EFFECT OF BINARY CHEMICAL REACTION ON UNSTEADY THREE...

ISSN 2277-2685 IJESR/July 2018/ Vol-8/Issue-7/5-13 S. Anuradha et. al., / International Journal of Engineering & Science Research

*Corresponding Author www.ijesr.org 5

EFFECT OF BINARY CHEMICAL REACTION ON UNSTEADY THREE-DIMENSIONAL MHD FLOW OF POWELL-EYRING NANO FLUID

S. Anuradha*1, R Praveena2

1Prof. & Head, PG & Dept. of Mathematics, Hindusthan College of Arts & Science, Coimbatore, India.

2Research Scholar, PG & Dept. of Mathematics, Hindusthan College of Arts & Science, Coimbatore, India.

ABSTRACT

In the present investigation, the effect of binary chemical reaction on three dimensional MHD flow of Powell-Eyring nanofluid over stretching sheet is analyzed numerically. Mathematical model has been formulated to construct continuity, momentum, energy and concentration equations. The governing boundary layer equations are reduced into system of nonlinear ordinary differential equations by using similarity transformation and then solved by Runge-kutta method. The influence of non-dimensional parameters Powell-Eyring fluid parameter ε, unsteady parameter S, magnetic parameter M, Prandtl number Pr, Lewis number Le, Thermophoresis number Nt, Brownian number Nb, non-dimensional energy E, Biot number Bi, temperature difference parameter δ, dimensionless reaction rate σ and fitted rate constant n respectively on velocity, temperature and concentration profiles are shown numerically and graphically with the help of graphs.

Keywords: MHD, Powell Eyring-fluid, stretching sheet, Binary chemical reaction, Activation energy.

1. INTRODUCTION

There is a significant role of non-Newtonian fluid flow in the field of Engineering and Industrial physics such as paper coating, plasma and mercury, nuclear fuel slurries etc. Further in non-Newtonian fluids, there is a nonlinear relationship between the stress and strain. Powell and Eyring studied non-Newtonian fluid based on the behaviour of high shear rates. Motivation of studies on Powell-Eyring model, many researchers gave attention to investigate this model. In view of these research, Eldabe et. al., [1] discussed the effect of couple stresses on the MHD of a non-Newtonian unsteady flow between two parallel porous plates. Horiuchi and Dutta [2] investigated the Joule heating effects in micro channel flows. Reza and Gupta [3] studied the steady two-dimensional oblique stagnation point flow towards a stretching surface. Singh [4] examined the heat and mass transfer with three-dimensional flow of a viscous fluid. Ariel [5] explored the three-dimensional flow past a stretching sheet by perturbation method. Xuet [6] studied the Series solutions of unsteady free convection flow in the stagnation-point region of a three dimensional body. Wang and Tan [7] discussed the double-diffusive convection of Maxwell fluid in a porous medium heated from below by Stability analysis. Analytical and numerical solutions of third grade fluid flow between micro-parallel plates can be found by Akgul and Pakdemirli [8]. Niuet et. al., [9] studied the thermal convection of a viscoelastic fluid in an open-top porous layer heated from below. Rashidi and Pour [10] investigated unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis. Bachoket et. al., [11] developed the flow and heat transfer at a general three-dimensional stagnation point in a nanofluid. Ali et. al., [12] examined unsteady MHD natural convective flow from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical reaction and radiation effects. Ahmad and Asghar [13] studied the second grade fluid subject to a transverse magnetic field with arbitrary velocities over a stretching sheet. Hayat et. al., [14] derived the three-dimensional flow of upper convected Maxwell (UCM) fluid. Fang et. al., [15] discussed the exact solutions of heat transfer over a generalized stretching/shrinking wall problem.

Hayat et. al., [16] examined the similar solution for the three dimensional flow in an Oldroyd-B fluid over a stretching surface. Bachok et. al., [17] discussed boundary layer stagnation-point flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid. Nadeem and Lee [18] studied boundary layer flow of nanofluid over an exponentially stretching surface. Nadeem et. al., [19] examined heat transfer analysis of

