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International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 Radiation and Thermo - Diffusion Effects on Mixed Convective Heat and Mass Transfer Flow af a Viscous Dissipated Fluid over a Vertical Surface in the Presence of Chemical Reaction with Heat Source D Chenna Kesavaiah1, P V Satyanarayana2, S Venkataramana3 1 Department of H & BS, Visvesvaraya College of Engineering & Technology, Greater Hyderabad, A.P, India chennakesavaiah@gmail.com 2 Fluid Dynamics Division, School of Advanced Science, VIT University, Vellore - 632 014, T N, India 3 Department of Mathematics, S V University, Tirupati - 517 502, A.P, India ABSTRACT very light molecular weight H 2 , He and the medium The present work analyzes the influence of chemical molecular weight N2 air) the diffusion – thermo reaction on MHD mixed convection heat and mass effects was found to be of a magnitude just it cannot be transfer for a viscous fluid past an infinite vertical neglected. plate embedded in a porous medium with radiation and heat generation. The dimensionless governing The heat and mass transfer simultaneously affecting each equations for this investigation are solved analytically other that will cause the cross diffusion effect. The heat using two - term harmonic and non-harmonic transfer caused by concentration gradient is called the functions. The effects of various parameters on the diffusion-thermo or Dufour effect. On the other hand, velocity, temperature and concentration fields as well mass transfer caused by temperature gradients is called as the skin-friction coefficient, Nusselt number are Soret or thermal diffusion effect. Thus Soret effect is presented graphically and Sherwood number is referred to species differentiation developing in an initial presented numerically. homogenous mixture submitted to a thermal gradient and the Dufour effect referred to the heat flux produced Keywords: Radiation, mass transfer, chemical reaction, by a concentration gradient. The Soret effect, for heat source and concentration instance has been utilized for isotope separation, and in mixture between gases with very light molecular weight H2 , He and of medium molecular weight N2 , air I. INTRODUCTION The phenomenon of heat and mass transfer has been the Soret effect [thermal-diffusion] refers to mass flux object of extensive research due to its applications in produced by a temperature gradient. These effects are Science and Technology. Such phenomena are observed neglected on the basis that they are of a smaller order of in buoyancy induced motions in the atmosphere, in magnitude than the effects described by Fourier’s and bodies of water, quasisolid bodies such as earth and so Fick’s laws. In view of the importance of this diffusion – on. This assumption is true when the concentration level thermo effect. Similarity equations of the momentum is very low. Therefore, so ever, exceptions the thermal energy and concentration equations are derived by diffusion effects for instance, has been utilized for introducing a time dependent length scale. Malsetty et. al isotropic separation and in mixtures between gases with [19] have studied the effect of both the Soret coefficient and Dufour coefficient on the double diffusive IJSET@2013 Page 56 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 convective with compensating horizontal thermal and spacecraft re-entry aerothermodynamics which operate solutal gradients. Saritha and Satya Narayana [25] at higher temperatures, radiation effects can be thermal diffusion and chemical reaction effects on significant. Alagoa et al. [3] studied radiative and free unsteady MHD free convection flow past a semi infinite convection effects on MHD flow through porous vertical permeable moving plate. Mohamed [21] Double medium between infinite parallel plates with time- - Diffusive convection - radiation Interaction on dependent