MHD Heat and Mass Transfer for Viscous flow over

Thermal Energy and Power Engineering

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MHD Heat and Mass Transfer for Viscous flow over Nonlinearly Stretching Sheet in a Porous Medium R. N. Jat, Gopi Chand and Dinesh Rajotia Department of Mathematics, University of Rajasthan, Jaipur – 302004, India. [email protected] , [email protected] & [email protected] Effect of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet was studied by Cortell [16]. Jat and Chaudhary [17-19] studied the MHD boundary layer flow over a stretching sheet for stagnation point, heat transfer with and without viscous dissipation and Joule heating. Ellahi et al. [20,21] obtained analytic solution for MHD flow in an annulus and MHD flow in a third grade fluid with variable viscosity. Radiation effects on the MHD flow near the stagnation point of a stretching sheet was studied by Jat and Chaudhary [22]. Vyas and Rai [23] studied radiative flow with thermal conductivity over a non-isothermal stretching sheet in a porous medium. Later, Numerical study of MHD free convective flow and mass transfer over a stretching sheet considering Dofour and Soret effects in the presence of magnetic field was investigated by Ahammad and Mollah [24]. Recently, Alinejad and Samarbakhsh [25] studied viscous flow over nonlinearly stretching sheet with effect of viscous dissipation. Slip effects on ordinary viscous fluid flow have been discussed by Ellahi et al. [26], whereas MHD flow have been studied by Ellahi and Hammed [27]. Recently, Ellahi [28] and Ellahi et al. [29] studied the effects of MHD and temperature dependent viscosity on the flow of nonNewtonian nanofluid in a pipe and flow through a porous medium between two coaxial cylinders with heat transfer and variable viscosity respectively. Zeeshan and Ellahi [30] obtained the Series solutions for nonlinear partial differential equations with slip boundary conditions for non-Newtonian MHD fluid in porous space. Realizing the increasing technical applications of MHD effects, the present paper studies the problem of MHD boundary layer flow over an nonlinearly stretching sheet in porous medium with viscous dissipation.

Abstract- The steady two-dimensional laminar flow of a viscous incompressible electrically conducting fluid past over a porous substrate attached to a non-linearly stretching sheet in the presence of a uniform transverse magnetic field with viscous dissipation in the porous medium is investigated. The boundarylayer equations are transformed to ordinary differential equations with the help of similarity transformed and solved numerically by standard techniques. The velocity and temperature profiles are computed and discussed numerically and presented through graphs for various parameters like, Magnetic and Permeability parameters, Prandtl number and Eckert number. Keywords- Viscous dissipation; Nonlinearly stretching sheet; MHD; Boundary layer flow; Porous Medium

I.

INTRODUCTION

The steady two-dimensional boundary-layer flow of a viscous incompressible fluid over a non-linearly stretching sheet is very important and it has many practical applications in several industries such as polymer sheet extrusion from a dye, aerodynamic extrusion of plastic sheets, glass-fiber production and many others. The two-dimensional boundarylayer flow caused by a moving rigid surface was first investigated by Sakiadis [1]. Later, Crane [2], extended this idea for the two dimensional flow over a stretching sheet problem. Gupta and Gupta [3] using similar solution method, analyzed heat and mass transfer in the boundary layer over a stretching sheet subject to suction or blowing. Banks [4] studied similarity solutions of the boundary layer equations for stretching wall. Vajarvelu and Hodjnicolaou [5], Vajarvelu and Nayfeh [6], Reptis [7], Vajarvelu [8] studied the heat transfer in a viscous fluid over a stretching sheet with viscous dissipation in porous medium and without porous medium. Prasad et al. [9] analyzed study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity. Mahapatra and Gupta [10] also discussed stagnation-point flow of a visco-elastic fluid towards a stretching surface. Viscous flow over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field was investigated by Raptis and Perdkis [11]. Cortell [12-13] has worked on viscous flow and heat transfer over a nonlinearly stretching sheet. Abbas and Hayat [14] investigated radiation effect on MHD flow in a porous space. Awang and Hashim [15] obtained series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field.

II.