S. Anuradha et. al., / International Journal of Engineering & Science Research

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water-based nanofluid over an exponentially stretching sheet. Pal et. al., [20] presented the flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation. Hsiao [21] presented nano fluid flow for conjugate mixed convection and radiation with multimedia physical features. Noghrehabadi et. al.,[22] analyzed the fluid flow and heat transfer of nanofluids over a stretching sheet near the extrusion slit. Alin [23] explored the Flow and heat transfer of Powell–Eyring fluid over a shrinking surface in a parallel free stream. Rosali [24] examined the effect of unsteadiness on mixed convection boundary-layer stagnation point flow over a vertical flat surface embedded in a porous medium. Mabood et. al., [25] have been founded MHD stagnation point flow and heat transfer impinging and transpiration on stretching sheet with chemical reaction. Pang et al. [26] discussed the Review on combined heat and mass transfer characteristics in nanofluids. Bahiraei and Hangi [27] examined heat transfer characteristics of magnetic nanofluids. Hsiao [28] studied the stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Mahanthesh et. al., [29] examined the Nonlinear radiative heat transfer in MHD three-dimensional flow of water based nanofluid over a nonlinearly stretching sheet with convective boundary condition. Mahanthesh et. al., [30] numerically investigated the Magneto hydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet. Marneniand Kamran [31] investigated MHD natural convection flow past an impulsively started infinite vertical porous plate with Newtonian heating in the presence of radiation. Hsiao [32] examined unsteady oblique stagnation point flow of elastic-viscous fluid due to sinusoidal wall temperature over an oscillating stretching surface with heat transport analysis. Hayat et al. [33] discussed mixed convective peristaltic flow of water based nanofluids with joule heating and convective boundary conditions. Joule heating effects on MHD mixed convection of a Jeffrey fluid over a stretching sheet with power law heat fluxcan be analysed numerically by Harish Babu et. al., [34]. Rehman et. al., [35] presented heat transfer analysis for three dimensional stagnation-point flow over an exponentially stretching surface. Khan et. al., [36] studied new modeling for 3D Carreau fluid flow considering nonlinear thermal radiation. Hayat et. al., [37] analyzed numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon water nanofluid. Rehman and Nadeem [38] studied heat transfer analysis for three-dimensional stagnation-point flow of water-based nanofluid over an exponentially stretching surface. Irfan et. al., [39] examined three dimensional unsteady flow of Carreau nano fluid with variable thermal conductivity and heat source/sink. Khan et al.[40] studied the modeling and simulation for three dimensional magneto Eyring-Powell nanomaterial subject to nonlinear thermal radiation and convective heating. Agbaje et. al., [41] investigated numerically the unsteady non-Newtonian Powell-Eyringnano fluid flow over a shrinking sheet with heat generation and thermal radiation. Ganesh Kumar et. al., [42] explored the unsteady squeezed flow of a tangent hyperbolic fluid in the presence of variable thermal conductivityover a sensor surface. Mamatha et. al., [43] investigated the unsteady Eyring Powell Cattaneo-Christovdusty nanofluid over sheet with heat and mass flux conditions. Kevin et. al., [44] analyzed modeling of Joule heating and convective cooling in a thick-walled micro-tube. Heat transfer of two-phase nanofluid flow between non-parallel walls considering Joule heating effect can be found analytically by Dogonchi and Ganji [45]. Sucharitha et. al., [46] presented Joule heating and wall flexibility effects on the peristaltic flow of magneto hydrodynamic nanofluid. Arif et. al., [47] examined combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder. Mahanthesh et. al., [48] discussed the three-dimensional MHD flow of Powell-Eyring nanofluid over a convectively heated stretching surface in the presence of radiation, viscous dissipation and Joule heating. In the present investigation, the effect of binary chemical reaction on three dimensional MHD flow of Powell-Eyring nanofluid over stretching sheet is analyzed numerically. Mathematical model has been formulated to construct continuity, momentum, energy and concentration equations. The governed boundary layer equations are converted into system of nonlinear ordinary differential equations by using similarity transformation and then solved by Runge-kutta method. The characteristics of non-dimensional parameters on velocity, temperature and concentration profiles are discussed numerically and graphically with the help of graphs.

2. MATHEMATICAL FORMULATION

Let us consider three-dimensional boundary layer flow of an electrically conducting Eyring-Powell fluid past a convectively heated stretching sheet. Let x,y, z be the Cartesian coordinates axis with the origin O and u,v,w



represent the velocity components in the directionsx,yandz respectively. At z=0, the sheet coincides with the plane and the flow occupies the region when z is greater than 0. By fixing the origin as O, the sheet is stretched in two directions x and y with the velocities u and v in the form:

( , )1w

axu x t

ta=

-and ( , )

1w

byv y t

ta=

- (1)

where a and b are positive constants

The convective temperature and concentration of nanoparticles at the sheet are fT and wC andT¥ and C¥ denote

the ambient fluid temperature and concentration. The transverse magnetic field is taken in the following form:

01/2(1 )

BB

ta=

- (2)

The Cauchy stress tensor T for Powell-Eyring fluid model is as follows

T pI t= - + (3)

.( )f i ija p J Br t s= -Ñ +Ñ + ´ (4)

In this Powell-Eyring fluid model, p is represent pressure, I is identity tensor and ijt extra stress tensor

11 1sinhi i

ijj j

u ux x

t mb g

-æ ö¶ ¶

= + ç ÷ç ÷¶ ¶è ø (5)

where b and g are the characteristic length.

