Magnetic Field and Chemical Reaction Effects on Convective Flow of

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1 Communications in Applied Sciences ISSN Volume 1, Number 1, 213, Magnetic Field and Chemical Reaction Effects on Convective Flow of Dust Viscous Fluid P. Mohan Krishna 1, Dr.V.Sugunamma 2, N.Sandeep 3 1Research Scholar, Department of mathematics, S.V.U College of Sciences, Tirupati, India 2Associate Professor, Department of mathematics, S.V.U College of Sciences, Tirupati, India 3Assistant Professor, Department of mathematics, SET, Jain Universit, Bangalore, India Corresponding author: Dr.V.Sugunamma, Associate Professor, Department of mathematics, S.V.U College of Sciences, Tirupati, India ABSTRACT: In this paper we analses the laminar convective flow of a dust viscous fluid of non conducting walls in the presence of transverse magnetic field with volume fraction and the first order chemical reaction is taken into consideration. An exact analtic solution is obtained for the temperature and concentration fields b using perturbative technique from these temperature and concentration fields, we obtain numerical solution for the velocit field and we analze the effects of velocit, temperature and concentration of the fluid at the various parameters as t (time), M (Magnetic parameter), Pr(Prandtl number),gr (Grashof number), K1(Chemical reaction parameter).from these we observed that increase in transverse magnetic field causes the decrease of velocit of the fluid and fluid temperature decreases b increase in prandtl number and Concentration profile increases b increase in chemical reaction parameter. Ke Words: Convective flow, MHD, Dust fluid, Concentration, Chemical Reaction. 1. Introduction The concept of heat transfer of a dust fluid has a wide range of applications in air conditioning,refrigeration, pumps, accelerators, nuclear reactors space heating, power generation, chemical processing,filtration and geothermal sstems etc. The good example of heat transfer is the radiator in a car, in which the hot radiator fluid is cooled b the flow of air over the radiator surface. The experimental and Copright 213 the authors. 161

2 162 Communications in Applied Sciences theoretical works on MHD flow with chemical reaction have been done extensivel in various areas i.e. sustain plasma confinement for controlled thermo nuclear fusion, liquid metal cooling of nuclear reactions and electromagnetic casting of metals. Keeping above facts man authors attracted in this field of stud of heat and mass transfer through dust fluids. Seddeek et al [1] examined the effect of chemical reaction and variable viscosit on hdromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media he has been analsed in the presence of radiation and magnetic field. Chambre and Yang [2] have worked on thermal diffusion of a chemicall reactive species in a laminar boundar laer flow. Muthucumaraswam R and P. Ganesan [3] studied the chemical reaction on the flow past an impulsivel started vertical plate with uniform heat and mass flux. The same tpe of problem with inclusion of constant wall suction was studied b Makinde.O.D. and P. Sibanda [4]. Soundalgekar et al [5] discussed MHD effects on impulsivel started vertical infinite plate with variable temperature in the presence of transverse magnetic field. Ghosh [6] studied the hdromagnetic flow of a dust viscoelastic Maxwell fluid through a rectangular channel for an arbitrar pressure gradient. Unstead hdromagnetic flow and heat transfer from a non isothermal stretching sheet immersed in a porous medium was discussed b Chamkha [7].Viscous heating of high Prandtl number fluids with temperature dependent viscosit. Eckert et al. [8] studied the effect of radiation on temperature and velocit in electrodnamics froth flow process. The concluded that the velocit and temperature increases as the radiation parameter increases. Attia [9] has investigated an unstead MHD Couette flow and heat transfer of dust fluid with variable phsical properties. Mishra et al. [1] have studied the transient conduction and radiation heat transfer with temperature dependent thermal conductivit on two dimensional flows. Chakrabort [11] studied MHD flow and heat transfer of a dust viscoelastic stratified fluid down an inclined channel in porous medium under variable viscosit. Nag and Jana [12] have discussed unstead couette flow of a dust gas between two infinite parallel plates, when one plate is kept fixed and the other plate

