International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

The Effects of Induced Magnetic Field and Radiation on MHD Mixed Convective Flow over a Porous Vertical Plate in the Presence of Viscous and Magnetic Dissipation

J. Prakash1, A. G. Vijaya Kumar2, B. Rushi Kumar2 and S. V. K. Varma3

 transfer have enormous applications. Radiative flows are Abstract—The objective of this investigation is to study the encountered in many industrial and environment processes effects of induced magnetic field and radiation on MHD heat and e.g. heating and cooling chambers, fossil fuel combustion mass transfer flow of a steady, viscous, incompressible, electrically energy processes, evaporation from large open water conducting, Newtonian fluid over a porous vertical plate in the reservoirs, astrophysical flows, solar power technology and presence of viscous and magnetic dissipation taking into account the space vehicle re-entry. Soundalgekar [1] investigated the induced magnetic field with a magnetic . The similarity solutions of the transformed dimensionless governing effects of magnetic field on hydromagnetic flow of a viscous equations are obtained by using perturbation technique. The incompressible fluid caused by uniformly accelerated motion influence of magnetic parameter (M), radiation parameter (R), near an infinite flat plate. The effects of transversely applied chemical reaction parameter (k), (Prm), magnetic field, on the flow of an electrically conducting fluid Schmidt number (Sc), thermal (Gr) and mass past an impulsively started infinite isothermal vertical plate Grashof number(Gm) on Non-dimensional velocity, temperature, studied by Soundalgekar et al. [2]. MHD effects on concentration and induced magnetic field have been shown impulsively started vertical plate with variable temperature in graphically. It is found that the effect of magnetic parameter and the presence of transverse magnetic field were considered by magnetic Prandtl number is to increase the velocity and temperature Soundalgekar et al. [3]. The dimensionless governing fields while the induced magnetic field is negatively increasing with equations were solved using Laplace-transform technique. them. Kumari and Nath [4] studied the development of the asymmetric flow of a viscous electrically conducting fluid in Keywords—Chemical reaction, MHD, Magnetic Prandtl the forward stagnation point region of a two-dimensional number, Heat and Mass transfer, , Magnetic Dissipation body and over a stretching surface were set into impulsive motion from the rest. The governing equations were solved I. INTRODUCTION using finite difference scheme. Transient MHD heat and mass HE study of Magneto hydrodynamics with heat and mass transfer flow with thermal diffusion in a rotating system was T transfer in the presence of radiation and diffusion has studied by Alam and Sattar [5]. Muthucumaraswamy and attracted the attention of a large of number of scholars due to Janakiraman [6] studied MHD and radiation effects on diverse applications. In astrophysics and geophysics, it is moving isothermal vertical plate with variable mass diffusion. applied to study the stellar and solar structures, radio Rajesh and Varma [7] studied thermal diffusion and radiation propagation through the ionosphere etc. In engineering, we effects on MHD flow past a vertical plate with variable find its applications like in MHD pumps, MHD bearings etc. temperature. Very recently, Kumar and Varma [8] The phenomenon of mass transfer is also very common in investigated thermal radiation and mass transfer effects on theory of stellar structure and observable effects are detectable MHD flow past an impulsively started exponentially on the solar surface. In free flow, the study of accelerated vertical plate with variable temperature and mass effects of magnetic field plays a major role in liquid metals, diffusion. electrolytes and ionized gases. In power engineering, the Diffusion rates can be altered tremendously with chemical thermal physics of hydro magnetic problems with mass reactions. The chemical reaction effects depend on whether the reaction is homogeneous or heterogeneous. This depends J. Prakash1 Department of Mathematics, University of Botswana, Private Bag on whether they occur at an interface or as a single phase 0022, Gaborone, Botswana: phone (0267) 355 2949; fax: (0267)318 5097; e- volume reaction. In well-mixed systems, the reaction is mail: [email protected]). A. G. Vijaya Kumar2, B. Rushi Kumar2 Division, School of heterogeneous, if it takes place at an interface and Advanced Sciences, VIT University, Vellore-632 014, TN, India (e-mails: homogeneous, if it takes place in solution. In most cases, a [email protected]; [email protected]). chemical reaction depends on the concentration of the species S. V. K. Varma3 Department of Mathematics, S.V. University, Tirupati-517 502, A.P., India, (e-mail: [email protected]). itself. A reaction is said to be of first order, if the rate of