Let us considering

3

1 1 1 1 1 1sinh , 1

6i i i i

j j j j

u u u ux x x xg g g g

-æ ö æ ö¶ ¶ ¶ ¶

@ - <ç ÷ ç ÷ç ÷ ç ÷¶ ¶ ¶ ¶è ø è ø (6)

The governing three-dimensional Eyring- Powell fluid equations are

0u v wx y z¶ ¶ ¶

+ + =¶ ¶ ¶

(7)

22 2 2 2

2 2 2 3 2

12f f f

u u u u u u u u Bu v w v u

t x y z z z z zm s

bgr bg r r¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶æ ö+ + + = + - -ç ÷¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø

(8)

22 2 2 2

2 2 2 3 2

12f f f

v v v v v v v v Bu v w v v

t x y z z z z zm s

bgr bg r r¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶æ ö+ + + = + - -ç ÷¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø

(9)

22

2T

m B

DT T T T T C T Tu v w D

t x y z z z z T za t

¥

ì ü¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ï ïæ ö æ ö+ + + = + +í ýç ÷ ç ÷¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø è øï ïî þ (10)

2 22

2 2 (C )a

n ETT

B r

DC C C C C T Tu v w D K C e

t x y z z T z Tk

æ ö-ç ÷è ø

¥¥ ¥

æ öæ ö¶ ¶ ¶ ¶ ¶ ¶+ + + = + - ç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶è ø è ø

- (11)



The above velocity components u, v and w along with x, y and z directions, v kinematic viscosity, m dynamic

viscosity,s electrical conductivity, mf f

kc

ar

= thermal diffusivity of the fluid, T and C are temperature and

volume fraction of nanoparticles, k thermal conductivity of the fluid, ( )

( )p

f

cc

rt r= ratio of the effective

heat capacity of nanoparticle and BD Brownian diffusion coefficient, TD thermophoretic diffusion coefficient.

2 (C )a

n E

Tr

TK C e

Tk

æ ö-ç ÷è ø

¥¥

æ ö- ç ÷

è øis the modified Arrhenius equation ; 2

rK = reaction rate, aE =activation energy, k =

8.61x10-5eV/K the Boltzmann constant and n =fitted rate constant and n lies between -1 and 1.

The boundary conditions are:

( , ), ( , ), 0, ( ),w w f f w

Tu u x t v v y t w k h T T C C

z¶

= = = = - =¶

at 0z =

0, 0, 0, 0, ,u v

u v T T C Cz z ¥ ¥¶ ¶

® ® ® ® ® ®¶ ¶

as z®¥ (12)

Similarity transformations are

( ) ( ) ( ) ( ) ( ) ( )

( )

' ', , ( ) ( ) ,1 1 1

( ) , ( ) ,1

f

f w f

v aax byu f v g w f g

t t t

T T C C az

T T C C v t

h h h ha a a

q h f h ha

¥ ¥

¥ ¥

= = = - +- - -

- -= = =

- - -

(13)

The governing Powell-Eyring equations are reduced into the set of non-linear ordinary differential equations as follows

''' '' ' 2 ' '' '' 2 ''' 2 '1

1(1 ) ( ) ( ) ( ) ( ) 0

2f f g f f S f f f f M fe h ed+ + + - - + - - = (14)

''' '' ' 2 ' '' '' 2 ''' 2 '2

1(1 ) ( ) ( ) ( ) ( ) 0

2g f g g g S g g g g M ge h ed+ + + - - + - - = (15)

'' ' ' ' '2Pr( ) Pr Pr 0f g Nb Ntq q fq q+ + + + = (16)

( ) 1(1 ) 0E

nNtLe f g Le e

Nbdqf f q s dq j

æ ö-ç ÷+è ø¢¢ ¢ ¢¢+ + + - + =

(17)

The reduced boundary conditions become

' ' '0, 0, 1, , ( 1), 1f g f g c Biq q f= = = = = - = at 0h =

' ' '' ''0, 0, 0, 0, 0, 0f g f g f q® ® ® ® ® ® ash ®¥ (18)

Where 1 2,d d ande are Eyring Powell parameter, c is stretching ratio parameter, 2M is magnetic parameter,

S is unsteady parameter, Pr is Prandtl number, Nb is Brownian motion parameter, Nt is thermophoresis



parameter, R is thermal radiation parameter, xEc is Eckert number along x direction, yEc is Eckert number

along y direction, Bi is Biot’s number and Le is Lewis number.