3 Communications in Applied Sciences 163 moves in its own plane b using Laplace Transform technique. Here the found that the dust velocit in the case of accelerated start of the plate is less than the fluid velocit. The carried out a similar calculation for the simpler case of a homogeneous flow without porous medium. The convective magnetohdrodnamic two phase flow and heat transfer of a fluid in an inclined channel was investigated b Malashett et al. [13]. Attia [14] has investigated an unstead MHD Couette flow and heat transfer of dust fluid with variable phsical properties. Keeping above facts and applications, in the present paper we discuss the laminar convective flow of a dust viscous fluid of non-conducting walls in the presence of transverse magnetic field with volume fraction and chemical reaction. Here we analze the effect of velocit, velocit of the particle phase, temperature of the dust fluid and concentration at the transverse magnetic field along with various phsical parameters. 2. Mathematical Formulation In Cartesian co-ordinate sstem, we consider unstead laminar flow of a dust, incompressible, Newtonian, electricall conducting, viscous fluid of uniform cross section h, when one wall of the channel is fixed and the other is oscillating with time about a constant non-zero mean. Initiall at t, the channel wall as well as the fluid is assumed to be at the same temperature T and concentrationc. When t>, the temperature of the channel wall is instantaneousl raised to T w and concentration raised to C w which oscillate with time and is thereafter maintained constant. Let x- axis be along the fluid flow at the fixed wall and -axis perpendicular to it. The transverse magnetic field is applied to the flow along direction with the first order chemical reaction. 2.1 Assumptions: The governing equations are written based on the following assumptions:

4 164 Communications in Applied Sciences The dust particles are solid, spherical, non-conducting, and equal in size and uniforml distributed in the flow region. (i) The densit of dust particles is constant and the temperature between the particles is uniform throughout the motion. (ii) The interaction between the particles, chemical reaction between the particles and liquid has been considered. (iii) The volume occupied b the particles per unit volume of the mixture, (i.e., volume fraction of dust particles) and mass concentration have been taken into consideration. 2.2 Governing equations of the flow The fluid flow is governed b the momentum and energ equations under the above assumptions are [ ] (1) * + (2) (3) (4) The boundar conditions to the problem are: for at (5) at

5 Communications in Applied Sciences 165 Where is the velocit of the fluid and is velocit of the dust particles, is the mass of each dust particle, is the number densit of the dust particle, is the temperature, is the initial temperature, is the raised temperature, C is the concentration, is the initial concentration, is the raised concentration, is the volume fraction of the dust particle, f is mass concentration of dust particle, volumetric coefficient of the thermal expansion, is the Stoke s resistance coefficient, is the electrical conductivit of the fluid, is the magnetic permeabilit, is the magnetic field induction, is the specific heat at constant pressure, is the thermal conductivit, is chemical reaction parameter, is dimensionless chemical reaction parameter. The problem is simplified b writing the equations in the following non dimensional. Here the characteristic length is taken to be h and the characteristic velocit is. (6) Substituting the above non-dimensional parameters equation (6) in the governing equations (1-4) (after removing asterisks), we get (7) * + (8) (9) (1) Where

6 166 Communications in Applied Sciences (Thermal Grashof number), (mass Grashof number),, (Magnetic parameter), (mass concentration of dust particles), (concentration resistance ratio) and (Prandtl number), (Schmidt number), (Chemical Reaction parameter). The corresponding non-dimensional boundar conditions are: for at (11) at 3. Solution of the Problem: To solve the equations (7-1) we use the below equations introduced b Soundalgekar and Bhat equations. (12) After substituting equation (1) in equations (6-8), we can write (13) [ ] (14) (15)

7 Communications in Applied Sciences 167 (16) (17) [ ] (18) (19) [ ] (2) The corresponding boundar conditions becomes at at (21) On solving equation (15) and (16) with the help of boundar conditions (21), we get (22) (23) Substituting equations (22) and (23) in equations (13) and (14) (24) * + (25) Substituting equation (25) in equation (24), we obtain (26) Where B solving equation (26) with the boundar conditions (21), we get

8 168 Communications in Applied Sciences (27) * + The first and second order partial derivatives of are * + (28) * + (29) Substituting equations (27) and (29) in equation (14), we obtain * + (3) Where B solving equations (19) and (2) with the boundar conditions (21), we get (31) (32) Substituting equations (31) and (32) in equations (17) and (18), we obtain (33) * + (34) Substituting equation (34) in equation (33) we obtain (35)