suction. Bestman and Adjepong [4] analyzed Unsteady MHD flow over a vertical moving porous plate unsteady hydromagnetic free convection flow with with heat generation and Soret effects. Bhupendra radiative heat transfer in a rotating fluid. Olanrewaju Kumar et.al [5] Three-Dimensional mixed convection et.al [23] further results on the Effects of Variable flow past an infinite vertical plate with constant surface Viscosity and Magnetic Field on Flow and Heat Transfer heat flux. Ahmed and Kalita [2] Soret and magnetic field to a Continuous Flat Plate in the Presence of Heat effects on a transient free convection flow through a Generation and Radiation with a Convective Boundary porous medium bounded by a uniformly moving infinite Condition. vertical porous plate in presence of heat source. Gireesh Kumar and Satyanarayana [15] Mass transfer effects on In all these investigations, the viscous dissipation is MHD unsteady free convective Walter's memory flow neglected. The viscous dissipation heat in the natural with constant suction and heat sink. convective flow is important, when the flow field is of extreme size or at low temperature or in high Hydromagnetic flows and heat transfer have become gravitational field. Gebhart [14] shown the importance more important in recent years because of its varied of viscous dissipative heat in free convection flow in the applications in agriculture, engineering and petroleum case of isothermal and constant heat flux at the plate. industries. Raptis [24] studied mathematically the case Israel Cookey et al. [17] investigated the influence of of time varying two-dimensional natural convective flow viscous dissipation and radiation on unsteady MHD free of an incompressible, electrically conducting fluid along convection flow past an infinite heated vertical plate in a an infinite vertical porous plate embedded in a porous porous medium with time dependent suction. Sreekantha medium. Soundalgekar [28] obtained approximate Reddy et.al [29] Effects of chemical reaction, radiation solutions for two-dimensional flow of an incompressible and thermo-diffusion on convective heat and mass viscous flow past an infinite porous plate with constant transfer flow of a viscous dissipated fluid in a vertical suction velocity, the difference between the temperature channel with constant heat flux. Md. Abdul Sattar and of the plate and the free stream is moderately large Md. Alam [20] Thermal diffusion as well as causing free convection currents. Takhar and Ram [32] transportation effect on MHD free convection and Mass studied the MHD free convection heat transfer of water Transfer flow past an accelerated vertical porous plate. at 4 C through a porous medium. Elbashbeshy [12] Satyanarayana et.al [26] viscous dissipation and thermal studied heat and mass transfer along a vertical plate radiation effects on an unsteady MHD convection flow under the combined buoyancy effects of thermal and past a semi-infinite vertical permeable moving porous species diffusion, in the presence of magnetic field. plate Sudheer Babu et. al [31] Effects of thermal radiation and chemical reaction on MHD convective In all these investigations, the radiation effects are flow of a polar fluid past a moving vertical plate with neglected. For some industrial applications such as glass viscous dissipation. production and furnace design and in space technology applications, such as cosmical sight aerodynamics The effects of radiation on MHD flow and heat transfer rocket, propulsion systems, plasma physics and problem have become more important industrially. IJSET@2013 Page 57 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 Many processes in engineering areas occur at high investigated the effect of the first order homogeneous temperature, and knowledge of radiation heat transfer chemical reaction on the process of an unsteady flow becomes very important for the design of the pertinent