Unsteady MHD Flow and Heat Transfer Over a Porous Stretching … · 2017. 5. 15. · Porous...

International Journal of Computational and Applied Mathematics.

ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 325-333

© Research India Publications

http://www.ripublication.com

Unsteady MHD Flow and Heat Transfer Over a

Porous Stretching Plate

1A. K. Jhankal*, 2R. N. Jat and 3Deepak Kumar

1Department of Mathematics, Army Cadet College, Indian Military Academy,

Dehradun – 248 007, India. 2,3 Department of Mathematics, University of Rajasthan, Jaipur-302 004, India.

*Corresponding author E-mail: [email protected]

Abstract

This paper considers the problem of unsteady MHD flow over a porous

stretching plate. The governing boundary layer equations are transformed into

ordinary differential equations using similarity transformation which are than

solved numerically using shooting technique. The effects of unsteadiness

parameter, magnetic parameter, porosity, Eckert number and Prandtl number

on the flow and heat transfer characteristics are presented and discussed.

Keywords: MHD, boundary layer, stretching plate, porosity, similarity

solution.

NOMENCLATURE

𝑎, 𝑏, 𝑐 Constants, [−]

𝐴 Unsteady parameter (𝑐 𝑎⁄ ), [−]

𝐵0 Constant applied magnetic field, [𝑊𝑏 𝑚−2]

𝐶𝑝 Specific heat at constant pressure, [𝐽 𝐾𝑔−1𝐾−1]

𝑓 Dimensionless stream function, [−]

𝐸𝑐 Eckert number (= 𝑢2 𝐶𝑝(𝑇𝑤 − 𝑇∞)⁄ ), [−]

mailto:[email protected]

326 A. K. Jhankal, R.N. Jat and Deepak Kumar

𝑘0 Permeability of porous medium, [Darcy]

𝑀 Magnetic parameter (= 𝜎𝑒𝐵02 𝜌𝑎⁄ ), [−]

𝑃𝑟 Prandtl number (= 𝜇𝐶𝑝 ⁄ ), [−]

𝑆𝑝 Porosity parameter (= 𝜈 𝜌𝑎⁄ ), [−]

𝑡 Dimensionless time, [𝑠]

𝑇 Temperature of the fluid, [𝐾]

𝑢, 𝑣 Velocity component of the fluid along the x and y directions, respectively,

[𝑚 𝑠−1]

𝑥, 𝑦 Cartesian coordinates along the surface and normal to it, respectively, [𝑚]

Greek symbols

𝜌 Density of the fluid, [𝐾𝑔 𝑚−3]

𝜇 Viscosity of the fluid, [𝐾𝑔 𝑚 𝑠−1]

𝜎𝑒 Electrical conductivity, [𝑚2 𝑠−1]

𝜂 Dimensionless similarity variable,

[= (𝑈∞ 𝜈𝑥⁄ )1/2𝑦]

𝜈 Kinematic viscosity, [𝑚2𝑠−1]

Thermal conductivity, [𝑊 𝑚−2𝐾−4]

𝛹 Stream function, [= (𝜈𝑥𝑈∞)1

2𝑓(𝜂)]

𝜃 Dimensionless temperature, [= (𝑇 − 𝑇∞) (𝑇𝑤 − 𝑇∞)]⁄

Superscript

Derivative with respect to 𝜂

Subscripts

𝑤 Properties at the plate

∞ Free stream condition

INTRODUCTION

The Study of MHD flow plays an important role in various industrial applications. Some

important applications are cooling of nuclear reactors, liquid metals fluid, power

generation system and aero dynamics. The problems of heat and mass transfer in the

Unsteady MHD Flow and Heat Transfer Over a Porous Stretching Plate 327

boundary layers on stretching surfaces have attracted considerable attention during the

last few decades. It is importance in connection with many engineering problems, such as

wire drawing, glass-fiber and paper production, drawing of plastic films, metal and

polymer extrusion and metal spinning. Both the kinematics of stretching and

simultaneous heating or cooling during such processes have a decisive influence on the

quality of the final products.

In his pioneering work, Sakiadis [1] developed the flow field due to a flat surface, which

is moving with a constant velocity in a quiescent fluid. Crane [2] extended the work of

Sakiadis [1] for the two-dimensional problem where the surface velocity is proportional

to the distance from the flat surface. As many natural phenomena and engineering

problems are worth being subjected to MHD analysis, the effect of transverse magnetic

field on the laminar flow over a stretching surface was studied by Pavlov [3]. Andersson

[4] then demonstrated that the similarity solution derived by Pavlov [3] is not only a

solution to the boundary layer equations, but also represents an exact solution to the

complete Navier-Stokes equations. Liu [5] extended Andersson’s results by finding the

temperature distribution for non-isothermal stretching sheet, both in the prescribed

surface temperature and prescribed surface heat flux conditions, in which the surface

thermal conditions are linearly proportional to the distance from the origin.

The heat transfer aspect for the problem posed by Crane [2] was studied by Grubka and

Bobba [6], who reported the solution for the energy equation in terms of Kummer’s

functions. Several closed form analytical solutions for specific conditions also reported.

