influence of thermal radiation on unsteady free

Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 2 (2015) © Research India Publications : http://www.ripublication.com

INFLUENCE OF THERMAL RADIATION ON UNSTEADY FREE CONVECTION FLOW OF WATER NEAR 4OC PAST A MOVING VERTICAL PLATE R. Srinivasa Raju1, M. Anil Kumar, K. Sarada2 and Y. Dharmendar Reddy* 1 Department of Mathematics, GITAM University, Hyderabad Campus, Rudraram, 502329, Telangana State, India. Department of Mathematics, Anurag Group of Institutions (formerly as C.V.S.R. College of Engineering), Ghatkesar, Ranga Reddy District, Telangana State, India. 2 Lecturer in Mathematics, Vivekananda Govt. Degree College, Vidyanagar, Hyderabad, 500010, Andhra Pradesh, India

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Corresponding Email address: [email protected]

vertical porous plate in presence of Soret and Dufour using finite element method was studied by Srinivasa Raju et al [6]. The effects of thermal radiation on non-darcy mixed convection flow in the presence of magnetic field was studied by Srinivasacharya et al [7]. Srinivasacharya and Swamy Reddy [8] studied the effects of thermal radiation on natural convection in porous medium saturated with power-law fluid in the presence of chemical reaction. The effect of chemical reaction on mixed convection in magnetohydrodynamic micropolar fluid flow in presence of thermal radiation was studied by Srinivasacharya and Upendar Mendu [9]. The purpose of this paper is to study is the effect of thermal radiation on an unsteady free convection flow of water near 4o C past a vertical moving porous plate. Finite element technique was used to solve dimensionalized non-linear coupled partial differential equations. Interesting findings are observed and were compared to the effects of thermal radiation reported by Raptis and Perdikis [3].

Abstract— The aim of this work is to study the influence of thermal radiation on unsteady free convection flow of water near 4oC (Pr = 11.40) past a moving vertical plate. The effect of the suction/injection parameter at the plate on the velocity is considered. The governing dimensionalized non-linear coupled partial differential equations are solved using Finite element technique. The numerical solutions of velocity and temperature of the fluid are obtained and discussed through graphs and the physical aspects of the problem are highlighted and discussed. Comparisons with previously published work on special cases of the problem are obtained and are observed to be in accord. Index Terms— Thermal radiation, Free convection, Water near 4o C, Finite element technique.

I. INTRODUCTION From a technological point of view, free convection flows over an infinite vertical plate has an important applications in fluid mechanics. It becomes a more attractive problem when the fluid is water at 4˚C which is electrically conductive, and the flow is focused to transverse and constant magnetic field. The steady mixed convective water flow at 4˚C over a vertical porous plate was studied by Ling et al [1]. Michalis Xenos et al [2] studied the effect of constant uniform suction on unsteady MHD free convection flow of water at 4˚C past an infinitely vertical moving plate with constant velocity. The unsteady free convection flow of water at 4˚C in the laminar boundary layer over a vertical moving plate embedded in a porous medium was investigated by Raptis and Perdikis [3]. In this study, fourth-order Runge - Kutta scheme was used for solving momentum equations numerically. Siva Reddy Sheri and Srinivasa Raju [4] demonstrated the effect of thermal diffusion (Soret) on unsteady hydrodynamic free convection flow past a semi-infinite vertical plate in the presence viscous dissipation. Sivaiah and Srinivasa Raju [5] studied the effects of Hall current and heat source on unsteady magneto hydrodynamic free convective flow in presence of viscous dissipation, heat and mass transfer using finite element method. The effect of thermal radiation on free convective flows have very important applications in space technology and design of pertinent equipments. Recent advances in nuclear power plants, gas cooled nuclear reactors, space vehicles, gas turbines and hypersonic flights have involved in this research field. The effects of thermal radiation and Heat source on magnetohydrodynamic free convective flow over an infinite

II. BASIC EQUATIONS Let us consider a two dimensional laminar free convection flow of water near 4o C past a moving vertical porous plate. The x and y axes are along and normal to the plate respectively. The flow governing equations of the problem are: Continuity Equation:

v 0 y

(1)

Equation of Momentum:

u u  2u  v   2  g (T   T )2 t y y (2) Equation of Energy: T  T     2T  qr   v =  t  y C p  y2 y  The corresponding boundary conditions of the flow are:

t   0 : u  0, T   T

for all y

  u  ct , v  vw t , T   Tw at y  0 t  0 :  u  0, T   T as y    237

(3)

(4)


Where C p , Specific heat at constant pressure; Gr , Grashof

III. NUMERICAL SOLUTION BY FEM

number for heat transfer; g , Gravity due to Acceleration; qr , Radiative heat flux;  , Thermal conductivity; y , Distance;

Finite element technique is used to solving the nondimensional momentum and energy equations (10) and (11) along with the imposed boundary conditions (12). Eight nodes quadrilateral elements were used in the finite element technique. The unknown field variables were approximated either by linear shape functions, defined by four corner nodes, or by quadratic shape functions, defined by eight nodes. Mesh sensitivity analysis was analyzed to arrive at the proper mesh sizes.