52 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

reaction is directly proportional to the concentration itself, studied by Chamkha and Camille [23]. Raptis et al. [24] Cussler [9]. A few representative areas of interest in which studied the hydromagnetic free convection flow of an heat and mass transfer combined along with chemical optically thin gray gas taking into account the induced reaction play an important role in chemical industries like in magnetic field in the presence of radiation and the analytical food processing and polymer production. Chambre and solutions were obtained by perturbation technique. MHD Young [10] have analyzed a first order chemical reaction in mixed free-forced heat and mass convective incompressible the neighborhood of a horizontal plate. Apelblat [11] studied steady laminar boundary layer flow of a gray optically thick analytical solution for mass transfer with a chemical reaction electrically conducting viscous fluid past a semi-infinite of first order. Das et al. [12] have studied the effect of vertical plate have been considered by Emad and Gamel [25]. homogeneous first order chemical reaction on the flow past Orhan and Kaya [26] examined MHD mixed convective heat an impulsively started vertical plate with uniform heat flux transfer along a permeable vertical infinite plate in the and mass transfer. Again, mass transfer effects on moving presence of radiation and solutions are derived using Kellar isothermal vertical plate in the presence of chemical reaction box scheme and accurate finite-difference scheme. Zueco et studied by Das et al. [13]. The dimensionless governing al. [27] investigated thermophoretic effects on magneto equations were solved by the usual Laplace Transform hydrodynamic heat and mass transfer in boundary layer flow technique. along a flat plate with viscous heating, Joule heating and wall In all the above studies the authors have considered a very suction by employing the network simulation method. Ghosh small magnetic Reynolds numbers, allowing magnetic et al. [28] presented an exact solution for the hydromagnetic induction effects to be neglected. But the effect of induced free convection flow past an infinite vertical flat plate in the magnetic field must be considered for relatively large values presence of transverse magnetic field taking into account the of the magnetic . Glauert [14] presented a induced magnetic field. Sahin Ahmed [29] investigated the seminal analysis for hydromagnetic flat plate boundary layers study of influence of thermal radiation and magnetic Prandtl with uniform magnetic field in the stream direction at the number on the steady MHD heat and mass transfer mixed plate. Raptis and Soundalgekar [15] examined a problem of convection flow of a viscous, incompressible, electrically- study flow of an electrically conducting fluid past a moving conducting Newtonian fluid over a vertical porous plate with vertical infinite plate in presence of constant suction and heat induced magnetic field. Recently, Ahmed [30] investigated flux in the presence of induced magnetic field. Alom et al. the effects of thermal radiation and viscous/magnetic [16] studied MHD mixed convection heat and mass transfer dissipation on the steady MHD heat transfer flow over a study flow along a moving vertical porous plate taking into porous vertical plate with induced magnetic field. account induced magnetic field, thermal diffusion, constant In this paper, it is proposed to study the effects of viscous heat and mass fluxes with non-linear coupled equations and and magnetic dissipation on steady two-dimensional the obtained governing equations were solved by shooting magneto-hydrodynamic mixed convection heat and mass iteration technique. Beg et al. [17] presented local non- transfer flow of a Newtonian, electrically conducting and similarity numerical solutions for the velocity, temperature viscous incompressible radiative fluid over a porous vertical distributions in forced convection MHD boundary layers in plate taking into the account of induced magnetic field and the presence of magnetic Prandtl numbers and induction. The magnetic Prandtl number. The governing equations for this study of heat and mass transfer on free convective three investigation are formulated and solved using perturbation dimensional flow of a viscous incompressible fluid over a technique. vertical porous plate was studied by Ahmed [18]. Again Ahmed and Liu [19] examined the effects of mass transfer on II. MATHEMATICAL FORMULATION a mixed convection three dimensional heat transfer flow of a In this problem, we consider the effects of viscous and viscous incompressible fluid past an infinite vertical porous magnetic dissipation on steady two-dimensional magneto- plate in the presence of transverse periodic suction velocity. hydrodynamic mixed convection heat and mass transfer flow And the governing non-linear coupled partial differential of a Newtonian, electrically conducting and viscous equations were solved analytically by using perturbation incompressible radiative fluid over a porous vertical plate technique. * England and Emery [20] investigated the effect of radiation with induced magnetic field. The x -axis is taken along the on an optically thin gray gas bounded by a stationary plate. plate in vertically upward direction and y* -axis is normal to Raptis and Massalas [21] examined the radiation effects on it into the fluid region. It is assumed that the wall is the oscillatory fluid flow of an absorbing/emitting gray gas * with induced magnetic field and the solutions were obtained maintained at a constant temperature Tw and concentration * * by perturbation technique. Hossain et al. [22] investigated C higher than the ambient temperature T and radiation effects on the free convection flow of an optically w  incompressible fluid along a uniformly heated vertical infinite * concentration C respectively. A uniform magnetic field of plate with a constant suction. The effects of heat strength H is assumed to be applied normal to the direction generation/absorption and thermophoresis on hydromagnetic 0 flow with heat and mass transfer over a flat surface were