FORMULATION OF THE PROBLEM

Let us consider a steady two dimensional laminar flow of a viscous incompressible electrically conducting fluid past over a porous substrate attached to a non-linearly stretching sheet. The x-axis is taken along the stretching surface in the direction of motion and y-axis is perpendicular to it. A uniform magnetic field of strength B0 is assumed to be applied normal to the stretching surface. The magnetic Reynolds number is taken to be small and therefore the induced magnetic field is neglected. Two equal and opposite forces are applied along the x-axis is so that the sheet is stretched keeping the origin fixed.

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The stretching sheet has uniform temperature Tw and a non-

2υ c ( n2+1) x f (η ) n +1

ψ ( x, y ) =

linear velocity U w = cx , where n (>1) is constant. All the fluid properties are assumed to be constant throughout the motion. Under the usual boundary layer approximations, the governing boundary layer equations of mass conservation, momentum and energy with Joule heating and viscous dissipation are: n

∂u ∂v + = 0 ∂x ∂y

u

T − T∞ = θ (η ) Tw − T∞ where

c(n + 1) ( n2−1) (6) x 2υ Then, the momentum and energy equations (2) and (3) are transformed to:

η=y

(1)

υ ∂u ∂u ∂ 2u σ B02 u− u +v = υ 2− K0 ρ ∂x ∂y ∂y

(2) 1 '  2n  ' 2  1   f ''' + ff '' −  0 ( f ) −  M +  f = n + 1 n + 1 K     

2

 ∂T  ∂u  ∂T  ∂ 2T 2 2 +v ρcp  u  = κ 2 + µ   + σ B0 u ∂y  ∂y  ∂x  ∂y 

(3)



(7)

2  M (8) f '  = 0 ( n 1) +     The corresponding boundary conditions are:



θ '' + Pr  f θ ' + Ec  f '' + 2

f = 0 ’ f ' =1, θ =1 f ' → 0; θ → 0 η →∞:

η = 0:

(9) Where prime (´) denote the differentiation with respect to η and dimensionless parameters are:

2σ B02 ρ cx ( n −1) u2 Ec = c p (Tw − T∞ )

M= Fig.1: Physical model for MHD flow past over a porous substrate attached to the stretching sheet.

Where u and v are the velocities in the x- and y- directions respectively, K 0 is the permeability of the porous substrate, ρ

Pr =

is the density, μ is the viscosity, ν = µ is the kinematic

coefficient c f and heat transfer rates i.e. the Nusselt number Nu are:

The boundary conditions are:

y →∞:

u →0;

v = 0; T → T∞

T = Tw cf =

(4)

III. ANALYSIS The equation of continuity (1) is identically satisfied if we = ⇒ cf choose the stream function ψ such that u=

∂ψ ∂y

,

v= −

∂ψ ∂x

(Prandtl number)

K=

viscosity, c p is the specific heat at constant pressure, κ is thermal conductivity of the fluid under consideration and T is the temperature.

u = cx n (n > 1) ;

(Eckert number)

K 0 c ( n −1) x (Permeability parameter) (10) 2υ The physical quantities of interest are the skin-friction

ρ

y = 0:

µcp κ

(Magnetic parameter)

τ

w = ρU w 2 2

 ∂u    ∂y  y =0

µ

ρU w 2 2

2υ (n + 1) − ( n2+1) '' = x f (0) c

2(n + 1) '' f (0) Re

(11)

and

(5)

 ∂T  x   ∂y  y =o Nu = − Tw − T∞

The momentum and energy equations can be transformed into the corresponding ordinary differential equations by introducing the following similarity transformations:



⇒ Nu = −

c(n + 1) x 2υ

( n +1) 2

− θ ' (0) =

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(n + 1) Re ' θ (0) 2

Re =

cx n +1

(Reynolds number)

ν

(12)

(13)

where 1.4

1.2

1 M = 0, 0.5, 1.0, 2.0

f '(η)

0.8

0.6

0.4

0.2

0

0

0.5

1.5

1

2.5

2

3

3.5

4

4.5

5

η

Fig. 2: Velocity profile against

η

for various values of Magnetic parameter M for n = 1.5 and K = 0.5.