9 Communications in Applied Sciences 169 On solving equation (35), with the boundar conditions (21), we get (36) The first and second order partial derivatives are given b (37) (38) Substituting equations (37) and (38) in equation (34), we obtain * + (39) and Substituting the equations (27) and (36) in the equation (12), we obtain * + * + (4) Substituting the equations (3) and (39) in the equation (12), we obtain * + * + (41) Substituting equations (22) and (31) in equation (12), we obtain (42)

10 17 Communications in Applied Sciences Substituting equations (23) and (32) in equation (12), we obtain (43) Hence the equations (4)-(43) represents velocit of the fluid, velocit of the dust particle, temperature and concentration respectivel. 4. Results and Discussion: In order to stud the behavior of fluid velocit, dust velocit, temperature and concentration fields, a comprehensive numerical computation is carried out for various values of the parameters that describe the flow characteristics, and the results are reported in terms of graphs as shown in Figures (1) (15). The variation of fluid velocit for different values of M is shown in Figure 1. It is observed that the velocit decreases with increases of magnetic parameter (M).The effect of fluid velocit for different Prandtl number are shown in Figure 2. In this case, the velocit decreases with increases of.figure 3 shows the variation of fluid velocit for different thermal Grashof number. It is clear that the velocit increases with increase of thermal Grashof number. Figure 4 represent the effect of fluid velocit for different values of Schmidt number. It is clear that the velocit increases with increases of Schmidt number. The effect of fluid velocit for different time is shown in Figure 5. We observe that, the velocit decreases with the increases of time. The effects of dust velocit for different values of M are shown in Figure 6. We observe that, the velocit increases graduall with respect to magnetic parameter (M). Figure 7 shows the dust velocit for different Prandtl number, we observe that the decreases of velocit with increases of. Figure 8 shows the variation of dust velocit for different thermal Grashof number. It is clear that the velocit increases graduall with respect to thermal Grashof number. Figure 9 represent the effect of dust velocit for different values of Schmidt number. It is clear that the velocit increases with increases of Schmidt number.

11 Communications in Applied Sciences 171 The effect of dust velocit at different time intervals is shown in Figure 1. We observe that, the velocit decreases with the increases of time The variations of concentration for different values of the chemical reaction parameter are shown in Figure 11. It is observed that the concentration increases graduall with respect to chemical reaction parameter. Figure 12 represents the effect of concentration for different values of Schmidt number. We observed that the concentration increases graduall with respect to Schmidt number. Figure 13 represents the effect of concentration for different values of time, it is observed that the concentration decreases with increases of time. Figure 14 represents the effect of Temperature for different values of Prandtl number, it is observed that the Temperature decreases with increases of.figure 15 represents the effect of Temperature for different values of time, it is observed that the Temperature decreases with increases of time.

12 u 172 Communications in Applied Sciences M=1 M=2 M=3 M= Fig.1: Variation of Fluid velocit for different values of M When =2, =.71, =.6, t=.2, f=.2, =2, =5.

13 u Communications in Applied Sciences Pr=.71 Pr=3 Pr=7 Pr= Fig.2: Variation of Fluid velocit for different values of Pr When =2, M=5, =.6, t=.2, f=.2, =2, =5.

14 u 174 Communications in Applied Sciences Gr=-15 Gr= Gr=1 Gr= Fig.3: Variation of Fluid velocit for different values of Gr When =2, =.71, =.6, t=.2, f=.2, =2, M=5.

15 u Communications in Applied Sciences Sc=.22 Sc=.6 Sc=.78 Sc= Fig.4: Variation of Fluid velocit for different values of Sc When =2, =.71, M=5, t=.2, f=.2, =2, =5.

16 u 176 Communications in Applied Sciences t=.2 t=.4 t=.6 t= Fig.5: Variation of Fluid velocit for different values of t When =2, =.71, =.6, M=5, f=.2, =2, =5.

17 v Communications in Applied Sciences M=1 M=2 M=3 M= Fig.6: Variation of dust velocit for different values of M When =2, =.71, =.6, t=.2, f=.2, =2, =5.

18 v 178 Communications in Applied Sciences 3 25 Pr=.71 Pr=3 Pr=7 Pr= Fig.7: Variation of dust velocit for different values of Pr When =2, M=5, =.6, t=.2, f=.2, =2, =5.

19 v Communications in Applied Sciences Gr=-15 Gr= Gr=1 Gr= Fig.8: Variation of dust velocit for different values of Gr When =2, =.71, =.6, t=.2, f=.2, =2, M=5.