53 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

of flow. We have also made the following assumptions in our Subject to the boundary conditions are analysis to the flow model. * * * HCCTTvvuy *****  0,,,,0:0 • The equation of conservation of electric charge is 0 w xw   * * *****  0     HCCTTUuy x  0,,,: JDiv  0 where  ,, JJJJ zyx  • The of the flow is not taken to (7) be small and hence the induced magnetic field is not The local radiant for the case of an optically thin gray gas is negligible. expressed by • The plate is constrained to a constant suction velocity.

• The Soret and Dufour effects are neglected due to the qr ** 4 *4  4   TTa  (8) concentration of the diffusing species is assumed to be very y* small in comparison with the other chemical species. • All the fluid properties are assumed to be constant except It is assumed that the temperature differences within the the influence of the pressure gradient in the body force term. 4 flow are sufficiently small and that T * may be expressed as • There is a first order chemical reaction between the a linear function of the temperature. This is obtained by diffusing species and the fluid. * * expanding in a Taylor series about T and neglecting • Since the flow is assumed to be in the direction of x -axis,  the higher order terms, thus we get therefore all the physical quantities are functions of space co- *4 *3 ** 4 * *   34 TTTT  (9) ordinates in y and t only. Using the following non-dimensional quantities (10) and Then under usual Boussinesq’s approximation along with with the help of equations (8) and (9), the governing the assumptions considered to the flow, the equations that equations (3), (4), (5) and (6) reduce to describe the physical situation are given by (10) CONSERVATION OF MASS 2 * ud du dB v * 2 M GmGr   0  0 which implies that  vv 0 =a constant dy dy dy y* * * ** ** vc (1) yv0 uvTTCC p y,,, u  **  ** ,Sc  ,Pr  , GAUSS’S LAW OF MAGNETISM u0 Tww T C C D * H ** *** y * vg Tww T  vg C C   H  0 which takes the solution as  HH =a constant Gr  ,Gm  ,Prv , M  e 0 y* y 0 22me U0 v 0 U0 v 0 v0 (2)  H *264av * T *3 Q* K v U BRe x ,   ,,,Q k l Ec  0 ,Pr  v U v22 c v v ** m 0 CONSERVATION OF MOMENTUM 0 0 p 0 0 Cp Tw  T  * 2 * * * uu***** exHH0  (11) v*  g  T  T   g  C  C   *2  * 22 y y  y d2 d R Pr du  EcPr  dH  2 Pr   Ec Pr     0 (3) dy dy 4 dy Pr dy CONSERVATION OF ENERGY  m   (12) 2 22 TT  11q v u  H  2 r x Bd du dB v 2       yyc  c c y c y 2  M m PrPr m  0 p  y p p  p   dy dy dy (4) (13) CONSERVATION OF MAGNETIC INDUCTION 2 * *2 * d 1 d  * H 1  H u  k v x  x  H (5) 2 * *2 0 * Scdy dy y  e y y (14) CONSERVATION OF SPECIES DIFFUSION The corresponding boundary conditions are C*  C*2 * ** (6) v *  D 2 l  CCK   y y*