1 0.9 0.8 0.7 K = 0.5, 1.0, 1.5, 2.0

f '(η)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

η

Fig. 3: Velocity profile against

η

for various values of Permeability parameter K for n = 1.5 and M = 1.0.

stress which is proportional to f ′′(0) and the rate of heat transfer which is proportional to θ′(0) are tabulated in Table 1.0 and Table 2.0 for different values of parameters respectively. It is observed from the table that the shear stress increase and heat transfer rate decrease (for Ec = -0.5) and increase (for Ec = 0.5) as Magnetic Parameter increases. Also the shear stress decrease and heat transfer rate increase as Permeability Parameter increases Also the Nusselt number decreases for increasing value of Ec for a given Pr, whereas it is increases for increasing value of Pr for a given value of Ec.

IV. RESULT AND DISCUSSION The set of nonlinear ordinary differential equations (7) and (8) with boundary conditions (9) were solved numerically using Runge - Kutta forth order algorithm with a systematic guessing of f ′′(0) and θ′(0) by the shooting technique until the boundary conditions at infinity are satisfied. The step size ∆𝜂𝜂 = 0.001 is used while obtaining the numerical solution and accuracy up to the seventh decimal place i.e.1 × 10−7 , which is very sufficient for convergence. In this method, we choose suitable finite values of η→∞, say η∞ , which depend on the values of the parameter used. The computations were done by a program which uses a symbolic and computational computer language Matlab. The shear

The velocity profile f (η ) for different values of the magnetic parameter M is shown in fig.2. It is observed that velocity boundary layer thickness increase with the increasing '



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values of M. The velocity profile f (η ) for different values of the permeability parameter K is shown in fig.3. It is '' TABLE 1.0: VALUES OF f (0)

observed that velocity boundary layer thickness decreases with the increasing values of K.

'

K

M

f '' (0)

0.5

0 0.5 1.0

-0.61016178 -0.46849653 -0.31536821

2.0 0 0.5

0.02209619 -0.85782905 -0.73987740

1.0 2.0

-0.61016178 -0.31536821

0 0.5 1.0

-0.93038545 -0.81977316 -0.69797228

2.0 0 0.5

-0.41869183 -0.96499581 -0.85782905

1.0 2.0

-0.73987738 -0.46849654

1.0

1.5

2.0

TABLE 2.0: VALUES OF

K

FOR DIFFERENT VALUES OF M AND K.

Pr 0.7

1.0 0.5 7.0

0.7

1.0 1.0 7.0

0.7

1.0 1.5 7.0

0.7

−θ ' (0) Ec -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5

FOR DIFFERENT VALUES OF THE PARAMETERS K, M, PR AND EC.

M 0 0.62268675 0.53664197 0.45059719 0.78073952 0.66331920 0.54589887 2.53027720 1.97085630 1.41143539 0.61996631 0.49727587 0.37458542 0.78720963 0.61772567 0.44824170 2.78623606 1.91931371 1.05239136 0.61079155 0.48588356 0.35097556 0.79099344 0.60413266 0.41727188 2.87380617 1.90383967 0.93387316 0.61142526 0.48054105 0.34965684

0.5 0.67711083 0.55836610 0.43962138 0.84862808 0.68781399 0.52699992 2.73631889 1.99915729 1.26199569 065841157 0.51608231 0.37375304 0.83536153 0.63974277 0.44412400 2.91753463 1.94414786 0.97076109 0.65544611 0.50332319 0.35120026 0.83466051 0.62485924 0.41505797 2.99121448 1.92737474 0.86353501 0.65442919 0.49727586 0.34012554

1.0 0.74897243 0.58060544 0.41223845 0.93897542 0.71258574 0.48619606 3.03308098 2.02864836 1.02421574 0.70864418 0.53664197 0.36463975 0.89827567 0.66331920 0.42836273 3.10311461 1.97085630 0.83859799 0.70014748 0.52276579 0.34538411 0.89079238 0.64745721 0.40412204 3.15233507 1.95285233 0.75336959 0.69667132 0.51608232 0.33549332

