Effects of magnetohydrodynamic flow past a vertical plate with variable

Appl. Math. Mech. -Engl. Ed. 31(3), 329–338 (2010) DOI 10.1007/s10483-010-0306-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2010

Applied Mathematics and Mechanics (English Edition)

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature ∗ Ibrahim A. ABBAS2,3 ,

G. PALANI1

(1. Department of Mathematics, S. S. Govt. Arts College, Tirutanni, Tirutanni-631-209, Tamil Nadu, India; 2. Department of Mathematics, Faculty of Sciences and Arts, Najran University, Najran, Kingdom of Saudi Arabia; 3. Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt) (Communicated by Zhe-wei ZHOU)

Abstract An analysis is performed to study the magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The dimensionless governing equations are unsteady, two-dimensional, coupled and non-linear governing equations. It is found that the magnetic field parameter has a retarding effect on the velocities of air and water. Key words

finite element, vertical plate, skin friction, velocity, Nusselt number

Chinese Library Classification O34 2000 Mathematics Subject Classification

76R10

Nomenclature a, Bo , g, Gr, M, N uX , N u, P r, T , T, , T∞ Tw ,

constant; magnetic field induction; acceleration due to gravity; thermal Grashof number; magnetic field parameter; dimensionless local Nusselt number; dimensionless average Nusselt number; Prandtl number; temperature; dimensionless temperature; temperature of fluid away from the plate; temperature of the plate;

t , t, u, v,

time; dimensionless time; velocity components in x, y-directions, respectively; U, V , dimensionless velocity components in X, Y directions, respectively; x, spatial coordinate along the plate; X, dimensionless spatial coordinate along the plate; y, spatial coordinate normal to the plate; Y, dimensionless spatial coordinate normal to the plate.

∗ Received Mar. 23, 2009 / Revised Nov. 24, 2009 Corresponding author G. PALANI, Professor, Ph. D., E-mail: [email protected]

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Ibrahim A. ABBAS and G. PALANI

Greek symbols α, β, τX , τ,

thermal diffusivity; volumetric coefficient of thermal expansion; local skin friction; average skin friction;

ν, σ, ρ,

kinematic viscosity; electrical conductivity; density.

∞,

free stream conditions.

Subscripts w,

1

conditions on the wall;

Introduction

Natural convection flows are frequently encountered in nature. There are many applications in science and technology for them. Extensive research works have been published on flow past a vertical plate under different conditions. The analytical method fails to solve the problem of unsteady two-dimensional natural convection flows past a semi-infinite vertical plate. The advent of advanced numerical methods and the developments in computer technology pave the way to solve such difficult problems. The unsteady natural convection flow past a semi-infinite vertical plate was first solved by Hellums and Churchill[1] , using an explicit finite difference method. Because the explicit finite difference scheme has its own deficiencies, a more efficient implicit finite difference scheme has been used by Soundalgekar and Ganesan[2] . A numerical solution of transient free convection flow with mass transfer on a vertical plate by employing an implicit method was obtained by Soundalgekar and Ganesan[3] . Takhar et al.[4] studied the transient free convection past a semi-infinite vertical plate with variable surface temperature using an implicit finite difference scheme of Crank-Nicolson type. The influence of magnetic field on electrically conducting viscous incompressible fluid is of importance in many applications such as extrusion of plastics in the manufacture of rayon and nylon, the purification of crude oil, and the textile industry, etc. In many process industries the cooling of threads or sheets of some polymer materials is important in the production line. The rate of the cooling can be controlled effectively to achieve final products of desired characteristics by drawing threads, etc., in the presence of an electrically conducting fluid subjected to magnetic field. The study of magnetohydrodynamic (MHD) plays an important role in agriculture, engineering and petroleum industries. The problem of free convection under the influence of the magnetic field has attracted the interest of many researchers in view of its applications in geophysics and astrophysics. The problem under consideration has important applications in the study of geophysical formulations, in the explorations and thermal recovery of oil, and in the underground nuclear waste storage sites. The MHD has also its own practical applications. For instance, it may be used to deal with problems such as the cooling of nuclear reactors by liquid sodium and induction flow meter, which depends on the potential difference in the fluid in the direction perpendicular to the motion and to the magnetic field. Soundalgekar et al.[5] analyzed the problem of free convection effects on Stokes problem for a vertical plate under the action of transversely applied magnetic field. Sacheti et al.[6] obtained an exact solution for the unsteady MHD free convection flow on an impulsively started vertical plate with constant heat flux. Shanker and Kishan[7] discussed the effect of mass transfer on the MHD flow past an impulsively started vertical plate with variable temperature or constant heat flux. Elbashbeshy[8] studied the heat and mass transfer along a vertical plate under the combined buoyancy effects of thermal and species diffusion, in the presence of the magnetic field. Ganesan and Palani[9] obtained a numerical solution of the unsteady MHD flow past a semi-infinite isothermal vertical plate using the finite difference method.