The non-dimensional parameters which are involved in this problem are

3 3 2 22 0

1 22 2

1, , , , , , , Pr ,

2 2

( )( ), , , , ,

w w r

B f m

f T fw B w a

u v B Kb v vc Le M

vxC vyC C a D a c

h D T TT T D C C ENb Bi Nt E S

T v T v T avk a

sd d e s

mb r a

tt adk

¥¥

¥

= = = = = = = =

-- ¥ -= = = = = =

¥

(19)

3. RESULTS AND DISCUSSION

In this section, the effect of binary chemical reaction on three dimensional MHD flow of Powell-Eyring nanofluid over stretching sheet is analyzed numerically. The governing boundary layer equations are reduced into system of nonlinear ordinary differential equations by using similarity transformation and then solved by Runge-kutta method. The influence of non-dimensional parameters Powell-Eyring fluid parameter ε, unsteady parameter S, magnetic parameter M, Prandtl number Pr, Lewis number Le, Thermophoresis number Nt, Brownian number Nb, non-dimensional energy E, Biot number Bi, temperature difference parameter δ, dimensionless reaction rate σ and fitted rate constant n respectively on velocity, temperature and concentration profiles are shown numerically and graphically with the help of graphs.

Figure 1 explores the various values of Powell-Eyring fluid parameter ε on velocity profiles f’ and g’. Increasing values of Powell-Eyring fluid parameter ε increase both profiles. Higher values of Powell-Eyring fluid parameter ε reduced the boundary layer thickness. Figure 2 depicts unsteady parameter S on velocity profiles f’ and g’ which reduces f’ and g’ initially. This is because of decreasing values in stretching rate. Influence of magnetic parameter M on velocity profiles f’ and g’ are presented in the figure 3 and it was observed that increasing values of magnetic parameter M decrease the profiles f’ and g’. Figure 4 illustrates the behaviour of unsteady parameter S on temperature profile and concentration profile respectively. Enhancing values of unsteady parameter S increase temperature profile and concentration profile. In figure 5, comparative numerical analysis of non-dimensional parameter Prandtl number Pr on temperature profile and Lewis number Le on concentration profile are shown. Increasing values of Prandtl number Pr decreases temperature profile which diminishes the boundary layer thickness. With increasing values of Lewis number Le,

Concentration profile decreases. Figure 6 and 7 demonstrates the effect of Thermophoresis number Nt, Brownian number Nb on temperature profile and concentration profile respectively. It is noticed that increasing values of Thermophoresis number Nt increase both profiles but increase in Brownian number Nb increases temperature profile and decrease the concentration profile. Increasing values of Biot number Bi increase temperature profile and concentration profile respectively in figure 8. The effect of non-dimensional parameters non-dimensional energy E, temperature difference parameter δ, dimensionless reaction rate σ and fitted rate constant n related to binary chemical reaction on concentration profile are revealed in the figure 9-10. Concentration profile increases with larger values of temperature difference parameter δ, non-dimensional energy E and dimensionless reaction rate σ and it can be observed that opposite reaction for fitted rate constant n. As a result, it can be observed that chemical reaction requires more activation energy when mass flux of stretching sheet is smaller.



4. CONCLUSION

Numerical solutions of three dimensional MHD flow of Powell-Eyring nanofluid over stretching sheet with binary chemical reaction are presented. The main outcome of the present study are as follows:

· Increasing values of Powell-Eyring fluid parameter ε increase both profiles. Higher values of Powell-Eyring fluid parameter ε reduced the boundary layer thickness.

· Unsteady parameter S on velocity profiles f’ and g’ which reduces f’ and g’ nitially. This is because of decreasing values in stretching rate.

· Increasing values of magnetic parameter M decrease the profiles f’ and g’.

· Enhancing values of unsteady parameter S increase temperature profile and concentration profile.

· Increasing values of Prandtl number Pr decreases temperature profile which diminishes the boundary layer thickness.

· With increasing values of Lewis number Le, Concentration profile decreases.

· It is noticed that increasing values of Thermophoresis number Nt increase both profiles but increase in Brownian number Nb increases temperature profile and decrease the concentration profile.

· Increasing values of Biot number Bi increase temperature profile and concentration profile respectively .

· Concentration profile increases with larger values of temperature difference parameter δ, non-dimensional energy E and dimensionless reaction rate σ and it can be observed that opposite reaction for fitted rate constant n

· It can be observed that chemical reaction requires more activation energy when mass flux of stretching sheet is smaller.

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