COMBINED EFFECTS OF RADIATION AND HEAT GENERATION ON MHD NATURAL CONVECTION FLOW ALONG A VERTICAL FLAT PLATE IN PRESENCE OF HEAT CONDUCTION

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1 BRAC University Journal, vol.vi, no., 9, pp 11- COMBINED EFFECTS OF RADIATION AND HEAT GENERATION ON MHD NATURAL CONVECTION FLOW ALONG A VERTICAL FLAT PLATE IN PRESENCE OF HEAT CONDUCTION Mohammad Mokaddes Ali Faculty o Lie Science Malana Bhashani Science and Technology University, Tangail-19, Bangladesh mmokaddesali@yahoo.com and Rosanara Akhter Department o Mathematics Bangladesh University o Engineering and Technology, Dhaka-1, Bangladesh ABSTRACT Study o the eects o radiation and heat generation on MHD natural convection lo o an incompressible viscous electrically conducting luid along a vertically placed lat plate in presence o heat conduction is considered. The governing equations o the lo are transormed into dimensionless orm ith appropriate transormations and then solved using the implicit inite dierence method ith Keller-Bo scheme. The resulting numerical solutions o transormed governing equations are presented graphically in terms o velocity proile, temperature distribution, local shear stress, local heat transer rate and surace temperature and the eects o magnetic parameter (M), radiation parameter (R), Prandtl number (Pr) and heat generation parameter (Q) on the lo and the graphs are discussed. Key ords: Radiation, heat generation, MHD, inite dierence method and vertical lat plate. Nomenclature b Plate thickness C Local skin riction coeicient Speciic heat at constant pressure T b T Temperature at outside surace o the plate Temperature o the luid Average temperature o porous plate C p T Dimensionless stream unction T Temperature o the ambient luid g Acceleration due to gravity u, v Velocity components G Grash o number u, v Dimensionless velocity components r h Dimensionless temperature, y Cartesian co-ordinates H Strength o magnetic ield, y k, k s Fluid and solid thermal conductivities β Coeicient o thermal epansion l Length o the plate Dimensionless similarity variable M Magnetic parameter θ Dimensionless temperature N Local Nusselt number μ Viscosity o the luid u p Conjugate conduction parameter ν Kinematic viscosity Pr Prandtl number ρ Density o the luid q Heat lu σ Electrical conductivity Q Heat generation parameter τ Shearing stress R Radiation parameter ψ Stream unction Dimensionless Cartesian co-ordinate

2 BRAC University Journal, vol.vi, no., 9, pp 11- I. INTRODUCTION Radiative heat transer on natural convection lo o an incompressible, viscous and electrically conducting luid in presence o transverse magnetic ield plays important roles in nuclear poer plants, cooling o transmission lines and electric transormer etc. Also, rom technological point o vie, MHD natural convection lo has signiicant applications in the ield o stellar and planetary magnetospheres, aeronautics, chemical engineering and electronics. Considering o its importance, these lo have been studied several research groups [1-3]. As the engineering processes closely related ith temperature, accordingly radiation heat transer has signiicant inluence on engineering. Due to its ide applications in space technology such as space lights, aerodynamic rockets, propulsion systems, plasma physics, spacecrat reentry aerodynamics and many researchers studied the eect o radiation on MHD ree convection lo. Takhar and Soundalgekar [4] studied the eect o radiation on MHD ree convection lo o a gas past a sami-ininite vertical plate using the Cogley-vincenti-Giles equilibrium model. The problem o natural convection-radiation interaction on boundary layer lo ith Rosscland diusion approimation along a vertical thin cylinder has been investigated by Hossain and Alim [5]. Radiation eect on ree convection lo o luid rom a porous vertical plate as studied by Hossain et al. [6]. Thermal radiation and buoyancy eects on MHD ree convection heat generating lo over an accelerating permeable surace ith temperature- dependent viscosity studied by Seddeek [7]. Abdel-naby et al. [8] studied the radiation eects on MHD unsteady ree convection lo over a vertical plate ith variable surace temperature. The study o heat generation in moving luids is important in problems dealing ith the chemical reactions and those concerned ith dissociating luids such as heat generation are resistance heating in ires, eothermic chemical reactions in a solid and nuclear reactions in nuclear uel rods here electrical, chemical and nuclear energies are converted to heat. Eperimental and theoretical orks on heat generation eect have been done etensively [9-13]. But, the eect o radiation and heat generation under the process o steady natural convection lo is studied less as ound in literature survey. Thereore, e consider the study is to the eect o radiation and heat generation on MHD natural convection lo o an incompressible, viscous and electrically conducting luid along a vertical lat plate under the inluence o transverse magnetic ield. The governing partial dierential equations are reduced to locally nonsimilar partial dierential orms by using appropriate transormations. The transormed boundary layer equations are solved numerically adopting implicit inite dierence method together ith Keller Bo Scheme technique [14, 15]. Here, the assumption is ocused on the evaluation o the surace shear stress in terms o local skin riction and the rate o heat transer in terms o local Nusselt number, velocity proiles and temperature distribution or some selected values o parameters consisting magnetic parameter M, radiation parameter R, Prandtl number Pr and heat generation parameter Q. II. MATHEMATICAL FORMULATION We consider a steady, laminar, incompressible, viscous and electrically conducting luid along a vertical lat plate o length l and thicknessb. The plate temperature is grater than ambient T T b temperature maintained constant at the outer surace o the plate and uniorm magnetic ield o strength H is imposed along the y -ais. The lo coniguration and the coordinates system are shon in Fig. 1. H T b H Insulator b l T s b Interace (,) T u Fig. 1. Physical model and coordinate system. The governing equation o such lo under the Bousinesq approimation e can be epressed ithin the usual boundary layer as [8, 13]: v T g y

