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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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  • 1. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913 Chemical Reaction And Radiation Effects On MHD Free Convective Flow Past An Exponentially Accelerated Vertical Plate With Variable Temperature And Variable Mass Diffusion K.Rajasekhar *, G.V.Ramana Reddy ** and B.D.C.N.Prasad*** *(Department of Mathematics, Usha Rama College of Engineering, Telaprolu, Krishna (India)-521109) ** (Department of Mathematics, RVR & JC College of Engineering, Guntur, India -522 019) ***(Department of Mathematics, PVPSIT, College of Engineering, Vijayawada, India-520007)ABSTRACT An analytical study is performed to MHD bearings etc. The phenomenon of mass transferinvestigate the effects of chemical reaction and is also very common in theory of stellar structure andradiation on unsteady MHD flow past an observable effects are detectable, at least on the solarexponentially accelerated vertical plate with surface. The study of effects of magnetic field on freevariable temperature and variable mass diffusion convection flow is important in liquid metals,in the presence of applied transverse magnetic electrolytes and ionized gases. The thermal physics offield. It is assumed that the effect of viscous hydro -magnetic problems with mass transfer is ofdissipation is negligible in the energy equation and interest in power engineering and metallurgy.there is a first order chemical reaction between thediffusing species and the fluid. The flow is Diffusion rates can be altered tremendouslyassumed to be in x’ ‐ direction which is taken by chemical reactions. The effect of a chemicalalong the infinite vertical plate in the upward reaction depend whether the reaction is homogeneousdirection and y’ ‐ axis is taken normal to the plate. or heterogeneous. This depends on whether theyAt time t’ > 0, the both temperature and species occur at an interface or as a single phase volumeconcentration levels near the plate are raised reaction. In well-mixed systems, the reaction islinearly with time t. A general exact solution of the heterogeneous, if it takes place at an interface andgoverning partial differential equations is obtained homogeneous, if it takes place in solution. In mostby usual closed analytical method. The velocity, cases of chemical reactions, the reaction rate dependstemperature and concentration fields are studied on the concentration of the species itself. A reaction isfor different physical parameters like thermal said to be of the order n, if the reaction rate isGrashof number (Gr), mass Grashof number proportional to the nth power of the concentration. In(Gm), Schmidt number (Sc), Prandtl number (Pr), particular, a reaction is said to be first order, if theradiation parameter (R), magnetic field parameter rate of reaction is directly proportional to the(M), accelerated parameter (a), heat absorption concentration itself. A few representative fields ofparameter, chemical reaction parameter (K) and interest in which combined heat and mass transfertime (t) graphically. along with chemical reaction play an important role in chemical process industries such as food processingKeywords – MHD, thermal radiation, chemical and polymer production.reaction, variable temperature, variable massdiffusion Chambre and Young [1] have analyzed a first order chemical reaction in the neighbourhood ofINTRODUCTION a horizontal plate. Gupta [2] studied free convection Free convection flows are of great interest in on flow past a linearly accelerated vertical plate in thea number of industrial applications such as fiber and presence of viscous dissipative heat usinggranular insulation, geothermal systems etc. perturbation method. Kafousias and Raptis [3]Buoyancy is also of importance in an environment extended this problem to include mass transfer effectswhere differences between land and air temperatures subjected to variable suction or injection. Freecan give rise to complicated flow patterns. Magneto convection effects on flow past an exponentiallyhydro-dynamic has attracted the attention of a large accelerated vertical plate was studied by Singh andnumber of scholars due to its diverse applications. In Naveen kumar [4]. The skin friction for acceleratedastrophysics and geophysics, it is applied to study the vertical plate has been studied analytically by Hossainstellar and solar structures, interstellar matter, radio and Shayo [5]. Basant kumar Jha [6] studied MHDpropagation through the ionosphere etc. In free convection and mass transform flow through aengineering it finds its applications in MHD pumps, porous medium. Later Basant kumar Jha et al. [7] analyzed mass transfer effects on exponentially accelerated infinite vertical plate with constant heat 907 | P a g e