Chen and Char [7] investigated the effects of suction and injection on the heat transfer

characteristics of a continuous, linearly stretching sheet for both the power law surface

temperature and the power law surface heat flux variations. Char [8] then studied the case

when the sheet immersed in a quiescent electrically conducting fluid in the presence of a

transverse magnetic field. The effect of thermal radiation on the heat transfer over a

nonlinear stretching sheet immersed in an otherwise quiescent fluid has been studied by

Bataller [9]. The effect of transverse magnetic field on the laminar flow over a stretching

surface was studied by number of researchers Chakrabarthi and Gupta [10], Chiam [11],

Ghaly [12], Raptis [13], Ishak et al. [14], Muhaimin et al. [15], Noor et al. [16], Jat and

Chaudhary [17], , Jhankal and Kumar [18] etc.

The unsteady boundary layer flow over a stretching sheet has been studied by Devi et al.

[19], Elbashbeshy and Bazid [20] and quit recently by Tsai et al. [21]. In the present

study, we consider the problem of unsteady flow over a porous stretching plate in

presence of transverse magnetic field. The governing partial differential equations are

transformed into ordinary differential equations using similarity transformations which

are than solved numerically using shooting technique.


MATHEMATICAL FORMULATION OF THE PROBLEM

Let us consider an unsteady, laminar two-dimensional boundary layer flow over a

continuously porous stretching plate. The fluid is an electrically conducting

incompressible viscous fluid. It is assumed that external fluid owing polarization of

charges and Hall-effect are neglected. At time t=0, the plate is impulsively stretched with

the velocity 𝑈∞(𝑥, 𝑡) along x-axis, keeping the origin fixed in the fluid of ambient

temperature 𝑇∞. The stationary Cartesian coordinate system has its origin located at the

leading edge of the plate with the positive x-axis extending along the plate, while y-axis

is measured normal to the surface of the plate. A transverse magnetic field of strength 𝐵0

is assumed to be applied in the positive y-axis, normal to the surface. Under the usual

boundary layer approximations, the governing equation of continuity, momentum and

energy (Pai [22], Schlichting [23], Bansal [24]) under the influence of externally imposed

transverse magnetic field (Jeffery [25], Bansal [26]) are:

𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦= 0 …(1)

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 𝜈

∂2u

∂y2−

σeB02u

ρ−

νu

k0 …(2)

𝜕𝑇

𝜕𝑡+ 𝑢

𝜕𝑇

𝜕𝑥+ 𝑣

𝜕𝑇

𝜕𝑦= 𝛼

∂2T

∂y2+

σeB02u2

ρCp …(3)

Accompanied by the boundary conditions:

𝑦 = 0: 𝑢 = 𝑈𝑤, 𝑣 = 0, 𝑇 = 𝑇𝑤

𝑦 → ∞: 𝑢 → 0, 𝑇 → 𝑇∞ …(4)

We assume that the stretching velocity 𝑈𝑤(𝑥, 𝑡) and surface temperature 𝑇𝑤(𝑥, 𝑡)

are of the form:

𝑈𝑤(𝑥, 𝑡) =𝑎𝑥

1−𝑐𝑡, 𝑇𝑤(𝑥, 𝑡) = 𝑇∞ +

𝑏𝑥

1−𝑐𝑡 …(5)

The governing partial differential equations (1) – (3) can be reduced to ordinary

differential equations by introducing the following transformation

𝜂 = (𝑈∞

𝜈𝑥)

1/2

𝑦, 𝛹 = (𝜈𝑥𝑈∞)1

2𝑓(𝜂), 𝜃(𝜂) =𝑇−𝑇∞

𝑇𝑤−𝑇∞ …(6)

The continuity equation (1) is satisfied by introducing a stream function Ψ such that

𝑢 =𝜕Ψ

𝜕𝑦 and v= −

𝜕Ψ

𝜕𝑥.

The transformed nonlinear ordinary differential equations are:

𝑓 ′′′ + 𝑓𝑓 ′′ − 𝑓′2 − 𝑀𝑓 ′ − 𝑆𝑝𝑓′ − 𝐴 (𝑓 ′ +

1

2𝑓 ′′𝜂) = 0 …(7)


𝜃′′

𝑃𝑟+ fθ′ − 𝑓 ′𝜃 + 𝐸𝑐𝑀𝑓 ′

2− 𝐴 (𝜃 +

1

2𝜂𝜃 ′) = 0 …(8)

The transformed boundary conditions are:

𝑓(0) = 0, 𝑓 ′(0) = 1, 𝜃(0) = 1 𝑎𝑛𝑑 𝑓 ′(∞) → 0, 𝜃(∞) → 0. …(9)

Where prime denotes differentiation with respect to η , 𝑀 =𝜎𝑒𝐵0

2

𝜌𝑎 is the magnetic

parameter, 𝑆𝑝 =𝜈

𝜌𝑎 is the porosity parameter, 𝑃𝑟 =

𝜇𝑐𝑝

𝜅 is the Prandtl number and

𝐸𝑐 =𝑢𝑤

2

𝐶𝑝(𝑇𝑤−𝑇∞) is the Eckert number

NUMERICAL SOLUTION AND DISCUSSION

The non-linear differential equations (7) and (8) subject to the boundary conditions

(9) are solved by Runge-Kutta fourth order scheme with a systematic guessing of ' 'f (0) and (0) by the shooting technique until the boundary conditions at infinity

are satisfied. The step size = 0.01 is used while obtaining the numerical solution

and accuracy upto the seventh decimal place i.e. 1 x 104, which is very sufficient

for convergence. The computations were done by a programme which uses a

symbolic and computer language Matlab.

