstrating a clear boundary layer behavior

As the angle α increases on negative side, the velocity profile rises, increasing the constant portion parallel to the horizontal axis While, in a divergent channel, the velocity profile is more arched near the centerline and increment in positive angle α reduces the velocity by increasing the curve of the profile

FIG 7 The effect of Hartman number Ha on temperature profiles in a divergent channel keeping α= 2.5 and Re = 20.

With the purpose of having a clear insight of the flow and heat transfer properties of the fluid, the system of governing equations is solved by Haar wavelet and. .. been investigated on flow and heat transfer in a wedge shaped region The effects of pertinent parameters have been observed through Figs 2-1 2and TablesI–II The results obtained by Haar wavelet. .. Pr = 6.2, Ec = 0.5, and Re = 20.

The overall structure and the effect of angle α on the fluid velocity and temperature distribution

of Eyring- Powell fluid flow over a wedge shaped