3 Combined eects o radiation and heat generation on MHD convection lo.. v + =, u σh u u + v = ν + gβ ( T T ), ρ T T k u + v = ρc here Γ = K λ = K K λ ( T p λ ) T eb λ T Q ( T T ) + ( T T ) 4Γ b ρcp dλ and is the mean absorption coeicient e bλ is Plank s unction and T is the temperature o the luid in the boundary layer. Where kinematics viscosityν, Thermal epression co-eicient β, Electrical conductivityσ, C p is the speciic heat due to constant pressure. The boundary conditions are: u = v =, T ( ) = T,, T k (4) s = ( T Tb ) at y =, > y b k u, T T > at y, We observe that the equations (1) to (3) together ith the boundary conditions (4) are nonlinear partial dierential equations. No e introduce the olloing dimensionless dependent and independent variables: =, y = l v l v = Gr ν 4Γl R = ν ( T T ) 3 g β l b Gr = ν σ H l M = Gr μ y l 1/ 4 Gr u l, u = Gr ν T T, θ = T T b,,, (5) here ν = ( μ ρ) is the kinematic viscosity, Gr is the Grasho number and θ is the non-dimensional (1) () (3) temperature. Then equations (1) to (3) can be ritten in dimensionless orm as: v (6) + =, u (7) u + v + Mu = +θ, θ θ 1 θ (8) u + v = R ( θ 1) + Qθ. Pr The corresponding boundary conditions are: θ u = v =, θ 1= p at y =, > (9) u, θ at y, > 1 / In the above, M = ( σ H l μ ) Gr is the magnetic parameter, R = ( 4Γl ν ) Gr is the radiation parameter, Pr = ( μ C p k ) is the Prandtl number and Q = ( Q l μ C p ) Gr is the heat generation parameter and 1/ p = k k b l Gr is a conjugate ( )( ) 4 s conduction parameter. The value o the conjugate conduction parameter p depends on ( b l), ( k k s ) and Gr but each o hich depends on the types o considered luid and the solid. Thereore in dierent cases p is dierent but not alays a small number. In the present analysis e have taken p=1.the stream unction and similarity variable and the dimensionless temperature are considered in the olloing orm to solve the equations (7) and (8) and or the boundary conditions described in equation (9): ψ = = y θ = 4 / 5 1/ 5 (1 + ) (, ), 1/ 5 (1 + ), 5 (1 + ) h(, ), (1) here ψ is the dimensionless stream unction hich is related to the velocity components such asu = ψ and ψ v = and h (, ) is a dimensionless temperature. By substituting equation (1) in equations (7) and (8) and the 13