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature

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In recent years, the effects of the transverse magnetic field on the flow of a viscous incompressible electrically conducting fluid have also been extensively studied by many research workers. However, the unsteady natural convection flow over a semi-infinite vertical plate with the MHD has not been given any attention to in the literature. Hence, it has been proposed to solve the transient free convection MHD flow past a semi-infinite vertical plate with variable surface temperature with the finite element method.

2

Mathematical analysis

We considered a two-dimensional unsteady flow of a viscous incompressible fluid past a semi-infinite vertical plate. It is assumed that the effects of viscous dissipation are negligible in the energy equation. Initially, it is also assumed that the plate and the fluid are of the same temperature. At time t > 0, the temperature of the plate is supposed to be suddenly raised and maintained at a higher temperature in the form Tw (x) = T∞ + axn . The x-axis is measured along the plate in the upward directions from the leading edge and the y-axis is measured normal to the plate. A uniformly transverse magnetic field is applied along the y-axis. It is also assumed that the induced magnetic field and viscous dissipation effects are negligible. Then, under the usual Boussinesq’s approximation, the boundary layer flow is governed by the following equations: ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ∂ 2 u σBo2 ∂u + u − T ) + ν − + v = gβ(T u, ∞ ∂t ∂x ∂y ∂y 2 ρ ∂T ∂2T ∂T ∂T + v = α + u . ∂t ∂x ∂y ∂y 2

(1) (2) (3)

The initial and boundary conditions are as follows: when t when t ⎧ ⎪ ⎨ u = 0, u = 0, ⎪ ⎩ u → 0,

≤ 0, u = 0, v = 0, T = T∞ ; > 0,

v = 0, T = T∞ T → T∞

T = T∞ + axn at x = 0, as y → ∞.

(4a)

at y = 0, (4b)

Introduce the following non-dimensional quantities: ⎧ x y uL − 1 ⎪ X= , Y = , U= Gr 2 , ⎪ ⎪ ⎪ L L ν ⎪ ⎪ ⎪ ⎪ vL − 1 νt 1 ⎪ ⎪ Gr 4 , t = 2 Gr 2 , V = ⎪ ⎪ ν L ⎪ ⎪ ⎪ ⎪ ⎨ T − T∞ , Tw (x) = Tw (L) − T∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ gβL3 (Tw (L) − T∞ ) ⎪ ⎪ Gr = , ⎪ 3 ⎪ ν ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ P r = ν , M = σBo L Gr− 12 . α ρν

(5)

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Governing equations reduce to the following form: ∂U ∂V + = 0, ∂X ∂Y ∂U ∂U ∂U +U +V = ∂t ∂X ∂Y ∂T ∂T ∂T +U +V = ∂t ∂X ∂Y The corresponding initial and boundary conditions

(6) ∂2U − M U, ∂Y 2 1 ∂2T . P r ∂Y 2 in a dimensionless form are given by T+

when t ≤ 0, U = 0, V = 0, T = 0 for all Y ; when t > 0, ⎧ ⎨ U = 0, V = 0, T = X n at Y = 0, U = 0, T = 0 at X = 0, ⎩ U → 0, T → 0 as Y → ∞.

