Cent. Eur. J. Eng. • 4(4) • 2014 • 391-397 DOI: 10.2478/s13531-013-0174-x

Central European Journal of Engineering

Effect of MHD and Injection through one side of a long vertical channel embedded in porous medium with transpiration cooling Research Article

K.Govardhan1 , K. Kaladhar2∗ , G. Nagaraju1 , B. Balaswamy3 1 Department of Engineering mathematics GITAM University, India 2 Department of Mathematics National Institute of Technology Puducherry Karaikal-609605, India 3 Department of Mathematics Osmania University, India

Received 19 January 2014; accepted 09 July 2014

Abstract: This paper examines the effect of MHD, and injection through one side of a long vertical channel embedded in porous medium with transpiration cooling. The governing nonlinear partial differential equations have been transformed by similarity transformation into a set of ordinary differential equations, which are solved numerically by Adam-moultan Predictor-Corrector method with Newton-Raphson Method for missing initial conditions. Proflles of dimensionless velocity, temperature and concentration are shown graphically for different parameters entering into the analysis. Also the effects of the pertinent parameters on the heat transfer rates are tabulated. An analysis of the results obtained shows that the flow field is influenced appreciably by emerging parameters of the present study. Keywords: Porous Medium • MHD • Heat Transfer • Adams Predictor-Corrector Method © Versita sp. z o.o.

1.

Introduction

The study of heat transfer in porous media is a well developed field of investigation because of its importance in a variety of situations occurring in geothermal systems, microelectronic heat transfer equipment, thermal insulation, and thermoacoustic engines, which can provide cooling or heating using environmentally benign gases as the working fluid. It is also of interest in the nuclear industry, particularly in the evaluation of heat removal ∗

E-mail: [email protected]

from a hypothetical accident in a nuclear reactor and to provide effective insulation. Comprehensive literature surveys concerning the subject of porous media can be found in the recent books by Pop and Ingham [1], Vafai [2], Bejan and Kraus [3] and Nield and Bejan [4]. There has been a renewed interest in MHD flow and heat transfer in porous and clear domains due to the important effect of magnetic field on the performance of many systems using electrically conducting fluids such as MHD power generators, cooling of nuclear reactors, accelerators, pumps, and flow meters. A survey of magnetohydrodynamics (MHD) studies in the technological field was reported by Moreau [5]. In view of applications, Shohel and Fraser [6] have studied 391

Effect of MHD and Injection through one side of a long vertical channel embedded in porous medium with transpiration cooling

the mixed convection-radiation interaction in a vertical porous channel with Entropy generation. Radiation effects on MHD flow in a porous space has been presented by Zaheer and Hayat [7]. Pal [8] investigated the magnetohydrodynamic non-Darcy mixed convection flow from a vertical heated plate embedded in a porous medium with variable porosity. Rashidi and Erfani [9] presented the analytical solution of MHD stagnationpoint flow in porous media with heat transfer. Later, Rashidi et al. [10] presented the simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk. Recently, Turkyilmazoglu and Senel [11] considered the heat and mass transfer of the flow due to a rotating rough and porous disk. Most recently, Srinivasacharya and Upendar [12] presented the soret and dufour effects on MHD mixed convection heat and mass transfer in a micropolar fluid. Wang and Skalak [13] were the first to present the solution for the three-dimensional problem of fluid injection through one side of a long vertical channel for a Newtonian fluid. They obtained a series solution valid for small values of the cross-flow Reynolds number and a numerical solution for both small and large cross-flow Reynolds numbers. Huang [14] has reexaminedWang and Skalak’s [13] problem by using quasilinearization technique. Later Baris et al. [15] extended the same to an incompressible non-Newtonian fluid with elastic properties. Guria et al. [16] studied the three dimensional viscous impressible fluid through a porous medium bounded by two vertical walls one wall being impermeable and other a permeable. The steady flow of an incompressible viscous fluid above an infinite rotating disk in a porous medium is studied with heat transfer was considered by Attia [17]. Mixed convection heat and mass transfer in three dimensional flow of a viscous incompressible fluid past a vertical porous plate through a porous medium with periodic permeability is analyzed by Das et al. [18]. Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method has been presented by Rashidi et al. [19]. Recently, a study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method has been reported by Rashidi et al. [20]. Many of the problems in the literature deal with three dimensional flow is focused in Newtonian and non-Newtonian fluids but much attention is not given to Magnetohydrodynamics with injection effects. So in the present study it is proposed to investigate the effects of MHD and injection on the three-dimensional convective flow of a viscous incompressible fluid through 392

a long vertical channel embedded in a porous medium whose one side is made of a porous plate while the other is impermeable plate. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless form by similarity transformations. Free convection flow on a stretching surface with suction and blowing presented numerically using similarity transformations by Rama Subba Reddy et al. [21]. The Adams Predictor-Corrector Method is employed to solve the governing nonlinear equations. In order to get a clear insight of the physical problem, behavior of the emerging flow parameters on the velocity and temperature are displayed through graphical illustrations.