4 Mohammad Mokaddes Ali and Roshanara Akhter boundary condition (9) e obtain the transormed equations: / 5 + M (1 + ) 1(1 + ) (11) 1/1 (1 + ) + h = ( ), / 5 1/1 h + h h R (1 + ) h Pr (1 + ) 5(1 + ).(1) 1/5 3/1 / 5 1/1 h + R (1 + ) + Q ( 1+ ) h= ( h ) and the boundary conditions are 1/ 4 (,) = (,) =, h (,) = (1 + ) 1/5 1/ + (1 + ) h(,) aty=. (13) (, ), h(, ) aty The set o equations (11) and (1) together ith the boundary condition (13) are solved numerically by applying implicit inite dierence method ith Keller-Bo (1978) Scheme. In practical point o vie, it is important to calculate the values o the rate o heat transer and the skin riction coeicient. This can be ritten in the dimensionless orm as ( 3 / 4 C ) = Gr l μν τ and 4 ( ( )) N = Gr k T T q u b, (14) T here τ = μ and q = k are the y = y= shearing stress and the heat lu Thus the local skin riction co-eicient and rate o heat transer are: / 5 3 / (15) C = (1 + ) (,) 4 N (1 ) 1/ u = + h (,) (16) III. RESULTS AND DISCUSSION Here e discuss graphically the numerical results obtained rom equation (11) and (1) together the boundary condition (13) using the mentioned method and also observed that radiation and heat generation dose eects in the lo region. We ocus our attention on the eect o magnetic parameter, Prandtl number, radiation parameter and heat generation parameter on the velocity and temperature ield and also the surace shear stress in terms o local skin riction coeicients and heat transer rate in terms o Nusselt number ith in the lo region. Velocity Temperature. M= M= M= M= (a) M= M= M= M= (b) The numerical value o the surace temperatufig.: re (a) Variation o velocity and (b) variadistribution are obtained rom the relation tion o temperature against or varying o M ith Pr=.733, R=1 and Q=1. 1/ 5 5 θ (,) = (1 + ) h(,) (17) The numerical results o velocity and temperature The numerical results obtained or the velocity or dierent values o magnetic parameter M hile proiles and temperature distributions or various Pr =.733, R = 1 and Q = 1 are illustrated values o Prandtl number, magnetic parameter, in Fig. (a) and Fig. (b), respectively. Fig. (a) radiation parameter and heat generation parameter shos that the velocity proiles decreases ith the are discussed in the olloing sections. increase o magnetic parameter due to interaction o the applied magnetic ield and loing luid 14

5 Combined eects o radiation and heat generation on MHD convection lo.. particle, hich produce Lorentz orce that oppose the motion o the luid. Moreover, or each value o M, the velocity is zero at the boundary all and increase to the peak velocity as increases and then turn to decrease and inally approach to zero. Furthermore, e have seen that the velocity proiles meet together ater certain position o and cross the side. This is because, the gradient o decreasing o velocity increases ith the increasing o magnetic parameter. In Fig. (b), it can be seen that the temperature increases ithin the boundary layer or the increasing values o magnetic parameter M due to the interaction. Moreover, the temperature decreases monotonically ith increasing o or a particular value o M. The maimum values o the temperature are 63, 81, 949 and.933 or M =,, and 1., respectively. Each o hich occurs at the surace o the plate. The eect o radiation parameter on the velocity and the temperature distributions together ith a certain value o Pr, M and Q is presented in Fig. 3 (a) and Fig. 3 (b), respectively. Fluid absorbed heat hile radiation imitates rom the heated plate, as a result the motion and the temperature o the luid increases ith in the lo region. That s hy the velocity and the temperature increase ith the increasing o R. It means that the velocity boundary layer and the thermal boundary layer thickness increase or large value o R. From Fig. 3 (a) and also numerical values e have seen that the position o the peak velocity moves toard the boundary layer or the increasing R. Pr=.733 Pr=.9 Pr= 1. Pr= R= 1 R= 1 R= 3 R= 5 Velocity. Velocity (a) (a) 1. Pr=.733 Pr=.9 Pr= 1. Pr= R= 1 R= 1 R= 3 R= 5 Temperature Temperature (b) Fig.3: (a) Variation o velocity and (b) variation o temperature against or varying o R ith Pr=.733, M= and Q= (b) Fig.4: (a) Variation o velocity and (b) variation o temperature against or varying o Pr ith M=, R=1 and Q=1. The variations o velocity proile and temperature distribution or dierent values o Prandtl number Pr ith M =, R = 1 and Q = 1 shon in Fig. 4(a) and Fig. 4 (b), respectively. The increasing values o Pr increase viscosity o the luid. Viscosity increase means that the density o 15