3

(7) (8)

(9a) (9b)

Finite element method

The governing equations (6)−(8) are unsteady, coupled, and nonlinear with the initial and boundary conditions (9). They are numerically solved by the finite element method (FEM). According to the FEM, the region of integration of the governing equations is divided into rectangular meshes formed by two sets of lines parallel to the coordinate axis. Here, the region of integration is considered as a rectangle with sides Xmax (= 1.0) and Ymax (= 14.0) where Ymax corresponds to (Y = ∞) which lies very well outside the momentum and thermal boundary layers. The numerical values of the dependent variables like the velocities U, V , and the temperature T are obtained at the interesting points, which are called degrees of freedom. The weak formulations of the non-dimensional governing equations are derived. The set of independent test functions to consist of the velocities U, V , and the temperature T is prescribed. The governing equations are multiplied by independent weighting functions and then are integrated over the spatial domain within the boundary. Applying integration by parts and making use of the divergence theorem reduce the order of the spatial derivatives and allow for the application of the boundary conditions. The same shape functions are defined piecewise on the elements. Using the Galerkin procedure, the unknown fields U, V, T, and the corresponding weighting functions are approximated by the same shape functions. The last step towards the finite element discretization is to choose the element type and the associated shape functions. Eight nodes of quadrilateral elements are used. The unknown fields are approximated either by the linear shape functions, which are defined by four corner nodes or by the quadratic shape functions, which are defined by all of the eight nodes (two-dimensional quadrilateral elements). On the other hand, the unknown fields are approximated either by the linear shape functions, which are defined by three corner nodes or by quadratic shape functions, which are defined by all of the six nodes (two-dimensional triangular elements). The shape function is usually denoted by the letter N and is usually the coefficient that appears in the interpolation polynomial. A shape function is written for each individual node of a finite element and has the property that its magnitude is 1 at that node and 0 for all other nodes in that element. We assumed that the master element has its local coordinates in the range [–1, 1]. In our case, the two-dimensional quadrilateral elements are used, which are given as follows: (i) Linear shape functions are 1 (1 − ξ)(1 − η), 4 1 N3 = (1 + ξ)(1 + η), 4 N1 =

1 (1 + ξ)(1 − η), 4 1 N4 = (1 − ξ)(1 + η). 4

N2 =


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(ii) Quadratic shape functions are N1 = N2 = N3 = N4 = N5 = N6 = N7 = N8 =

1 (1 − ξ)(1 − η)(−1 − ξ − η), 4 1 (1 + ξ)(1 − η)(−1 + ξ − η), 4 1 (1 + ξ)(1 + η)(−1 + ξ + η), 4 1 (1 − ξ)(1 + η)(−1 − ξ + η), 4 1 (1 − ξ 2 )(1 − η), 2 1 (1 + ξ)(1 − η 2 ), 2 1 (1 − ξ 2 )(1 + η), 2 1 (1 − ξ)(1 − η 2 ). 2

We now study the local and average skin frictions, and the local and average Nusselt numbers. In the non-dimensional quantities, they are given by ∂U 3 4 , (11) τX = Gr ∂Y Y =0 1 ∂U 3 4 dX, (12) τ¯ = Gr ∂Y Y =0 0 ∂T 1 4 N uX = −XGr (13) TY =0 , ∂Y Y =0 1 ∂T 1 4 N u = −Gr TY =0 dX. (14) ∂Y Y =0 0

4

Results and discussions

In order to ascertain the accuracy of the numerical results, the present results are compared with the previous results of Takhar et al.[4] for P r = 0.7, M = 0, and n = 0 in Fig. 1. They are found to be in an excellent agreement.

Fig. 1

Comparison of steady state velocity profiles at X =1.0 (P r=0.7, M =0.0, n=0.0)

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The temperature and velocity profiles at X = 1.0 are shown in Figs. 2–7 for different values of parameters occurring into the problem and time t. It is observed that the temperature and velocity increase with time and reach a steady state. From Fig. 5, we observe that the velocity gradient for air (P r = 0.71) is always greater than that for water (P r = 7.0). Physically, this

Fig. 2

Temperature profiles at X =1.0 for different Pr and n (M =2.0)

Fig. 3

Temperature profiles at X =1.0 for different Pr and M (n=0.5)

Fig. 4

Temperature profiles at X =1.0 for different time t (P r=0.71, M =2.0, n=0.5)

Fig. 5

Velocity profiles at X =1.0 for different Pr and n (M =2.0)

Fig. 6

Velocity profiles at X =1.0 for different Pr and M (n=0.5)

Fig. 7

Velocity profiles at X =1.0 for different time t (P r=0.71, M =2.0, n=0.5)