  • 2. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913flux and uniform mass diffusion. Das et al. [8] have t  0 : u   0, T  T , C   C   forall ystudied the effect of homogeneous first order u   u0 exp(a t ), T   T  Tw  T  At ,   chemical reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux t   0,  C  C  (Cw  C ) At  at y   0   and mass transfer. Again, mass transfer effects on u   0, T   T  , C   C  as y      moving isothermal vertical plate in the presence ofchemical reaction studied by Das et al. [9]. The (4) 2dimensionless governing equations were solved by u where A  , u  - velocity of the fluid in 0the usual Laplace Transform technique.Muthucumaraswamy and Senthil Kumar [10] studied heat and mass transfer effects on moving vertical the x  direction, y  - the dimensional distanceplate in the presence of thermal radiation. Recently along the co-ordinate axis normal to the plate,Muthucumaraswamy et al. [11] studied mass transfer g  the gravitational acceleration, ρ - the fluideffects on exponentially accelerated isothermalvertical plate. density,  and   are the thermal and concentration The object of the present paper is to study expansion coefficients respectively, B0 - thethe effects chemical reaction and a uniform transverse magnetic induction, T  - temperature of the fluid inmagnetic field (fixed relative to the plate) on the freeconvection and mass transform flow past an the x  direction, C  the correspondingexponentially accelerated vertical plate with variable concentration, C  concentration of the fluid fartemperature and mass diffusion. The dimensionless away from the plate, σ is the electric conductivity,governing equations are solved using the closed C p - the specific heat at constant pressure, D -theanalytical method. diffusion coefficient, q r - the heat flux, Q0 -the2. Formulation of the Problem dimensional heat absorption coefficient, and K r is We consider the unsteady hydro magneticradiative flow of viscous incompressible fluid past an the chemical reaction parameter.exponentially accelerated infinite vertical plate with The local radiant for the case of an optically thin grayvariable temperature and also with variable mass gas is expressed by qr  4a T4  T 4 diffusion and heat source parameter in the presence ofchemical reaction of first order and applied transverse (5) ymagnetic field. Initially, the plate and the fluid are at  It is assumed that the temperature differences withinthe same temperature T in the stationary condition the flow are sufficiently small and that T  may be 4 with concentration level C at all the points. At expressed as a linear function of the temperature. Thistime t   0 , the plate is exponentially accelerated is obtained by expanding T  in a Taylor series about 4with a velocity u  u0 exp( at ) in its own plane  T and neglecting the higher order terms, thus we getand at the same time the temperature of the fluid near T 4  4T 3T   3T3   (6)the plate is raised linearly with time t and species From equations (5) and (6), equation (2) reduces toconcentration level near the plate is also increased T   2T linearly with time. All the physical properties of the C p   16a T3 T  T     (7)fluid are considered to be constant except the y  y 2influence of the body‐force term. A transverse On introducing the following non-dimensionalmagnetic field of uniform strength B0 is assumed to quantitiesbe applied normal to the plate. Then under usual u t u 2 y u0 T   T  C   C  u  ,t  0 , y  ,  ,  ,Boussinesq’s approximation, the unsteady flow is u0    Tw  T   Cw  Cgoverned by the following set of equations. a    g  (Tw  T )   g   (Cw  C ) u   2u   B0 2 a , Gr  , Gm  ,  g  T   T   g    C   C      u 2 3 3 u0 u0 u0 t  y 2  C p   B02  16a 2 T3 Pr  , Sc  ,M  ,R  ,(1) k D  u02  u02 T   2T  q K r  Q0 CP  k 2  r  Q0 T   T   (2) Kr  2 ,Q   C p u0 2 , t  y  y  u 0 (8)C   2C   D 2  K r  C   C   (3) we get the following governing equations which are t  y  dimensionlesswith the following initial and boundary conditions: 908 | P a g e