2.

Formulation of the Problem

Consider the steady Magnetohydrodynamic flow of a viscous incompressible fluid through a porous medium bounded by two infinite vertical plates in such a way that the plate at y = 0 is impermeable and the fluid is driven in through the plate at y = d with the uniform velocity V . A Cartesian coordinate system is introduced with the plates lying vertically on the xz plane, the z axis is taken along the direction of the flow, and the y axis is normal to the plates. The fluid flows out through the side and bottom of the plate and due to gravity which flows in z−direction also. Hence the flow becomes three dimensional. Let u, v, and w be the velocity components in the directions of the x, y and z axes, respectively. Let the temperature of the plates y = 0 and y = d be T0 and T1 respectively. Assume that the length of the plate is much greater than the breadth and breadth is the gap between the plates. Due to this assumption the edge effects are ignored and the isobars are parallel to the z− axis. The physical model and coordinate system are shown in Figure 1. A uniform magnetic field (B0 ) is applied perpendicular to the porous walls. Assume that the flow is steady and magnetic Reynolds number is very small so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Further, assume that all the fluid properties are constant. With the above assumptions, the equations governing the steady flow of an incompressible fluid, under usual MHD approximations are ∂v ∂w ∂u + + =0 ∂x ∂y ∂z ∂u ∂u ∂u 1 ∂P +v +w =− ∂x ∂y ∂z ρ ∂x  2  ∂ u ∂2 u ∂ 2 u ν σ B02 +ν + + u − u ∂x 2 ∂y2 ∂z 2 K ρ

(1)

u

(2)

K.Govardhan, K. Kaladhar, G. Nagaraju, B. Balaswamy

Introducing the following similarity transformations [13] y K V , S = 2 , u = xf 0 (η), d d d d2 g h(η), T − T0 = (T1 − T0 )θ(η) v = −V f(η), w = ν  ρ AV 2 2 2νV 0 2νV P =− x + V 2 (f(η))2 + f (η) − 2 d2 d K  Z − f(η)dy η=

(7) in Eqs. (2)-(5), we get the following nonlinear system of differential equations f 000 − Re(f 02 − f f”) + ReA −

0



h” + Refh −



 1 + M f0 = 0 S

 1 +M h+1=0 S

θ” + RePrfθ 0 = 0

Figure 1.

u

(3)

∂w ∂w ∂w +v +w = ∂x ∂y ∂z  2  1 ∂p ∂ w ∂2 w ∂2 w ν σ B02 − +ν + + − w− w ρ ∂z ∂x 2 ∂y2 ∂z 2 K ρ (4) where u, v and w are the velocity components in x, y and z directions respectively, T is the temperature, C is the concentration, g is the acceleration due to gravity, ρ is the density, µ is the dynamic coefficient of viscosity, P is the pressure, K is the permeability, k is the coefficient of thermal conductivity, σ is the electrical conductivity, B0 is the uniform magnetic field. The boundary conditions are

u = 0, v = −V , w = 0, T = T1

at

y=0

as y = d

(10)

f 0 (0) = 0, f(0) = 0, h(0) = 0, θ(0) = 0 f 0 (1) = 0, f(1) = 1, h(1) = 0, θ(1) = 1

u

u = 0, v = 0, w = 0, T = T0

(9)

where the primes indicate partial differentiation with ρCp is the Prandtl number, respect to η alone, Pr = k Vd Re = is the Reynolds number, A is a constant to ν be determined, S is the permeability parameter, M is the magnetic parameter, f is the normal velocity profile and h is the axial velocity profile. Boundary conditions (6) in terms of f, h and θ become

Physical model and coordinate system.