54 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

uy  B  0,1,1,0:0 Aexp  m y  A exp  2Pr y 18 2  8 m  uy  B  0,0,0,1: A9 exp  2 m1 y   A 10 exp   2 m2 y   A 11 exp   2 m3 y  (15)   (y ) exp  myEcA2   12 exp  mmyA1  2   13 exp  mmy2  3   Now, in order to solve the equations (11)-(14) with boundary  conditions given by (15), we apply the perturbation technique. A14 exp  m1  m 3  y  A15 exp  m1  Prm  y 2  By By  EcBy  OEc Aexp  m  Pr y  A exp  m  Pr y  01       16  2 mm   17  3   uy u y  Ecuy  OEc2  01       (27) 2 uy( ) 1  A1 exp  myA1   2 exp   myA2   3 exp   my3  y  01  y   Ec   y   O Ec  (16) Aexp  m y  Aexp  m y  A exp  2Pr y  2  30 3  19 2  20 m   y  01  y   Ec   y   O Ec  Aexp  2 m y   A exp   2 m y   A exp   2 m y  21 1 22 2 23 3 Ec  Aexp  m  m  y  Aexp  m  m  y Using (16) in equations (11)-(14) and equating (16) the 24  1 2  25  2 3  coefficients of the same degree terms and neglecting terms of  A26 exp  m1  m 3  y  A27 exp  m1  Prm  y 2  O Ec  , the following ordinary differential equations are Aexp  m  Pr y  A exp  m  Pr y 28  2 mm   29  3    obtained for u0,,, u 1 B 0 B1,,,, 0  1  0  1 : (28)    u0 u 0   Gr0  Gm 0  MB 0 (17) ByA( )7 exp  Prm yA  4 exp   myA1   5 exp   myA2   6 exp   my3   u1 u 1   Gr 1  Gm 1  MB 1  A43 exp Prm y   A31 exp   m3 y   A 32 exp   m2 y  (18)  A exp  2Pry   A exp   2 m y   A exp   2 m y    33 m 34 1 35 2 B0 Prmm B0   M Pr u0  Ec  A36 exp  2 m3 y   A 37 exp   m1  m 2  y (19)  Aexp  m  m y  Aexp  m  m y BPr B    M Pr u 38  2 3   39  1 3   1 mm1 1  Aexp  m  Pr y  A exp  m  Pr y Aexp  m  Pr y (20) 40  1 m   41  2 mm   42  3   R Pr Pr      0 (21) 0 0 0 (29) 4 GRAPHS AND TABLES   R Pr 2216 1Pr  1  1  PruB0    Pr/ Prm 0  4 14 (22)   12 0Sc  0  kSc  0  0 (23) 10 M=0.20, Ec=0.001   M=0.24, Ec=0.001 1Sc  1  kSc  1  0 (24) 8 M=0.26, Ec=0.001 M=0.20, Ec=0.00 where the primes denote differentiation with respect to y only. (u) Velocity 6 M=0.20, Ec=0.005 M=0.20, Ec=0.009 The boundary conditions (21) reduce to 4