past a vertical plate with a constant heat and mass equipment. Nuclear power plants, Gas turbines and transfer. Chamkha [7] studied the MHD flow of a various propulsion devices, for aircraft, missiles, numerical of uniformly stretched vertical permeable satellites and space vehicles are examples of such surface in the presence of heat generation/ absorption engineering areas. Shercliff [27] and Ferraro and and a chemical reaction. Muthucumaraswamy [22] Plumpton [13]. Hossian and Rees [16] examined the presented heat and mass transfer effects on a effects of combined buoyancy forces from thermal and continuously moving isothermal vertical surface with mass diffusion by natural convection flow from a uniform suction by taking into account the homogeneous vertical wavy surface. Combined heat and mass transfer chemical reaction of first order. in MHD free convection from a vertical surface has been Kesavaiah Ch et.al [18] Effects of the chemical reaction studied by Chein-Hsin-Chen [8]. Further, the effect of and radiation absorption on an unsteady MHD Hall current on the fluid flow with variable convective heat and mass transfer flow past a semi- concentration has many applications in MHD power infinite vertical permeable moving plate embedded in a generation, in several astrophysical and meteorological porous medium with heat source and suction studies as well as in plasma flow through MHD power Keeping the above application in view we made attempt generators. From the point of application, model studies in this paper to study the present work analyzes the on the Hall Effect on free and forced convection flows influence of a first - order homogeneous chemical have been made by several investigators. Datta and Jana reaction on MHD mixed convection heat and mass [10], Acharya et al.[1] and Biswal and Sahoo [6] have transfer for a viscous fluid past an infinite vertical plate studied the Hall effect on the MHD free and forced embedded in a porous medium with radiation and heat convection heat and mass transfer over a vertical surface. generation. The dimensionless governing equations for Stanford Shateyi et.al [30] The effects of thermal this investigation are solved analytically using two - radiation, hall currents, Soret and Dufour on MHD flow term harmonic and non-harmonic functions by mixed convection over a vertical surface in porous media. II. FORMULATION OF THE PROBLEM The growing need for chemical reactions in chemical We consider the mixed convection flow of an and hydrometallurgical industries requires the study of incompressible, viscous, electrically conducting viscous heat and mass transfer with chemical reaction. There are fluid embedded in a uniform porous medium in the many transport processes that are governed by the presence of thermal diffusion, chemical reaction, combined action of buoyancy forces due to both thermal radiation, thermal and concentration buoyancy effects and mass diffusion in the presence of the chemical such that x -axis is taken along the plate in upwards reaction effect. These processes are observed in nuclear direction and y -axis is normal to it. A transverse reactor safety and combustion systems, solar collectors, constant magnetic field is applied i.e. in the direction of as well as metallurgical and chemical engineering. Their y - axis. Since the motion is two dimensional and other applications include solidification of binary alloys length of the plate is large therefore all the physical and crystal growth dispersion of dissolved materials or particulate water in flows, drying and dehydration variables are independent of x . Let u and v be the operations in chemical and food processing plants, and components of velocity in x and y directions, combustion of atomized liquid fuels. Dekha et al. [11] respectively, taken along and perpendicular to the plate. IJSET@2013 Page 58 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 The