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.
328 A. K. Jhankal, R.N. Jat and Deepak Kumar
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)
Unsteady MHD Flow and Heat Transfer Over a Porous Stretching Plate 329
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).
330 A. K. Jhankal, R.N. Jat and Deepak Kumar
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
Unsteady MHD Flow and Heat Transfer Over a Porous Stretching Plate 331
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
332 A. K. Jhankal, R.N. Jat and Deepak Kumar
- 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.
REFERENCES
[1] Sakiadis, B. C., 1961, “Boundary-layer behavior on continues solid surfaces:
II. The boundary layer on continuous flat surface”, AIChE J., Vol.7, pp. 221-
225.
[2] Crane, L., 1970, “Flow past a stretching plate”, ZAMP, Vol.21, pp. 645-647.
[3] Pavlov, K. B., 1974, “Magnetohydrodynamic flow of an incompressible
viscous fluid caused by the deformation of a plane surface”, Magn. Gidrondin,
Vol.4, pp. 146-152.
[4] Andersson, H. I., 1995, “An exact solution of the Navier-Stokes equation for
magnetohydrodynamic flow”, Acta Machanica, 113, pp. 241-244.
[5] Liu, C., 2005, “A note on heat and mass tranfer for a hydromagnetic flow
over a stretching sheet”, International Communications in Heat and Mass
Transfer, 32, pp. 1075-1084.
[6] Grubka, L. J. and Bobba, K. M., 1985, “Heat transfer charactertics of a
continuous, stretching surface with variable temperature”, J. Heat Transfer,
107, pp. 248-250.
[7] Chen, C. K. and Char, M. I., 1988, “Heat transfer of a continous stretching
surface with suction or blowing”, J. Mathematical Anal. Appl., 135, pp. 568-
580.
[8] Char, M. I., 1994, “Heat transfer in a hydromagnetic flow over a stretching
sheet”, Heat Mass Transfer, 29, pp. 495-500.
[9] Bataller, R. C., 2008, “Similarity solution for flow and heat transfer of a
quiescent fluid over a nonlinearly stretching surface”, Mater. Process.
Technol., 203, pp. 176-183.
[10] Chakrabarthi, A. and Gupta, A. S., 1979, “A note on MHD flow over a
stretching permeable surface”. Q. Appl.Math., 37, pp. 73-78.
[11] Chiam, T., 1993, “Magneto hydrodynamic boundary layer flow due to a
continuous moving flat plate”, Comput. Math. Appl., 26, pp. 1-8.
Unsteady MHD Flow and Heat Transfer Over a Porous Stretching Plate 333
[12] Ghaly, A. Y., 2002, “Radiation effects on a certain MHD free convection
flow”, Chaos, Solitons and Fractals, 13, pp. 1843-1850.
[13] Raptis, A., 2004, “Effect of thermal radiation on MHD flow”, Applied
Mathematics and Computation, 153, pp. 645-649.
[14] Ishak, A., Nazar, R. and Pop, I., 2008, “Hydromagnetic flow and heat transfer
adjacent to a stretching vertical sheet”, Heat Mass Transfer, 44, pp. 921-927.
[15] Muhaimin, Kandasamy, R., Hashim, I. and Khamis, A. B., 2009, “On the
effect of chemical reaction, heat and mass transfer on nonlinear MHD
boundary layer past a porous shrinking sheet with suction”, Theoret. Appl.
Mech., Vol (36) No (2), pp. 101-117.
[16] Noor, N. F. M., Abdulaziz, O. and Hashim, I., 2010, “MHD flow and heat
transfer in a thin liquid film on an unsteady stretching sheet by the homotopy
analysis method”, International Journal for Numerical Methods in Fluids, 63,
pp. 357–373.
[17] Jat, R. N. and Chaudhary, S., 2010, “Hydromagnetic flow and heat transfer
on a continuous moving surface”, Applied Mathematical Science, 4, pp. 65-
78.
[18] Jhankal, A. K. and Kumar, M., 2013, “MHD Boundary Layer Flow Past a
Stretching Plate with Heat Transfer”, International J. of Engineering and
Science, Vol 2 (3), pp. 9-13.
[19] Devi, C. D. S., Takhar, H. S. and Nath, G., 1991, “Unsteady mixed convection
flow in stagnation region adjacent to a vertical surface”, Heat Mass Transfer,
26, pp. 71-79.
[20] Elbashbeshy, E. M. A. and Bazid, M. A. A.,2004, “Heat transfer over an
unsteady stretching surface”, Heat Mass Transfer, 41, pp. 1-4.
[21] Tsai, R., Huang, K. H. and Huang, J. S., 2008, “Flow and heat transfer over an
unsteady stretching surface with non-uniform heat source”, Int. Commun.
Heat Mass Transfer, 35, pp. 1340-1343.
[22] Pai, S. I., 1956, “Viscous Flow Theory: I Laminar Flow”, D.VanNostrand
Co., New York, USA.
[23] Schichting, H.,1964, “Boundary Layer Theory”, sixth ed., McGraw-Hill, New
York.
[24] Bansal, J. L., 1977, “Viscous Fluid Dynamics”, Oxford & IBH Pub. Co.,
New Delhi, India.
[25] Jeffery, A., 1966, “Magnetohydrodynamics”, Oliver and Boyed, New York,
USA.
[26] Bansal, J. L., 1994, “Magnetofluiddynamics of Viscous Fluids”, Jaipur
Publishing House, Jaipur, India.
334 A. K. Jhankal, R.N. Jat and Deepak Kumar