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is true because the increase in the Prandtl number is due to an increase in the viscosity of the fluid which makes the fluid thick and hence causes a decrease in the velocity of the fluid. The increase in the value of n reduces the velocity on the surface up to height of the plate X = 1.0. Therefore, the velocity decreases with the increase of the value of n. It is observed that the temperature gradient along the plate near the leading edge varies inversely with n. Due to this, the impulsive force along the plate increases with the decrease of the value of n. From the numerical results, it is observed that a higher value of velocity is noted for M = 0. The effect of a transverse magnetic field on an electrically conducting fluid gives rise to a resistive type force called Lorentz force. This force has a tendency to slow down the motion of the fluid and to increase its temperature. The temperature distribution decreases as the Prandtl number of the fluid increases. Local skin frictions are shown in Figs. 8–10. The local wall shear stress decreases as M increases. It is because the velocity gradient decreases near the plate as M increases, which is shown in Fig. 6. It is also observed that the local wall shear stress is more for lower Prandtl number compared with the higher Prandtl number of the fluid. It is observed that by increasing n, the skin friction can be reduced.

Fig. 8

Fig. 9

Local skin friction for different Pr and n (M =2.0)

Local skin friction for different Pr and M (n=0.5)

Fig. 10

Local skin friction for different time t (P r=0.71, M =2.0, n=0.5)

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In Figs. 11–13, the values of the local Nusselt number are plotted for various values of exponent n, Prandtl number, and magnetic field parameter. It is observed that the Nusselt number increases with n. However, it is observed that the above trend is reversed near the leading edge. The local Nusselt number increases with P r while it decreases as M increases.

Fig. 11

Fig. 12

Local Nusselt number for different Pr and n (M =2.0)

Local Nusselt number for different Pr and M (n=0.5)

Fig. 13

Local Nusselt number for different time t (P r=0.71, M =2.0, n=0.5)

The average skin friction and the Nusselt number are plotted in Figs. 14–17. The average skin friction decreases as n increases. This is because the velocity gradient near the plate decreases

Fig. 14

Average skin friction for different Pr and n (M =2.0)

Fig. 15

Average skin friction for different Pr and M (n=0.5)


Fig. 16

Average Nusselt number for different Pr and n (M =2.0)

Fig. 17

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Average Nusselt number for different Pr and M (n=0.5)

as n increases. The average skin friction increases with time t and then remains stationary for large values of t. The average skin friction decreases with increasing P r throughout the transient period. The average skin friction gets reduced with the increasing value of M . Figure 17 shows that there is no change in the average Nusselt number in the initial period with respect to n. This reveals that initially heat transfer is due to conduction only. The average Nusselt number decreases with time t and becomes steady after some time. The average Nusselt number increases with increasing P r. The average Nusselt number gets reduced by the increasing value of M .

5

Conclusions

An unsteady natural convection MHD flow past a semi-infinite vertical plate with variable surface temperature is considered here. The governing partial differential equations are transformed into a set of dimensionless governing equations, which are solved numerically using the FEM. The conclusions of the study are as follows: (i) The magnetic field parameter M has a retarding effect on the velocity. (ii) The velocity decreases with the increase value of exponent n. (iii) Temperature decreases as the Prandtl number of the fluid increases. (iv) Due to the increase in the value of M, local skin friction is found to decrease. (v) Rate of heat transfer decreases as M increases.

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[6] Sacheti, N. C., Chandran, P., and Singh, A. K. An exact solution for unsteady magnetohydrodynamic free convection flow with constant heat flux. International Communications in Heat and Mass Transfer 29, 1465–1478 (1986) [7] Shanker, B. and Kishan, N. The effects of mass transfer on the MHD flow past an impulsively started infinite vertical plate with variable temperature or constant heat flux. Journal of Energy, Heat and Mass Transfer 19, 273–278 (1997) [8] Elbashbeshy, E. M. A. Heat and mass transfer along a vertical plate with variable surface tension and concentration in the presence of magnetic field. International Journal of Engineering Science 34(5), 515–522 (1997) [9] Ganesan, P. and Palani, G. Numerical solution of unsteady MHD flow past a semi-infinite isothermal vertical plate. Proceedings of the Sixth ISHMT/ASME Heat and Mass Transfer Conference and Seventeenth National Heat and Mass Transfer Conference, Kalpakkam, India, 184–187 (2004)