∂w ∂w ∂w +v ∂y + w = ∂x ∂w ∂z  2  2 1 ∂P ∂ v ∂ v ∂2 v ν − +ν + 2 + 2 − v ρ ∂y ∂x 2 ∂y ∂z K

(8)

3. Numerical problem

solution

of

(11)

the

The above equations (8)-(10) along with the boundary conditions (11) are solved by reducing to initial value problem. Since these equations are non-linear, solutions of this problem is not possible. For that assume f = y1

(12)

y01 = y2

(13)

y02 = y3

(14)

y03 = y4

(15)

θ 0 = y5

(16)

(5)

(6)

393

Effect of MHD and Injection through one side of a long vertical channel embedded in porous medium with transpiration cooling

Further more, in order to integrate (12) and (13) as an initial value problem we require y2 (0) = 0 i.e. f”(0) = 0, y4 (0) i.e. h0 (0) and y5 (0) i.e. θ 0 (0). Due to the limited data the suitable guess for f”(0), h0 (0) and θ 0 (0) are chosen and then integration is carried out. The calculated values of f 0 , h and θ at η = 1 are compared with the given boundary conditions f 0 (1) = 0, h(1) = 0 and θ(1) = 1 and then adjusted the estimated values of f”(0), h0 (0) and θ 0 (0) to give better approximation for the solution. A series of values for f”(0), h0 (0) and θ 0 (0) are traced, then applied the Adam-Moulton predictor corrector method with step size h = 0.002. To improve the solutions, N-R method has been used. The above procedure is repeated until the results up to the desired degree of accuracy 10−5 .

4.

Results and Discussions

Figure 2 reflects the normal velocity profiles for different values of the permeability parameter S at Re = 2. As the permeability S increases the magnitude of the flow velocity reduces up to the half of the distance between the plates from the impermeable wall and then onwards reverse effect is observed when the fluid particles go nearer to the permeable plate. This because there is a sink in the fluid near the middle of the plates. Figure 3 represents the axial velocity profile for various values of S when Re = 5. It can be seen from Figure 3 that the axial velocity increases with the higher permeability which is due to the gravity. For lower injection rate and permeability it is greatly reduced. The maximum velocity of the fluid occurs at the middle of the plates. Figure 4 depicts the velocity profile f 0 (η) for various values of S at Re = 0.5. It can be observed that the flow velocity increases as in increase in permeability parameter (S). Figure 5 displays the effect of Magnetic parameter M on normal velocity profile when Re = 0.2. It is observed that as the magnetic parameter M increases the magnitude of the flow velocity reduces up to the half of the distance between the plates from the impermeable wall and then reverse trend is observed as the fluid particles go nearer to the permeable plate. There is a sink in the fluid near the middle of the plates. Figure 6 represents the axial velocity profile for various values of M at Re = 0.2. It can seen that the axial velocity increases as an increase in the magnetic field parameter, which is due to the gravity. Figure 7 shows that the tangential velocity profile f 0 (η) for various values of M when Re = 0.5. It is clear that as the magnetic parameter increases, the flow velocity increases. The effect of Pr on the temperature profile is shown in Figure 8. From this figure it is clearly seen that the temperature of the fluid increases significantly 394

Figure 2.

Effect of S on normal velocity at Re = 0.2.

Figure 3.

Effect of S on axial velocity at Re = 0.5.

with the increase of Prandtl number. Heat transfer θ 0 (η) at η = 0 (impermeable plate) and at η = 1 (porous plates) is plotted against cross flow Reynolds number for different values of S and Pr, is shown in Figure 9. It is observed that for small permeability the heat transfer is more in case of water (Pr = 0.5) than that of air (Pr = 0.7) on the impermeable plate. However, a reverse tendency is observed on the porous plate. For

K.Govardhan, K. Kaladhar, G. Nagaraju, B. Balaswamy

Figure 4.

Effect of S on flow velocity at Re = 0.5.

Figure 5.

Effect of magnetic parameter M on normal velocity when Re = 0.2.

higher permeability parameters the heat transfer remains the same on the walls and also for air and water. It is concluded that for higher Prandtl numbers the heat transfer on the impermeable plate is considerably higher than that of the porous plate. The effect of S on the non-dimensional heat transfer coefficients at both the plates η = 0, 1 with for fixed

Figure 6.

Effect of magnetic parameter M on axial velocity when Re = 0.5.

Figure 7.

Effect of magnetic parameter M on ï±Ćow velocity when Re = 0.5.

Pr = 0.7, Re = 0.2 is presented in Table 1. It can be noticed that the heat transfer coefficients are decreases with the increase of S. The effect of Reynolds number Re on the non-dimensional heat transfer coefficients is presented in Table 1. It can be observed that, for fixed values of Pr = 0.7 and S = 0.01, the non-dimensional heat transfer coefficient at η = 0 increase with the increase of Re and reverse trend is observed at η = 1. 395

Effect of MHD and Injection through one side of a long vertical channel embedded in porous medium with transpiration cooling

Table 1.

Effect of Permeability parameter (S), Reynolds number (Re), Prandtl number (Pr) on heat transfer coefficients.