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Pr , Prandtl number; t , time; a , Stefan Boltzmann constant; T  , Temperature of the fluid at the plate; Tw , Wall temperature; T , At infinity the temperature of the fluid; vw , Velocity of injection at the plate; u , Velocity of the fluid in the x  direction; v , Component of the velocity normal to the plate; u  , Component of the velocity along the plate;  , Thermal Conductivity;  , Density of the fluid; x, y , Spatial coordinates along and normal the plate respectively; y ,  , Dimensionless temperature; R , Radiation parameter;  , the coefficient of thermal expansion;  , Velocity of the plate; From equation (1), let we take

IV. RESULTS AND DISCUSSIONS 0.4

0.3 0.2

1 2

v (5) v  vw t   a   t  Where the constant a characterizes the phenomenon of suction at the plate when a  0 and injection when a  0 . The local radiant for the case of an optically thin gray gas is articulated by q (6)  r  4a* T4  T 4 y It is assumed that the differences of temperature are 4 sufficiently small such that T  may be uttered as a linear 4 function of the temperature. This is complete by rising T  in a Taylor series about T and ignoring higher - order terms, thus (7) T 4  4T3T   3T4 By using Equations (6) and (7), (3) trim down to  T  T     2T  (8)  v =  16a*T3 T  T   2 t  y C p  y 



0.1 0 0

y

2

3

parameter

1

θ 0.5

a = – 0.8, – 0.4, 0.0, 0.4, 0.8

0 0

1

y

2

3

Figure 2. Temperature profiles for different values of

Introducing the following dimensionless quantities

into equations (2) and (8), we get u a u  2u  1  Gr 2  t y y 2 t2  a  1  2 R  1    2 t  y Pr  y Pr 2 t The boundary conditions of the problem are t  0 : u  0,   0 for all y   u  t ,   1 at y  0   t  0 :  u  0 ,   0 as y    

1

Figure 1. Velocity profiles for different values of Suction/Injection



1 1   c 2  3 T   T  c  3 u  t  t  2  , y  y  2  , u  ,   , 1 Tw  T      c3  2 2 3 2 3 T   T   vg , Pr  vCp , R  16a * T     Gr  w 3  c        

a = – 0.8, – 0.4, 0.0, 0.4, 0.8

u

Suction/Injection parameter

The governing equations (10) and (11) subjected to the boundary conditions (12) are solved mathematically for numerical results with the following ranges of the main parameters: Gr  1, Pr  11.40 (This value of Pr

(9)

corresponds to water at 4 C ), a  -0.8, -0.4, 0.0, 0.4, 0.8 and R  0.0, 0.2, 0.5, 0.8, 1.0. The step size y  0.0005 is used to obtain the numerical solution and the boundary condition y   is approximated by ymax  3, which is properly large for the velocity to approach the relevant free stream velocity with a five-decimal accuracy as the criterion for convergence. Figures 1 and 2 represent the decrease in fluid velocity and fluid temperature when the suction parameter a is increased i.e. more fluid withdrawn through the plate. Both, boundary layer and thermal boundary layer thicknesses diminishes with the increase in suction parameter a . The influence of the thermal radiation parameter R on the o

(10) (11)

(12)

238


velocity is shown in the figure 3. The thermal radiation parameter R defines the relative contribution of conduction heat transfer to thermal radiation transfer. Thus the rise in the radiation parameter results in decreasing velocity within the boundary layer. 0.4 0.3

of the current numerical method is verified by comparing the results, with those reported earlier by Raptis and Perdikis [3] when R  0 . The results of these comparisons are shown in figure 6. It can be seen from this figure that the results are excellent in accord. 0.4

a = – 0.8, – 0.4, 0.0, 0.4, 0.8

0.3

R = 0.0, 0.2, 0.5, 0.8, 1.0

u

0.2

u

0.2

0.1

0.1

0

0 0

1

y

2

Present results Results of Raptis and Perdikis [3]

0

3

1

y

2

3

Figure 6. Comparison of present numerical results with the results of Raptis and Perdikis [3] when R  0 .

Figure 3. Velocity profiles for different values of thermal radiation parameter

V. CONCLUSIONS

1

θ

In this present paper, the effect of thermal radiation on unsteady free convection flow of water near 4o C past a moving vertical plate. The effect of the suction/injection is taken into account. Numerical solutions are carried out for momentum and energy equations. The effects of important parameters dominating the velocity and temperature profiles are depicted and discussed briefly. When wall suction a  0 is measured both the velocity and temperature fields are reduced. However, it was observed that the behaviour was in the reverse direction if wall injection a  0 is considered. The thermal radiation parameter is defined the relative contribution of heat transfer conduction to thermal radiation transfer. It is observed that an increase in the radiation parameter causes enhanced in escalating velocity and temperature within the boundary layer. As in the limit R  0 , our results coincides with the results of Raptis and Perdikis [3].

R = 0.0, 0.2, 0.5, 0.8, 1.0

0.5

0 0

1

y

2

3

Figure 4. Temperature profiles for different values of thermal radiation parameter

0.4

0.3

u

t = 0.1, 0.2, 0.3, 0.4

0.2

REFERENCES [1]

0.1 0 0

1

y

2

3 [2]

Figure 5. Velocity profiles for different values of time t

The temperature profiles for different values of the thermal radiation parameter are shown in Figure 4. It is observed that the temperature profiles are reducing as R increases. This shows that the buoyancy effect on the temperature distribution o is very significant in water at 4 C ( Pr  11.40). It is known that the radiation parameter and Prandtl number plays an important role in flow phenomena because, it is a measure of the relative magnitude of viscous boundary layer thickness to the thickness of the thermal boundary layer. Figure 5 depicts the velocity profiles for different dimensionless times t = 0.1, 0.2, 0.3, 0.4. In this case, the velocity profiles increases away from the plate, as time increases. In addition, the correctness

[3]

[4]

239

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