6 Mohammad Mokaddes Ali and Roshanara Akhter the luid increase, hich results luid does not move reely. It can be seen that the velocity decreases gradually and the peak velocity moves toards the interace or the increasing Pr. Moreover, the velocity is zero at the all and increases to the peak as increases and inally approaches to zero. These are epected behavior because it supports the no-slip condition at the all and the luid motion outside the boundary layer. Fig. 4 (b) shos that the temperature distribution over the hole boundary layer decreases due to the increase o Pr. It agrees the physical act that the temperature at the solid luid interace is reduced because, temperature at the plate considered constant. As a result, the thermal boundary layer thickness as ell as velocity boundary layer decreases ith the increasing o Pr. temperature or some values o Q hereas Pr =.733, M = and R =1. From Fig. 5(a) e conclude that the velocity proiles increases slightly due to the increasing value o heat generation parameter. This is because; increased value o Q produces more heat in the solid hich increase the motion o the luid. We have also seen that near the surace o the plate the velocity increases to maimum ith increase o heat generation parameter Q then ater the peak position start to decrease and inally approaches to zero. On the other hand rom Fig. 5(b), e observed that the same result holds or temperature distributions ithin the boundary layer due to increasing o Q. 1.6 Q= 1 Q= 4 Q= 7 Q= Skin riction 1. Velocity (a) M= M= M= M= (a) M= M= M= M=1. 1. Q= 1 Q= 4 Q= 7 Q= Heat transer. Temperature (b) Fig.5: (a) Variation o velocity and (b) variation o temperature against or varying o Q ith Pr=.733, M= and R=1. Fig. 5 (a) and Fig. 5(b), respectively depict the numerical results o the velocity and the (b) Fig.6:(a) variation o skin riction and (b)variation o heat transer against or varying o M ith Pr=.733, R=1 and Q=1. Fig. 6(a) and Fig. 6(b), reveal that the skin riction coeicient and the rate o heat transer or some values o M ith Pr =.733, R = 1and Q =1.The velocity decreases as shon in the Fig. (a), due to the increasing M. Accordingly, the skin riction on the plate decreases as observed in Fig. 6(a). But the temperature ithin the boundary layer 16

7 Combined eects o radiation and heat generation on MHD convection lo.. increases (Fig. (b)) or the increasing M. As a result, the heat transer rate rom the plate to luid decreases as shon in igure 6(b) Skin riction.9 Skin riction.9 R=1 R=1 R=3 R= (a) (a) Pr=.733 Pr=.9 Pr=1. Pr=1.446 Pr=.733 Pr=.9 Pr=1. Pr=1.446 R=1 R=1 R=3 R=5 Heat transer. Heat transer (b) Fig.7:(a) variation o skin riction and (b)var -iation o heat transer against or varying o R ith Pr=.733, M= and Q=1. The variation o the local skin riction coeicient C and local rate o heat transer N u or dierent values o R associated ith Pr =.733, M = and Q = 1 are illustrated in Fig. 7(a) and Fig. 7(b). Increased value o R accelerate the luid motion as mentioned in Fig. 3(a) and increases the shear stress at the all, or hich local skin riction increase ith the increasing o R. This phenomenon demonstrated in Fig. 7(a). Similarly increased value o the radiation parameter increases the temperature (Fig. 3(b)) hich ater decrease the rate o heat transers along the -direction (b) Fig.8:(a) variation o skin riction and (b)var -iation o heat transer against or varying o Pr ith M=, R=1 and Q=1. The eects o Prandtl number on the skin riction C and heat transer rate N u ith the increasing o aial distance or the ied value o magnetic parameter, radiation parameter and heat generation parameter are shon in Fig. 8 (a) and Fig. 8 (b), respectively. The values o Pr are proportional to the viscosity o the luid. So or the increase values o Pr the skin riction decreases on the plate hich is shon in Fig. 8(a). For a particular value o Pr the local skin riction coeicient increases monotonically due to the increasing o. From Fig. 8(b), it is observed that heat transer rate increases due to increase o Pr. Furthermore or a particular value o Pr the local heat transer rate decreases monotonically due to the increasing o. 17

8 Mohammad Mokaddes Ali and Roshanara Akhter Skin riction 1. Q=1 Q=4 Q=7 Q= (a) Surace temperature.9 M= M= M= M= (a) Heat transer. - Q=1 Q=4 Q=7 Q= (b) Fig.9:(a) variation o skin riction and (b)var -iation o heat transer against or varying o Q ith Pr=.733, M= and R=1. Fig. 9(a) and Fig. 9(b) illustrates the local skin riction coeicients and rate o heat transer or dierent values o Q against ith controlling parameter Prandtl number Pr =.733, radiation parameter R = 1 and magnetic parameter M =. It can be seen that an increase heat generation parameter Q increase the luid velocity ithin the boundary layer that shon in Fig.5 (a). So, the corresponding skin riction coeicients increase ith the increasing o Q. The opposite result is observed or heat transer distribution along ith the increasing o heat generation parameter. Surace temperature 1..9 R=1 R=1 R=3 R= (b) Fig.1:(a) Variation o surace temperature against or varying o M and (b) variation o surace temperature against or varying o R. The inluence o magnetic parameter M, radiation parameter R and heat generation parameter Q on surace temperature are depicted in Fig. 1(a), Fig. 1 (b) and 1(d) respectively. It is noted that the surace temperature increase due to the increase value o M, R and Q along the direction. This is to be epected because the thermal boundary layer thickness rises or the increasing o M, R and Q as observed in Fig. (b), Fig. 3(b) and 5(b), respectively. 18