Figure 1 and 2 show the velocity profile for different values of magnetic parameter

(M) and unsteady parameter (A) respectively, when the other parameter is fixed.

From both the figures, it is observed that the velocity gradient at the surface

increases (in magnitude) with M and A. Thus, the magnitude of skin friction

coefficient |𝑓 ′′(0)| increases as A increases (table 1). Physically, negative values of

𝑓 ′′(0) means the solid surface exerts a drag force on the fluid. This is not surprising

since the development of the velocity boundary layer is caused solely on the

stretching plate. Further, the velocity is found to decreases as the distance increases

from the surface increases and reaches the boundary condition at infinity

asymptotically.

Figure 3, which illustrate the porosity parameter Sp on the velocity profile. We infer

from this figure that the velocity profile decreases with increasing values of

porosity parameter Sp but quit slowly. This phenomenon corresponds with the

assumption of pure Darcy flow.

Figure 4 and 5 show that the temperature gradient at the surface increases as (in

magnitude) as M and A increases respectively, which impales an increase of heat

transfer rate at the surface |−𝜃 ′(0)| as A increases (table 2).


Figure 6 which illustrate the effect of Prandtl number (Pr) on the temperature

profiles. We infer from this figure that the temperature decreases with an increase in

Prandtl number, which implies viscous boundary layer thickness than the thermal

boundary layer. From these plots it is evident that large values of Prandtl number

result in thinning of the thermal boundary layer. In this case temperature

asymptotically approaches to zero in free stream region.

Figure 7, which is a graphical representation of the temperature profiles for

different values of Eckert number (Ec) versus η. It is evident from these plots that

temperature of the fluid decreases with increases in Eckert number. Physically it

means that the heat energy is stored in the fluid due to the frictional heating.

Table 1. Numerical values of Skin

friction coefficient, when M =0.5 and Sp =0.4

A f''(0)

1 -1.773

2 -1.951

3 -2.121

Table 2. Numerical values of Nusselt number

when Pr = 1.0, Ec = 0.05, M = 0.5, Sp = 0.4

A -'(0)

1 -1.8051

2 -2.0280

3 -2.2300

Figure 1: Velocity profile for various values of M

when A=1 and Sp =0.4.

Figure 2 : Velocity profile for various values of A

when M=0.5 and Sp=0.4.

Figure 3 : Velocity profile for various values of Sp

when A=1.0 and M=0.5.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

f ' (

)

M = .5 , 1 ,1.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

f ' (

)

A = 1 , 2 , 3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

f ' (

)

Sp = .4 , .8 , 1.2


Figure 4 : Temperature profile for various values of

M when Ec=0.05, Pr=1.0, and A=1.0.

Figure 5 : Temperature profile for various values of

A when Ec=0.05, Pr=1.0, and M=0.5.

Figure 6: Temperature profile for various values of

Pr when Ec=0.05, A=1.0, and M=0.5.

Figure 7: Temperature profile for various values of

Ec when Pr=1.0, A=1.0, and M=0.5.

CONCLUSION

In the present study, we consider the problem of unsteady flow over a porous

stretching plate in presence of transverse magnetic field.The governing partial

differential equations are transformed into ordinary differential equations by means

of similarity transformations. The resulting non-linear ordinary differential

equations are solved using Runge-Kutta fourth order method along with shooting

technique. The velocity and temperature distributions are discussed numerically and

presented through graphs. The numerical values of Skin-friction coefficient and

Nusselt number are derived, for various values of unsteady parameter A and

presented through tables. From the study, following conclusions can be drawn:

- Velocity gradient at the surface increases (in magnitude) with M and A.

- Velocity profile decreases with increasing values of porosity parameter Sp

but quit slowly.

- Temperature gradient at the surface increases as (in magnitude) as M and A

increases respectively.

- Temperature decreases with an increase in Prandtl number, which implies

viscous boundary layer thickness than the thermal boundary layer.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(

)

M = .5 ,1.5 , 2.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(

) A = 1 , 2 ,3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(

)

Pr = .7 ,1 , 1.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(

) Ec = .05 , .1 , 1.5


- Temperature of the fluid decreases with increases in Eckert number.

- Shear stress and Nusselt number increase (in magnitude) due to increase

parameter A.

ACKNOWLEDGEMENTS

One of the authors (D.K.) is grateful to the UGC for providing financial support in

the form of BSR Fellowship, India.

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