governing equations of continuity, momentum and qr energy for a flow of an electrically conducting fluid 4(T T ) I (7) y along a hot, non-conducting porous vertical plate in the eb presence of concentration and radiation is given by where I K w 0 T d , K w is the absorption v coefficient at the wall and eb is Planck’s function, I 0 (1) y is absorption coefficient v v0 (Constant) (2) The boundary conditions are p 0 p is independent of y (3) u 0, T Tw , C C ; y 0 y (8) u* 0, T T , C C ; y u 2u v 2 g T T Introducing the following non-dimensional quantities y y (4) u v0 y T T u , y , T g * C C B02u u v0 Tw T K C C g Tw T T T 2 C , Gr C p v k 2 Q0 T T C w C 3 v0 y y (5) v0 2 4 I 2 Ec , R u Tw T q C p v02 r y y C p Q0 Kr Q , Kr 2 , Pr C 2C C p v0 Kr C C 2 v0 k v DM y y 2 DT Tw T (6) B0 2 2 T M , S0 (9) Cw C 2 DT 2 v0 2 y * g Cw C Here, g is the acceleration due to gravity, T the Gm 3 , Sc v0 D temperature of the fluid near the plate, T the free stream III. SOLUTION OF THE PROBLEM temperature, C concentration, the coefficient of In the equations (4), (5), (6) and (8), we get thermal expansion, k the thermal conductivity, P the pressure, C p the specific heat of constant pressure, B0 2u u 1 the magnetic field coefficient, viscosity of the fluid, M u GrT Gm C (10) y 2 y K q the radiative heat flux, the density, the magnetic 2T T permeability of fluid V0 constant suction velocity, the Pr F Q Pr T y 2 y kinematic viscosity and D molecular diffusitivity. 2 (11) u Pr Ec The radiative heat flux qr is given by equation (5) in y the sprit of Cogly et.al [9] 2C C 2T Sc ScKrC ScS0 2 (12) y 2 y y IJSET@2013 Page 59 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 where Gr is Grashof number, Gm is the mass Groshof u0 0, u1 0, T0 1 number, Pr is Prandtl number, M is magnetic y0 T1 0, C0 1, C1 0 parameter, R is Radiation parameter, Sc is Schmidt (21) u0 0, u1 0, T0 0 number, Q is heat source parameter, Ec is the Eckert y T1 0, C0 1, C1 0 number, M is magnetic parameter Kr is Chemical Solving equations (15) to (20) with the help of (21), we reaction parameter and S 0 is Soret number. get The corresponding boundary condition in dimensionless u y A1e m 2 y A2e m 4 y A3e A4e m6 y m2y form are reduced to u 0, T 1, C 1 y0 (13) Ec A5e m 8 y A6 e 2 m 6 y A7 e 2 m 2 y u 0, T 0, C 0 y A9 e 2 m2 y A10 e m2 m6 y A8e 2m 4 y The physical variables u, T and C can expand in the A11e m2 m4 y A12 e m2 m4 y power of Eckert number Ec (E). This can be possible A13e m2 m6 y A14 e m4 m6 y A15e 2 m 2 y physically as Ec for the flow of an incompressible fluid is always less than unity. It can be interpreted physically A16 e m 8 y A17 e m 8 y A18e 2 m 6 y as the due to the Joules dissipation is super imposed on A19 e 2 m 2 y A20 e 2 m 4 y A21e 2 m 2 y the main flow. A22 e m 2 m 6 y A23e m 2 m 4 y u y u0 y Ec u1 y O E 2 A25e m 2m 6 y A26e m 4m 6 y A24 e m 2m 4 y T y T y Ec T y O E (14) 2 0 1 A27 e 2 m 2 y A28e m 10 y A29e m 12 y u y C y Ec C y O E 0 1 2 Using equation (14) in equations (10)–(12) and equating the coefficient of like powers of E, we have y em y 2 Ec B e 1 2 m6 y B2e 2 m2 y B3e 2 m4 y B4 e 2 m2 y B5e m2 m6 y 1 u0 u0 M u0 Gr T0 Gm C0 (15) K B6 e m2 m4 y B7 e m2 m4 y B8e m2 m6 y T0 Pr T0 F Q Pr T0 0 (16) B9e m4 m6 y B10e 2 m2 y B11e m8 y C0 ScC0 KrScC0 ScS0T0 (17) C y D1e m2y D2e m4y Ec D3e m8 y 1 u1 u1 M u1 Gr T1 Gm C1 (18) K D4 e 2 m6 y D5e 2 m2 y D6e 2 m4 y D7 e 2 m2 y D8e m2 m6 y D9e m2 m4 y T1 Pr T1 F Q Pr T1 Pr u02 (19) D11e m2 m6 y D10 e m2 m4 y C1 ScC1 KrScC1 ScS0T1 (20) The corresponding boundary conditions are D12 e m4 m6 y D13e 2 m2 y D14 e m10 y Skin – friction: The skin-friction coefficient at the plate is given by IJSET@2013 Page 60 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 u Final results are computed for the main physical A1m2 A2 m4 A3m2 A4 m6 parameters which are presented by means of graphs. The y y 0 influence of the thermal Grashof number Gr , solutal Ec A5 m8 2 A6 m6 2 A7 