9 Combined eects o radiation and heat generation on MHD convection lo.. Surace temperature Surace temperature 1..9 Pr=.733 Pr=.9 Pr=1. Pr= (c ) Q=1 Q=4 Q=7 Q= (d) Fig.1:(c) Variation o surace temperature against or varying o Pr and (d) variation o surace temperature against or varying o Q. The temperature ithin the boundary layer decreases or increasing Pr as illustrated in Fig. 4(b) hich results a decrease o interacial temperature as observed in Fig.1 (c). It can be seen that the interacial temperature increases monotonically or a selected value o Pr ith the increasing o along the upard direction. IV. CONCLUSION In this analysis the eect o radiation and heat generation on magnetohydrodynamic (MHD) natural convection lo along a vertical lat plate in presence o heat conduction has been investigated or some selected values o pertinent parameters including magnetic parameter, radiation parameter, Prandtl number and heat generation parameter. From the present investigation, it may be concluded that the velocity o the luid and the skin riction at the interace decrease ith the increasing magnetic parameter and Prandtl number hile they increase ith the increasing o radiation parameter and heat generation parameter.the temperature o the luid and also the surace temperature increases ith the increasing magnetic parameter, radiation parameter and heat generation parameter but decrease or increasing Prandtl number. Moreover, the rate o heat transer decreases ith the increasing o magnetic parameter, radiation parameter and heat generation parameter but increases or increasing o Prandtl number. REFERENCES [1] E. M. Sparro, R. D. Cess: Eect o magnetic ield on ree convection heat transer, Int. J. Heat and Mass Transer, 3, pp (1961) [] K. R. Sing, T. G. Coling: Thermal conduction in magnetohydrodynamics, J. Mech. Appl. Math., 16, pp.1-5. (1963) [3] N. Riley: Magnetohydrodynamic ree convection, J. Fluid Mech.18, pp ( 1964) [4] H. S. Takhar, V.M. Soundalgekar: Radiation eects on MHD ree convection lo o a gas past a simi- ininite vertical plate, Applied Scientiic Research, 36, pp (198) [5] M. A. Hossain, M.A. Alim: Natural convection-radiation interaction on boundary layer lo along a thin vertical cylinder, J. o Heat and Mass Transer 3, pp (1997) [6] M. A. Hossain, M.A. Alim, D.A.S. Ress: The eect o radiation on ree convection rom a porous vertical plate, Int. J. o Heat and Mass Transer, 4, pp (1999) [7] M. A. Seddee: Thermal radiation buoyancy eects on MHD ree convective heat generating lo over an accelerating permeable surace ith temperature dependent viscosity, Canadian Journal o Phys., 79, pp (1) [8] Abd El-Naby, Elsayed M. E.Elbarbaryand Nader Y. Abdelazem: Finite dierence solution o radiation eect on MHD 19

10 Mohammad Mokaddes Ali and Roshanara Akhter unsteady ree convection lo over a vertical plate variable surace temperature, Journal o Applied Mathematics,, pp (3) [9] M. M. Molla, M. A. Taher, M. M. K. Chodhury, M. A Hossain: Magnetohydrodynamic natural convection lo on a sphere in presence o heat generation eect, J. o nonlinear analysis modeling and control, 1, pp (5) [1] M. M. Molla, M. A. Taher, M. A. Hossain: Magnetohydrodynamic natural convection lo on a sphere ith uniorm heat lu in presence o heat generation, Acta Mech.186, pp (6) [11] M. M. Alam, M. A. Aim, M. M.K. Chodhury: Viscous dissipation on MHD natural convection lo over a sphere in presence o heat generation, J. o Nonlinear Analysis, Modelling and control,1, pp (7) [1] A. A. Mamun, Z. R. Chodhury, M.A.Azim, M.A. Maleque: Conjugate heat transer or a vertical lat plate ith heat generation eect, J. o nonlinear analysis modeling and control,13,pp-1-11(8) [13] A. A. Mamun, Z. R. Chodhury, M. A. Azim, M. M. Molla: MHD-conjugate heat transer analysis or a vertical lat plate in presence o viscous dissipation and heat generation, Int. communications in heat and mass transer,35,pp (8) [14] H. B. Keller: Numerical methods in the boundary layer theory, Annu. Rev. Fluid Mech., 1, pp (1978) [15] T. Cebeci, P. Bradsho: Physical computational aspects o convective heat transer, Springer, Ne York. (1984)