  • 3. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913 2u  u   Gr  Gm  Mu (9)  u( y, t )  A3e N3 y  A1e N2 y  A2e N1 y t (23)   t y 2   y, t   te N1 y (24) 1  2 R     Q (10)  ( y, t )   t e  N2 y  (25) t Pr y 2 Pr It is now important to calculate the physical 2  1   quantities of primary interest, which are the local wall   K r (11) shear stress, the local surface heat, and mass flux. t Sc y 2 Given the velocity field in the boundary layer, we canThe initial and boundary conditions in dimensionless now calculate the local wall shear stress ( i.e., skin-form are as follows: friction) is given byt  0 : u  0,   0,   0 forall y   u   w     , and in dimensionless form, we u  exp(at ),   t ,   t at y  0 (12)  y   y  0t  0,  u  0,   0,   0 as y   obtain where  u  Cf      N 3 A3  N 2 A1  N1 A2  tM , Gr , Gm, R, Pr, Q, Sc and K r denote the  y  y  0magnetic field parameter, thermal Grashof number, From temperature field, now we study themass Grashof number, radiation parameter, Prandtl rate of heat transfer which is given in non -number, heat absorption parameter, Schmidt number dimensional form as:and chemical reaction parameter respectively.   3. SOLUTION OF THE PROBLEM: Nu      tN1 In order to reduce the above system of  y  y  0partial differential equations to a system of ordinary From concentration field, now we study theequations in dimensionless form, we may represent rate of mass transfer which is given in non -the velocity, temperature and concentration as dimensional form as:u( y, t )  u0 ( y)eit (13)    it Sh     tN 2 ( y , t )   0 ( y )e (14)   y  y  0 it ( y, t )  0 ( y)e (15) whereSubstituting Eqns (13), (14) and (15) in Eqns (9), (10) N1  Sc  K r  i  , N 2  Pr i  R ,and (11), we obtain: u0  N3u0   Gr0  Gm0  2  N3  M  i , A1  Gr / N 2  N3 , 2    (16) 0  N 2 0  0 2 (17)   A2  Gm / N12  N3 , A3  eat  t  A1  A2  2 C0  N12C0  0 (18) Results and DiscussionHere the primes denote the differentiation with In order to get physical insight into therespect to y. problem, we have plotted velocity profiles forThe corresponding boundary conditions can be different values of the physical parameterswritten as a(accelerated parameter), M(magnetic field parameter) R(radiation parameter), Gr(thermalu0  e a i t ,  0  te  it , 0  te  it at y  0 Grashof number), Gm(mass Grashof number), (19)u0  0,  0  0, 0  0 as y   Pr(Prandtl number), Sc(Schmidt number), t(time) andThe analytical solutions of equations (16) – (18) with the chemical reaction parameter (K) is presentedsatisfying the boundary conditions (19) are given by graphically. The value of Sc (Schmidt number) is  u0 ( y)  A3e N3 y eit  A1e N2 y  A4e N1 y teit taken to be 0.6 which corresponds to the water‐vapor. Also, the value of Pr (prandtl number) are chosen (20) such that they represent air (Pr=0.71).0  y   te   N2 y e it (21) The effect of the magnetic parameter M is shown in Fig.1. It is observed that the tangential velocity of the0 ( y )   t e  N1 y  e  it (22) fluid decreases with the increase of the magnetic field number values. The decrease in the tangential In view of the above solutions, the velocity, velocity as the magnetic parameter M increases istemperature and concentration distributions in the because the presence of a magnetic field in anboundary layer become electrically conducting fluid introduces a force called 909 | P a g e
  • 4. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913the Lorentz force, which acts against the flow if the the fluid and therefore, heat is able to diffuse awaymagnetic field is applied in the normal direction, as in from the heated surface more rapidly for higherthe present study. This resistive force slows down the values of Pr . Hence in the case of smaller Prandtlfluid velocity component as shown in Fig.1. number as the thermal boundary later is thicker andFig.2 shows the effect of the accelerated parameter a the rate of heat transfer is reduced.on the velocity distribution. It is found that thevelocity increases with an increase in an accelerated For different values of thermal radiation Rparameter a . the temperature profiles are shown in Fig.11. It is For various values of the thermal Grashof observed that an increase in the thermal radiationnumber Gr and mass Grashof number Gm , the parameter, the decrease of the temperature profiles.velocity profiles ‘ u ’ are plotted in Figs. (3) and (4). The effect of the Schmidt number Sc on theThe thermal Grashof number Gr signifies the concentration is shown in Fig.12. As the Schmidtrelative effect of the thermal buoyancy force to the number increases, the concentration decreases. Thisviscous hydrodynamic force in the boundary layer. As causes the concentration buoyancy effects to decreaseexpected, it is observed that there is a rise in the yielding a reduction in the fluid velocity. A reductionvelocity due to the enhancement of thermal buoyancy in the concentration distribution is accompanied byforce. Also, as Gr increases, the peak values of the simultaneous reduction in the concentration boundary layers. The effect of chemical reactionvelocity increases rapidly near to the plate and thendecays smoothly to the free stream velocity. The mass parameter K r on the concentration profiles as shownGrashof number Gm defines the ratio of the species in Fig.13. It is observed that an increase in leads to abuoyancy force to the viscous hydrodynamic force. decrease in the values of concentration.As expected, the fluid velocity increases and the peak 1.4value is more distinctive due to increase in the speciesbuoyancy force. The velocity distribution attains a 1.2 Sc=0.60;Pr=0.71;R=1;Kr=5;distinctive maximum value in the vicinity of the plate t=0.1;a=0.2;Gr=5;Gm=5; =1;and then decreases properly to approach the free 1stream value. It is noticed that the velocity increaseswith increasing values of the mass Grashof number.Fig.5 shows the effect of the time t on the velocity 0.8 Velocityfield. It is found that the velocity increases with anincrease in time t . Fig.6. illustrates the velocity 0.6 M=1, 2, 3, 4profiles for different values of radiation parameter R , 0.4the numerical results show that the effect ofincreasing values of radiation parameter result in a 0.2decreasing velocity. For different values of thePrandtl number Pr the velocity profiles are plotted in 0Fig.7. It is obvious that an increase in the Prandtl 0 0.5 1 1.5 2 2.5 ynumber Pr results in decrease in the velocity withinthe boundary layer. Fig.8 illustrates the behavior Fig. 1. Effects of magnetic parameter on velocityvelocity for different values of chemical reaction profiles.parameter K r . It is observed that an increase in leads 2to a decrease in the values of velocity. For different 1.8 R=5;Sc=0.60;M=4;Pr=0.71; t=0.4;Kr=5;Gr=20;Gm=5; =1;values of the Schmidt number Sc the velocity 1.6profiles are plotted in Fig.9. It is obvious that an 1.4increase in the Schmidt number Sc results in 1.2 a = 0.2, 0.5, 0.9, 1.5decrease in the velocity within the boundary layer. VelocityFor different values of frequency of excitation  on 1 0.8the temperature profiles are shown in Fig.9. It isnoticed that an increase in the frequency of excitation 0.6results a decrease in the temperature field. Fig.10 0.4shows the behavior of the temperature for different 0.2values Prandtl number Pr . It is observed that an 0 0 0.5 1 1.5 2 2.5increase in the Prandtl number results a decrease of ythe thermal boundary layer thickness and in general Fig. 2. Effects of accelerated parameter on velocitylower average temperature within the boundary layer. profiles.The reason is that smaller values of Pr areequivalent to increase in the thermal conductivity of 910 | P a g e
  • 5. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913 1.4 1.2 1.4 R=5;Sc=0.60;M=4;Pr=0.71; 1 t=0.4;Kr=5;a=0.2;Gm=5; =1; 1.2 Sc=0.60;Pr=0.71;M=4;Kr=5; 0.8 t=0.1;a=0.2;Gr=20;Gm=5; =1; Velocity 1 Gr = 5, 10, 15, 20 0.6 0.8 Velocity 0.4 0.6 0.2 R=1, 2, 3, 4 0.4 0 0 0.5 1 1.5 2 2.5 0.2 yFig. 3. Effects of Thermal Grashof number on 0 0 0.5 1 1.5 2 2.5velocityprofiles. y Fig.6. Effects of radiation parameter on velocity 1.4 profiles. R=5;Sc=0.60;M=4;Pr=0.71; 1.2 t=0.4;Kr=5;a=0.2;Gr=20; =1; 1.4 1 R=5;Sc=0.60;M=4;Kr=5; 1.2 t=0.4;a=0.5;Gr=20;Gm=5; =1; Gm = 5, 10, 15, 20 0.8 Velocity 1 0.6 0.8 Velocity Pr = 0.71, 1.00 3.00, 7.00 0.4 0.6 0.2 0.4 0 0 0.5 1 1.5 2 2.5 0.2 yFig. 4. Effects of mass Grashof number on velocity 0 0 0.5 1 1.5 2 2.5profiles. y Fig.7. Effects of Prandtl number on velocity profiles. 1.5 R=5;Sc=0.60;M=4;Pr=0.71; Gm=5;Kr=5;a=0.2;Gr=20; =1; 1.4 R=5;Sc=0.60;M=4;Pr=0.71; 1.2 t=0.4;a=0.5;Gr=20;Gm=5; =1; t = 0.2 0.4 0.6 0.8 1 1 Velocity 0.8 Velocity Kr =1, 5, 10, 15 0.5 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 y 0 0 0.5 1 1.5 2 2.5Fig.5. Effects of time on velocity profiles. y Fig.8. Effects of chemical reaction parameter on velocity profiles. 911 | P a g e