2.0 0.95870448 0.62473794 0.29077140 1.20540944 0.76126525 0.31712107 3.97595248 2.09869193 0.20343138 0.85959050 0.58060544 0.30162038 1.08853921 0.71258574 0.33663227 3.72576536 2.02864836 0.33153137 0.83436493 0.56575114 0.29713736 1.05979295 0.69606782 0.33234270 3.68986946 2.00887589 0.32788229 0.82312645 0.55836610 0.29360575



2.0

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-0.5 0 0.5 -0.5 0 0.5

1.0

7.0

0.79305297 0.59768365 0.40231432 2.91708202 1.89641790 0.87575377

The temperature profiles for different values of M, Pr and Ec are presented in figure 4 to figure 7. It is observed from the figures that the boundary conditions are satisfied asymptotically in all the cases, which supporting the accuracy of the numerical results obtained. All the figures shows that increasing value of any parameter, result is decrease the thermal boundary layer except the Eckert number, result is increase the thermal boundary layer, whereas increase in

0.83478867 0.61772567 0.40066267 3.02904923 1.91931371 0.80957819

0.88794845 0.63974277 0.39153710 3.18010127 1.94414787 0.70819447

1.04717401 0.68781399 0.32845398 3.67963858 1.99915728 0.31867599

Eckert number is to increase the thermal boundary layer. It is observed that for small Prandtl number (Pr < 1, Fig. 6), there is a very low difference at the end of diagram between the curves with and without heat dissipation and in case with large Prandtl number (Pr > 1, Fig. 7) and negative Eckert number, the dimensionless temperature θ gains a negative value after reaching zero and at the end of path, it reaches zero again.

1.2

1

θ (η)

0.8

M = 0, 0.5, 1.0, 2.0

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

η Fig. 4: Temperature distribution against

η

for various values of Magnetic parameter M for Pr = 0.7, Ec = 0.1, K = 0.5, n = 1.5.

1.2

1

θ (η)

0.8

Pr = 0.5, 1.0, 2.0, 7.0

0.6

0.4

0.2

0

0

0.5

1.5

1

2

2.5

3

3.5

4

4.5

5


η

for various values of Prandtl number Pr for M = 1.0, Ec = 0.1, K = 1.0, n = 1.5.



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1.2

1

0.8

θ (η)

Ec = 0.5, 0, - 0.5 0.6

0.4

0.2

0

0

0.5

1

1.5

2

3

2.5

3.5

4

4.5

5


η

for various values of Eckert number Ec for M= 1.0, Pr = 0.71, K = 1.0, n = 1.5.

1.2

1

0.8 Ec = 0.5, 0, - 0.5

θ (η)

0.6

0.4

0.2

0

-0.2

0

0.5

1

1.5

2

3

2.5

3.5

4

4.5

5


V.

η

for various values of Eckert number Ec for M= 1.0, Pr = 7.0, K = 1.0, n = 1.5.

ii.

CONCLUSIONS

This paper extends the boundary layer problem of an electrically conducting fluid over a non-linear stretching porous surface by considering joule heating and viscous dissipation terms in the thermal boundary layer in the presence of magnetic field. Similarity equations are derived and solved numerically. The effects of different parameters like Magnetic parameter M, Permeability parameter K, Prandtl number Pr and Eckert number Ec are studies in detail.

REFERENCES

Sakiadis, B.C.: Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal 7(1), 26–28, (1961). Crane, L. J.: Flow past a stretching sheet. Z. Angew. Math. Phys. 21(4), 645–647, (1970). Gupta, P.S., Gupta, A.S.: Heat and mass transfer on stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, 744-746, (1977). Banks, W.H.H.: Similarity solutions of the boundary layer equations for stretching wall. J. Mec. Theor. Appl. 2, 375, (1983).

ACKNOWLEDGEMENTS

i.

This work has been carried out with the financial support of CSIR in the form of J.R.F awarded to one of the author (Gopi Chand).

The authors wish to express their sincere appreciation to the learned referee for careful reading of the manuscript and valuable suggestions.



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