S

Re

Pr

θ 0 (0)

θ 0 (1)

0.01

0.2

0.7

1.0219

0.9527

0.1

0.2

0.7

1.0213

0.9519

1

0.2

0.7

1.0212

0.9517

4

0.2

0.7

1.0212

0.9517

0.01

0.2

0.7

1.0219

0.9527

0.01

0.4 1

0.7

1.0439

0.907

0.01

0.6

0.7

1.0660

0.8631

0.01

0.8

0.7

1.0882

0.8206

0.01

1.0

0.7

1.1109

0.7784

0.01

0.2

1

1.0313

0.9330

0.01

0.2

2

1.0626

0.8697

0.01

0.2

5

1.1563

0.7009

0.01

0.2

50

2.2942

0.0154

0.01

0.2

100

3.0579

0.0001

Figure 8.

Effect of Prandtl number Pr on temperature profile.

Figure 9.

5.

The effect of Prandtl number Pr on the heat transfer coefficients for fixed values of Re = 0.2 and S = 0.01 is also presented in Table 1. It can be seen that θ 0 (0) increases where as θ 0 (1) decreases with an increase in the Prandtl number. 396

Heat transfer rate on the walls for different values of Pr&S.

Conclusions

In this study, the effects of MHD and injection on the three-dimensional flow of an incompressible fluid through a long vertical channel embedded in a porous medium whose is presented. Using the similarity transformations, the governing equations are cast into dimensionless form where numerical solution has been presented for a wide range of parameters. It is observed that as the

K.Govardhan, K. Kaladhar, G. Nagaraju, B. Balaswamy

permeability S increases the magnitude of the normal velocity reduces up to the half of the distance between the plates from the impermeable wall and then onwards reverse effect and the axial and tangential velocities increases with the higher permeability. It is observed that as the magnetic parameter M increases the magnitude of the normal velocity reduces up to the half of the distance between the plates from the impermeable wall and then reverse trend is observed as the fluid particles go nearer to the permeable plate. The flow and axial velocities increases as an increase in the magnetic parameter. The temperature of the fluid increases significantly with the increase of Prandtl number. As Prandtl effect increases, the heat transfer on the impermeable plate is considerably higher than that of the porous plate. The heat transfer coefficients are decreases with the increase of S. The nondimensional heat transfer coefficient at η = 0 increase with the increase of Re and reverse trend is observed at η = 1 and θ 0 (0) increases where as θ 0 (1) decreases with an increase in the Prandtl number.

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[10] Rashidi M. M. and Hayat T. and Erfani E. and Mohimanian Pour S. A. and Awatif Hendi A., Simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk, Communications in Nonlinear Science and Numerical Simulation, 16(11), 2011, 4303-4317 [11] Turkyilmazoglu M. and Senel P., Heat and mass transfer of the flow due to a rotating rough and porous disk, Int. J. of Thermal Sciences, 63, 2013, 146-15 [12] Srinivasacharya D. and Upendar M., Soret and dufour effects on MHD mixed convection heat and mass transfer in a micropolar fluid, Cent. Eur. J. Eng., 3(4), 2013, 679-689 [13] Wang C. Y. and Skalak F. J., Fluid injection through one side a long vertical channel, AICHE Journal, 20(3), 1974, 603-605 [14] Huang C. L., Application of quasilinearization technique to the vertical channel flow and heat convection, Int. J. Non-Linear Mech., 13, 1978, 5560 [15] Banis S. and Turkey I., Injection of a non-Newtonian fluid through one side of a long vertical channel, Acta Mechanica., 151, 2001, 163-170 [16] Guria M., Ghosh S. K. and Pop I., Three Dimensional Free convection flow in a vertical channel filled with a porous medium, Journal of Porous Media, 12, 2009, 985-995 [17] Attia H. A., Steady flow over a Rotating disk in Porous medium with heat transfer, Nonlinear Analysis: Modeling and Control, 14(1), 2009, 21-26 [18] Das S. S., Sataparthy A., Das J. K. and Panda I. P., Mixed Convection heat and Mass transfer in three dimensional flow of a Viscous incompressible fluid past a vertical porous medium with periodic permeability, Int J of Applied Engineering Research, 3(8), 2008, 1105-1120 [19] Rashidi M. M., Mohimanian Pour S. A., Hayat T. and Obaidat S., Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method, Computers & Fluids, 54, 2012, 1-9 [20] Rashidi M. M., Hayat T., Keimanesh T. and Yousefian H., A study on heat transfer in a second- grade fluid through a porous medium with the modified differential transform method, Heat Transfer-Asian Research, 42(1), 2013, 31-45 [21] Rama G. Subbareddy and Ibrahim S., Free convection on vertical stretching surface with suction and blowing, Applied Science Research, 52(3), 1994, 247257

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