m2 Grashof number Gm , the magnetic field parameter 2 A8 m4 2 A9 m2 m2 m6 A10 m2 m4 A11 m2 m4 A12 M , heat source parameter Q , thermal radiation m2 m6 A13 m4 m6 A14 parameter F , Prandtl number Pr , Porosity 2 A15 m2 A16 m8 A17 m8 Heat parameter K chemical reaction parameter Kr , 2 A18 m6 2 A19 m2 2 A20 m4 Soret number S0 and Schmidt number Sc on the 2 A21m2 m2 m6 A22 velocity, temperature and concentration profiles can be A23 m2 m4 m2 m4 A24 analyzed from Figures (1) - (26). The values of Gr & Gm are taken to be both positive as this value m2 m6 A25 m4 m6 A26 represent respectively cooling and the Eckert number 2 A27 m2 2 A28 m10 A29 m22 Ec is taken to be 0.001 . Transfer: Velocity Profiles The rate of heat transfer in terms of Nusselt number at the plate is given by The velocity profiles for different values of the Grashof T number Gr and the modified Grashof numbers Gc Nu m2 Ec 2 B1m6 2m2 B2 y y 0 are defined in Figures (1) and (2) respectively. It can be observed that an increase in Gr or Gc leads to the rise 2m4 B3 2m2 B4 m2 m6 B5 in the values of velocity, Here, the positive values of Gr m2 m4 B6 m2 m4 B7 correspond to a cooling of the surface by natural m2 m6 B8 m4 m6 B9 convection. In addition, the curves show that the peak value of the velocity increases rapidly near the wall of 2m2 B10 m8 B11 the porous plate as Gr or Gc increases, and then Sherwood number decays to the free stream velocity. The effect of thermal radiation parameter on the velocity field has been C illustrated in Figure (3). It is seen that as the thermal Sh m 2 D1 m 4 D2 Ec m8 D3 radiation parameter increases the velocity field y y 0 decreases. For various values of the permeability 2m6 D4 2m2 D5 2m4 D6 parameter K , the profiles of the velocity across the 2m2 D7 m2 m6 D8 boundary layer are shown in Figure (4). The velocity m2 m4 D9 m2 m4 D10 increases for increasing values of the permeability m2 m6 D11 m4 m6 D12 parameter K . Figure (5) illustrate the variation in velocity distributions across the boundary layer for 2m2 D13 m10 D14 various values of the chemical reaction parameter Kr . IV. RESULTS AND DISCUSSION It can be seen that the velocity decreases in the IJSET@2013 Page 61 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 destructive reaction Kr 0 . The MHD and the profiles for different values of the Prandtl number Pr concentration boundary layer become thin as the reaction is shown in Figure (12). The results show that an parameters. For different values of the magnetic field increase of the Prandtl number results in a decrease in parameter M , the velocity profile is plotted in Figure the thermal boundary layer thickness and a more uniform temperature distribution across the boundary (6). It is obvious that the effect of increasing values of layer. The reason is that the smaller values of Pr are the parameter M results in a decreasing velocity equivalent to increasing the thermal conductivities, and distribution across the boundary layer. Figure (7) present therefore heat is able to diffuse away from the heated the velocity profiles for different values of the Prandtl surface more rapidly than for the larger values of Pr . number Pr . The results show that the effect of Hence, the thicker the boundary layer is the shower the increasing values of Pr results in a decrease of the rate of heat transfer. At high Prandtl fluid has low velocity that physically is true because the increase in velocity, which in turn also implies that at lower fluid the Prandtl number is due to the increase in the velocity the specie diffusion is comparatively lower and viscosities of the fluid which makes the fluid thick and hence higher specie concentration is observed at high hence causes a decrease in the velocity of fluid. The Prandtl number. Figures (13) present the decreasing effects of internal heat generation parameter Q on the result of temperature when heat source parameter is increasing. Figures (14) - (16) shows the temperature velocity are displayed in Figure (8). It is clear that as the profiles for different values of Gm, Gr and Sc . It is parameter Q increases, the velocity profiles lead to a observed that an increasing values of Gm, Gr and fall. Figure (9) depicts the effect of Soret number Sr Sc the results decreases the thermal boundary layer on the fluid velocity and we observed an increase in the thickness across the plate. fluid velocity as Soret number Sr increases. This is Concentration Profiles because an increase in the volumetric rate of generation connotes increase in buoyancy force thereby increasing fluid velocity. For different values of the Schmidt The radiation F effects shows in figure (17). It is number Sc , the velocity profiles are plotted in Figure observed that and increasing values of F the concentration profiles increases. Figures (18) and (19) (10). It is obvious that the effect of increasing values of depict the effect of various values of the mass and Sc results in a decreasing velocity distribution. thermal Grashof numbers Gm, Gr on the Temperature profiles concentration boundary layer thickness. It is interesting to note that increase in this parameters bring a small Figure (11) show the effect of radiation parameter on the increase across the plate. The effect of permeability temperature profile. A rise in F causes a significant fall parameter K on the concentration profile is illustrated in the temperature values from the highest value at the in Figure (20). These results show that the with wall y 0 across the boundary layer to the free increasing permeability parameter concentration profiles decreases. Figure (21) show the effect of the chemical stream. Thus, greater value of F corresponds to smaller reaction parameter on concentration profiles radiation flux and the minimum temperature is observed. respectively. It is noticed that species concentration and Thermal radiation thereby reduces the rate of energy thermal boundary layer are decreasing, as the values of transport to the fluid. The variation of the temperature chemical reaction are increasing. Figure (22) shows the IJSET@2013 Page 62 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 effects of the magnetic parameter M . It is observed that chemical reaction and radiation. In the light of present the concentration decreases with increasing values of investigation, it is found that the concentration of the M . The influence of Prandtl number and heat source fluid increased during a generative chemical reaction and parameter are on the concentration field is seen in Figure decreased during a destructive chemical reaction. (23) and (24). It is noticed that the increase in the Prandtl Further, the momentum boundary layer thickness Sherwood number Sh number and decreases while thermal boundary layer thickness and heat source concentration profiles decreases with increasing Sc Sh F Sh parameter permeability parameter. Increasing Soret number 0.22 - 0.5 -0.8783 though reduce the thermal boundary layer thickness, while 0.4078 increases the reverse trend is seen in concentration profiles. 0.30 - 1.0 -0.8533 concentration of 0.4985 the fluid. The Table: (1) 0.38 - 1.5 -0.8302 influence of S 0 0.5842 on the 0.46 - 2.0 -0.8085 concentration of 0.6664 the fluid medium is seen in Figure (25). In general it is noted that increase in Soret number contributes to increase in concentration of the medium. Further the effect is found to be diminishing as we move away from the plate. Figure (26) reflects that with increase in Schmidt number Sc the fluid concentration decreases. Figure REFERENCES (27) and (28) shows the effects of Prandtl number and magnetic field on skin friction and Nusselt number versus thermal Grashof number. 