20 v 18 Communications in Applied Sciences Sc=.22. Sc=.6 Sc=.78 Sc= Fig.9: Variation of dust velocit for different values of Sc When =2, =.71, M=5, t=.2, f=.2, =2, =5.

21 v Communications in Applied Sciences t=.2 t=.4 t=.6 t= Fig.1: Variation of dust velocit for different values of t When =2, =.71, =.6, M=5, f=.2, =2, =5.

22 C 182 Communications in Applied Sciences k=.2 k=2 k=5 k= Fig.11: Variation of Concentration for different values of When M=5, =.71, =.6, t=.2, f=.2, =2, =5.

23 C Communications in Applied Sciences Sc=.22 Sc=.6 Sc=.78 Sc= Fig.12: Variation of Concentration for different values of Sc When =2, =.71, M=5, t=.2, f=.2, =2, =5.

24 C 184 Communications in Applied Sciences 1.5 t=.2 t=.4 t=.6 t= Fig.13: Variation of Concentration for different values of t When =2, =.71, =.6, M=5, f=.2, =2, =5.

25 T Communications in Applied Sciences Pr=.71 Pr=3 Pr=7 Pr= Fig.14: Variation of Temperature for different values of Pr When =2, M=5, =.6, t=.2, f=.2, =2, =5.

26 T 186 Communications in Applied Sciences 1.5 t=.2 t=.4 t=.6 t= Y Fig.15: Variation of Temperature for different values of t When =2, =.71, =.6, M=5, f=.2, =2, =5.

27 Communications in Applied Sciences 187 References: [1]. M.A.Seddeek, A.A. Darwish, M.S. Abdelmeguid, Effects of Chemical Reaction and Variable Viscosit on Hdromagnetic Mixed Convection Heat and Mass Transfer for Hiemenz Flow Through Porous Media with Radiation, Commun. Nonlinear Sci. Numer. Simulat. V12, (27),pp [2]. P.L.Chambre and J. D. Young. Diffusion of a Chemicall Reactive Species in a Laminar Boundar Laer Flow, Phsics of fluids, Vol. 1, (1958), pp [3]. R.Muthucumaraswam, P. Ganesan. First order chemical reaction on flow past an impulsivel started vertical plate with uniform heat and mass flux, Acta Mechanica 147, (21), [4]. O.D Makinde, P. Sibanda. MHD mixed-convective flow and heat and mass transfer past a vertical plate in a porous medium with constant wall suction, Journal of Heat transfer, 13, (28),11262/1-8. [5]. V.M.Soundalgekar, Gupta SK. and Birajdar NS.Effects of Mass transfer and free convection currents on MHD Stokes problem for a vertical plate. Nuclear Engineering Design, vol. 53, (1979),pp [6]. B.C. Ghosh, hdromagnetic flow of a dust viscoelastic Maxwell fluid through a rectan gular channel, Int. J. Appl.Mech. Eng. 7 (2) [7]. A. J. Chamkha,Unstead hdromagnetic flow and heat transfer from a non-isothermal stretching sheet immersed in a porous medium, International Communications in Heat and Mass Transfer, 25 (1998) [8]. E.R.G. Eckert and M.Faghri, Viscous heating of high Prandtl number fluids with tem peraturedependent viscosit, Int l Journal of Heat and Mass Transfer 29 (1986) [9]. H.A.Attia, N.A. Kotb, MHD flow between two parallel plates with heat transfer, Acta Mech. 117 (1996) [1]. S.C. Mishra, P. Talukdar, D. Trimis, and F. Durst, Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivit, International Communications in Heat and Mass Transfer, 32 (25) , [11]. S.Chakrabort, MHD flow and heat transfer of a dust viscoelastic stratified fluid down an inclined channel in porous medium under variable viscosit, Theoretical and Applied Mechanics, 26 (21) [12]. S.K.Nag and R.N. Jana Couette flow of a dust Gas, Acta Mechanica, Springer verlag. 33 (1979) [13]. M.S.Malashett, J. C. Umavathi, and J. Prathap Kumar, Convective magnetohdrodnamic fluid flow and heat transfer in an inclined channel, Heat and Mass Transfer, 37 (21) [14]. H. A. Attia,Unstead MHD couette flow and heat transfer of dust fluid with variable phsical properties,applied Mathematics and Computation, 177 (26)