  • 6. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913 1.4 0.2 R=5;Pr=0.71;M=4;Kr=5; 0.18 Pr=0.71;Sc=0.60;M=4; =1; 1.2 t=0.4;a=0.5;Gr=20;Gm=5; =1; Gm=5;Kr=5;a=0.2;Gr=20;t=0.2; 0.16 1 0.14 0.12 Temperature 0.8 Velocity Sc = 0.22, 0.30, 0.60, 0.78 0.1 0.6 R=1, 2, 3, 4 0.08 0.4 0.06 0.04 0.2 0.02 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 y yFig.9. Effects of Schmidt number on velocity profiles. Fig.12. Effects of radiation parameter on temperature profiles. 0.2 0.18 0.2 R=1;Sc=0.60;M=4;Pr=0.71; Gm=5;Kr=5;a=0.2;Gr=20;t=0.2; 0.18 Pr=0.71;Sc=0.60;M=4;R=5; 0.16 Gm=5;Kr=5;a=0.2;Gr=20;t=0.2; 0.14 0.16 0.12 0.14 Temperature 0.12 Concentration 0.1 =1, 2, 3,4 0.08 0.1 0.06 0.08 =1, 2, 3, 4 0.04 0.06 0.02 0.04 0 0.02 0 0.5 1 1.5 2 2.5 y 0 0 0.5 1 1.5 2 2.5Fig.10. Effects of frequency of excitation on ytemperature profiles. Fig.13. Effects of frequency of excitation on concentration profiles. 0.2 0.18 R=1;Sc=0.60;M=4; =1; 0.2 Gm=5;Kr=5;a=0.2;Gr=20;t=0.2; 0.16 0.18 Pr=0.71;Kr=5;M=4;R=5; Gm=5; =1;a=0.2;Gr=20;t=0.2; 0.14 0.16 0.12 0.14 Temperature 0.12 Concentration 0.1 0.08 Pr = 0.71, 1.00, 3.00, 7.00 0.1 Sc= 0.22, 0.30, 0.60, 0.78 0.06 0.08 0.04 0.06 0.02 0.04 0 0.02 0 0.5 1 1.5 2 2.5 y 0 0 0.5 1 1.5 2 2.5Fig.11. Effects of Prandtl number on temperature yprofiles. Fig.14. Effects of Schmidt number on concentration profiles. 912 | P a g e
  • 7. K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.907-913 with chemical reaction, The Bulletin of 0.2 GUMA, Vol.5, pp.13-20, 1999. 0.18 Pr=0.71;Sc=0.60;M=4;R=5; [10]. R. Muthucumaraswamy, G. Senthil kumar, 0.16 Gm=5; =1;a=0.2;Gr=20;t=0.2; Heat and Mass transfer effects on moving 0.14 vertical plate in the presence of thermal 0.12 radiation, Theoret. Appl. Mech., Vol.31, Concentration No.1, pp.35-46, 2004. 0.1 [11]. R. Muthucumaraswamy, K.E. Sathappan, R. 0.08 Kr =1,2, 3, 4 Natarajan, Mass transfer effects on 0.06 exponentially accelerated isothermal vertical 0.04 plate, Int. J. of Appl. Math. and Mech., 4(6), 0.02 19-25, 2008. 0 0 0.5 1 1.5 2 2.5 yFig.15. Effects of chemical reaction parameter onconcentration profiles.REFERENCES [1]. P. L. Chambre, J. D. Young, On the diffusion of a chemically reactive species in a laminar boundary layer flow, The Physics of Fluids, Vol.1, pp.48-54, 1958. [2]. A. S. Gupta, I. Pop, V. M. Soundalgekar, Free convection effects on the flow past an accelerated vertical plate in an incompressible dissipative fluid, Rev. Roum. Sci. Techn. -Mec. Apl., 24, pp.561-568, 1979. [3]. N. G. Kafousias, A.A. Raptis, Mass transfer and free convection effects on the flow past an accelerated vertical infinite plate with variable suction or injection, Rev. Roum. Sci. Techn.- Mec. Apl., 26, pp.11-22, 1981. [4]. A. K. Singh, N. Kumar, Free convection flow past an exponentially accelerated vertical plate, Astrophysics and Space science, 98, pp. 245-258, 1984. [5]. M. A. Hossain, L. K. Shayo, The Skin friction in the unsteady free convection flow past an accelerated plate, Astrophysics and Space Science, 125, pp. 315-324, 1986. [6]. B.K. Jha, MHD free-convection and mass transform flow through a porous medium, Astrophysics and Space science, 175, 283- 289, 1991. [7]. B.K. Jha, R. Prasad, S. RAI, Mass transfer effects on the flow past an exponentially accelerated vertical plate with constant heat flux, Astrophysics and Space Science, 181, pp.125-134, 1991. [8]. U. N. Das, R. K. Deka, V. M. Soundalgekar, Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forschung in Ingenieurwsen, Vol.60, pp.284-287, 1994. [9]. U. N. Das, R. K. Deka, V. M. Soundalgekar, Effects of mass transfer on flow past an impulsively started infinite vertical plate 913 | P a g e

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