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IJSET@2013 Page 65 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 2 4ScS0 m4 B3 APPENDIX D6 4m2 2Scm2 KrSc 2 2 4ScS0 m2 B4 Pr Pr 2 4 Pr F Q D7 m2 4m2 2Scm2 KrSc 2 2 ScS0 m2 m6 B5 D8 Sc Sc 2 4 KrSc m2 m6 Sc m2 m6 KrSc 2 m4 2 ScS0 m2 m4 B6 D9 m2 m4 Sc m2 m4 KrSc 2 1 1 4N m6 2 ScS0 m2 m4 B7 D10 m2 m4 Sc m2 m4 KrSc 2 Pr Pr 2 4 N m8 1 ScS0 m2 m6 B8 2 D11 m2 m6 Sc m2 m6 KrSc 2 1 N1 Pr F Q , N M ScS0 m4 m6 B7 K D12 m4 m6 Sc m4 m6 KrSc 2 Sc Sc 2 4 KrSc m10 2 2 4ScS0 m2 B10 D13 2 4m2 2Scm2 KrSc 2 2 4N m12 D14 D3 D4 D5 D6 D7 D8 2 2 D9 D10 D11 D12 D13 ScS0 m2 D1 2 Gr GmD2 m2 Scm2 KrSc A1 , A2 2 m m2 N 2 2 m4 m4 N 1 D2 1 D1 , N M A3 2 GmD1 , A4 A1 A2 A3 K m2 m2 N 2 ScS0 m8 B11 D3 2 A5 GrB11 , A6 GrB1 m8 Scm8 KrSc m m8 N 2 4m6 2m6 N 2 8 2 ScS0 m8 B11 GrB2 D3 A7 m8 Scm8 KrSc 2 4m2 2m2 N 2 2 4ScS0 m6 B1 D4 4m6 2Scm6 KrSc 2 GrB3 A8 2 4ScS0 m B2 4m 2m4 N 2 D5 2 4 4m 2Scm2 KrSc 2 GrB4 2 A9 4m 2m2 N 2 2 GrB5 A10 m2 m6 Pr m2 m6 N 2 IJSET@2013 Page 66 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 GrB6 A29 A5 A6 A7 A8 A9 A10 A11 m2 m4 Pr m2 m4 N 2 A11 A12 A13 A14 A15 A16 A12 GrB7 A17 A18 A19 A20 A21 A22 m2 m4 Pr m2 m4 N A23 A24 A25 A26 A27 A28 2 GrB8 2 2 2 Pr m2 A12 A13 B1 2 Pr m6 A4 B2 2 m2 m6 Pr m2 m6 N 2 4m6 2 Pr m6 N1 4m2 2 Pr m2 N1 GrB9 2 2 Pr m4 A2 A14 B3 2 m4 m6 Pr m4 m6 N 2 4m4 2 Pr m4 N1 GrB11 2 Pr m2 A32 A16 B4 m8 m8 N 2 4m2 2 Pr m2 N1 2 GrB10 GmD3 2 Pr m6 m4 A1 A4 A15 A17 2 B5 4m2 2m2 N m8 m8 N m2 m6 Pr m2 m6 N1 2 2 GmD4 GmD5 A18 A19 4m6 m6 N 2 4m2 m2 N 2 B6 2 Pr m2 m6 A1 A2 m2 m4 Pr m2 m4 N1 2 GmD6 A20 4m4 2 m4 N 2 Pr m2 m4 A3 A2 B7 m2 m4 Pr m2 m4 N1 2 GmD7 A21 4m2 2 m2 N 2 Pr m2 m6 A3 A4 B8 GmD8 m2 m6 Pr m2 m6 N1 2 A22 m2 m6 m2 m6 N 2 2 Pr m4 m6 A2 A4 B9 GmD9 m4 m6 Pr m4 m6 N1 2 A23 m2 m4 m2 m4 N 2 2 2 Pr m2 A1 A3 GmD10 B10 A24 4m2 2 Pr m2 N1 2 m2 m4 m2 m4 N 2 B11 B1 B2 B3 B4 B5 GmD11 A25 B6 B7 B8 B9 B10 m2 m6 m2 m6 N 2 GmD12 A26 m4 m6 m4 m6 N 2 GmD13 GmD14 A27 A28 2 4m 2m2 N 2 2 m10 m10 N IJSET@2013 Page 67 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 2 1.2 sc=0.65, S 0=1.0,Ec=0.001,Pr=0.025,Gr=5.0 1 1.5 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.75,Gr=5.0 Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.8 Gm=5.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 1 Gm=1.0,2.0,3.0,4.0 0.6 u u 0.5 0.4 0 0.2 0 Kr=0.5,1.0,1.5,2.0 -0.5 0 2 4 6 8 10 y -0.2 0 1 2 3 4 5 6 Figure (1): Velocity profiles for different values of Gm y Figure (5): Velocity profiles for different values of Kr 1 Sc=0.65,S =1.0,Ec=0.001,Pr=0.025,Gr=5.0 2 0 0.8 Gm=1.0,F=3.0,Q=1.0, Kr=0.5,K=1.0, M=2.0 Sc=0.65,S 0 =1.0,Ec=0.001,Pr=0.025,Gr=5.0 1.5 Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.6 1 u 0.4 Gr=1.0,2.0,3.0,4.0 u 0.5 0.2 0 M=0.5,1.0,1.5,2.0 0 -0.5 0 5 10 15 -0.2 y 0 5 y 10 15 Figure (6): Velocity profiles for different values of M Figure (2): Velcoity profiles for different values of Gr 1.2 1.4 1 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0 1.2 Sc=0.65, S 0=1.0;Ec=0.001,Pr=0.025,Gr=5.0 Gm=1.0,F=3.0,Q=1.0, Kr=0.5, K=1.0,M=2.0 Gm=2.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.8 1 0.6 0.8 u 0.6 0.4 u 0.4 0.2 0.2 0 F=1.0,2.0,3.0,4.0 Pr=0.025,0.50,0.075,0.1 0 -0.2 0 5 10 15 -0.2 y 0 5 10 15 Figure (7): Velocity profiels for different values of Pr y Figure (3): Velocity profiles for different values of F 1.2 2 1 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0 Gm=2.0,F=3.0,Q=1.0,Kr=0.5, K=1.0, M=2.0 1.5 Gm=2.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.8 0.6 1 u K=1.0,2.0,3.0,4.0 0.4 u 0.5 0.2 0 0 Q=2.0,4.0,6.0,8.0 -0.2 0 5 10 15 -0.5 y 0 5 10 15 y Figure (8): Velocity profiles for different values of Q Figure (4): Velocity profiles for different values of K IJSET@2013 Page 68 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 0.7 1 0.6 Sc=0.65, S 0=1.0,Ec=0.001,Pr=0.75,Gr=5.0 Ec=0.001,Pr=0.025,Q=1.0Gr=5.0,Gm=2.0 Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.8 0.5 0.6 0.4 u 0.3 F=1.0,2.0,3.0,4.0 S0=1.0,2.0,3.0,4.0 0.4 0.2 0.2 0.1 0 0 0 1 2 3 4 5 6 7 8 0 5 10 15 20 y y Figure (9): Velocity profiles for different values of S 0 Figure (11): Temperature profiles for different values of F 1 1.2 Ec=0.001,F=2.0,Q=1.0,Gr=5.0,Gm=2.0 0.8 1 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0 Gm=1.0, F=3.0, Q=1.0,Kr=0.5,K=1.0,M=2.0 0.6 0.8 Pr=0.025,0.050,0.075,0.10 0.6 0.4 u 0.4 0.2 0.2 Sc=0.22,0.30,0.60,0.78 0 0 0 2 4 6 8 10 12 14 16 18 y -0.2 Figure (12): Temperature profiles for different values of Pr 0 5 10 15 y Figure (10): Velocity profiles for different values of Sc 1 Ec=0.001,Pr=0.025,F=2.0,Gr=5.0,Gm=2.0 0.8 0.6 Q=1.0,2.0,3.0,4.0 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 y Figure (13): Temperature profiles for different values of Q 1.2 1 Ec=0.001,Pr=0.71,F=2.0,Q=5.0,Gr=5.0 0.8 0.6 Gm=1.0,2.0,3.0,4.0 0.4 0.2 0 -0.2 0 0.5 1 1.5 2 y Figure (14): Temperature profeiles for different values of Gm IJSET@2013 Page 69 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 1 1 Sc=0.65, S 0=0.5, Ec=0.001, Pr=0.75 0.8 Ec=0.001,Pr=0.71,F=3.0,Q=1.0,Gm=10.0 0.8 F=2.0,Q=5.0,Kr=0.5,K=0.5,M=2.0,Gm=50.0 0.6 0.6 C Gr=5.0,10.0,15.0,20.0 0.4 Gr=5.0:5.0:20.0 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 y y Figure (15): Temperature profiles for different values of Gr Figure (19): Concentration profiles for different values of Gr 1.4 1 1.2 Ec=0.001,Pr=0.75,F=1.0,Q=4.0,Gr=4.0 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=50.0 Gm=10.0, M=1.0,K=0.5, Kr=0.5,S 0=3.0 0.8 F=2.0, Q=50.0, Kr=0.5, M=2.0, Gm=100.0 1 0.8 0.6 C K=0.5,1.0,1.5,2.0 0.6 Sc=0.22,0.08,0.38,0.46 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 y y Figure (16): Temperature profiles for different values of Sc Figure (20): Concentration profiles for different values of K 1 1 Sc=0.65,S0=2.0, Ec=0.001, Pr=0.025 Sc=0.65,S0=2.0,Ec=0.001,Pr=0.025,Gr=5.0 0.8 Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0 0.8 F=1.0,Q=1.0, Kr=0.5,K=0.5,M=2.0,Gm=2.0 0.6 0.6 C C 0.4 F=1.0,2.0,3.0,4.0 Kr=0.5,1.0,1.5,2.0 0.4 0.2 0.2 0 0 2 4 6 8 10 0 y 0 5 10 15 Figure (17): Concentration profiles for different values of F y Figure (21): Concentration profiles for different values of Kr 1 1 Sc=0.65, S 0=0.5, Ec=0.001, Pr=0.75 0.8 Gr=50.0,F=2.0,Q=5.0,Kr=0.5,K=0.5,M=2.0 Sc=0.65,S =0.5,Ec=0.001,Pr=0.75,K=0.5 0 0.8 Gr=60.0, F=2.0, Q=5.0, Kr=0.5, Gm=20.0 0.6 Gm=5.0,10.0,15.0,20.0 0.6 C 0.4 C 0.4 0.2 0.2 0 0 1 2 3 4 5 M=0.5,1.0,1.5,2.0 y 0 Figure (18): Concentration profiles for different values of Gm 0 1 2 3 4 5 y Figure (22): Concentration profiles for different values of M IJSET@2013 Page 70 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 1 Sc=0.65, S 0=0.5, Ec=0.001, Gr=5.0 F=2.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=10.0 0.8 0.6 2.2 Sc=0.65,S 0=1.0,Ec=0.001,Pr=0.71,Gr=5.0 C Pr=0.025,0.050,.075,0.10 2 0.4 Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 1.8 0.2 1.6 1.4 0 0 1 2 3 4 5 y 1.2 Figure (23): Concentration profiles for different values of Pr 1 Pr=0.5,1.0,1.5,2.0 1 0.8 1 1.5 2 2.5 3 3.5 4 4.5 5 Sc=0.65, S 0=2.0, Ec=0.001, Pr=0.025 Gr 0.8 F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0 Figure (27): Effect of Prandtl number on skin friction versus Gr 1 0.6 C Q=1.0,2.0,3.0,4.0 0.9 0.4 0.8 0.2 Nu 0.7 M=0.5,0.6,0.7,0.8 0 0.6 0 2 4 6 8 10 y Figure (24): Concentration profiles for different values of Q Ec=0.001,Pr=0.75,F=1.0,Q=4.0,Gr=4.0,M=2.0 0.5 Gm=10.0, K=0.5, Sc=0.65, Kr=0.5, S 0=2.0 1 0.4 1 1.5 2 2.5 3 3.5 4 4.5 5 Sc=0.65, S 0=2.0, Ec=0.001, Pr=0.025 Gr 0.8 F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0 Figure (28): Effect of Magnetic field on Nusselt number 0.6 C S0=1.0,2.0,3.0,4.0 0.4 0.2 0 0 5 10 15 y Figure (25): Concentration profiles for different values of S 0 1 S0=2.0,Ec=0.001,Pr=0.025,Gm=2.0 0.8 F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0 0.6 C Sc=0.22,0.30,0.60,0.68 0.4 0.2 0 0 5 10 15 y Figure (26): Concemtration profiles for different values of Sc IJSET@2013 Page 71 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.2, Issue No.2, pg : 56-72 1 Feb. 2013 IJSET@2013 Page 72