0 1 0 BBuuy 1  0,0,0,0:0 2

BBuuy  0,0,0,1: 0 0 1 0 1 0 1 2 3 4 5 6 7 8 9 10 (25) y The solution of (17) to (24) under the transformed boundary conditions given by (25) yield Fig.1 Velocity profiles against coordinate y for different values of M and Ec, Sc=0.6, Pr=0.71, Prm=2.0, R=0.5, k=5, Gr=2 and Gm=2 (y ) exp m1 y  (26)

55 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

14 0.5

12

0 10 R=0.0, Prm=2.0 R=0.4, Prm=2.0 8 R=0.7, Prm=2.0 -0.5 R=0.5, Prm=2.5 6 R=0.5, Prm=3.0 R=0.0, Prm=2.0 R=0.4, Prm=2.0 Velocity(u) -1 4 R=0.7, Prm=2.0 R=0.5, Prm=2.5 2 -1.5 R=0.5, Prm=3.5 R=0.5, Prm=5.0 0 field(B) Magnetic Induced

-2 -2 0 1 2 3 4 5 6 7 8 y -2.5 Fig. 2 Velocity profiles against coordinate y for different values of R 0 1 2 3 4 5 6 7 8 and Prm, Sc=0.6, Pr=0.71, Ec=0.001, M=0.2, k=5, Gr=2 and Gm=2 y Fig. 5 Induced magnetic field against coordinate y for different 12 values of R and Prm with Sc=0.6, Pr=0.71, M=0.2, Ec=0.001, k=5, Gr=2 and Gm=2 10 Gr=2, Gm=2, Sc=0.22, K=5.0 Gr=2, Gm=2, Sc=0.30, K=5.0 0.2 Gr=2, Gm=2, Sc=0.60, K=5.0 8 0 Gr=2, Gm=2, Sc=0.60, K=1.0

Gr=2, Gm=2, Sc=0.60, K=3.0 -0.2 6 Gr=2.5, Gm=2, Sc=0.60, K=5.0 Gr=2, Gm=2.5,Sc=0.60,K=5.0 -0.4

4 Velocity(u) -0.6

-0.8 2 Gr=2.0, Gm=2.0, Sc=0.22, K=5.0 Gr=2.0, Gm=2.0, Sc=0.30, K=5.0 -1 Gr=2.0, Gm=2.0, Sc=0.60, K=5.0 0 -1.2 Gr=2.0, Gm=2.0, Sc=0.60, K=1.0

Induced Magnetic field(B) Magnetic Induced Gr=2.0, Gm=2.0, Sc=0.60, K=3.0 -2 -1.4 Gr=2.5, Gm=2.0, Sc=0.60, K=5.0 0 1 2 3 4 5 6 7 8 Gr=2.0, Gm=2.5, Sc=0.60, K=5.0 y -1.6 Fig. 3 Velocity profiles against coordinate y for different -1.8 values of Sc and k, Pr=0.71, Prm=2.0, Ec=0.001, M=0.2, 0 1 2 3 4 5 6 7 8 y k=5, Gr=2 and Gm=2 Fig.6 Induced magnetic field against coordinate y for different values of Gr, Gm, Sc and k with Pr=0.71, Prm=2.0, R=0.5, M=0.2, 0 Ec=0.001 and k=5

-0.5 1.4

-1 1.2 M=0.20, Ec=0.001 -1.5 M=0.24, Ec=0.001 1 M=0.26, Ec=0.001

M=0.20, Ec=0.0 )  M=0.20, Ec=0.001 -2 M=0.20, Ec=0.005 0.8 M=0.24, Ec=0.001 M=0.20, Ec=0.009 M=0.26, Ec=0.001 -2.5 M=0.20, Ec=0.004 Induced Magnetic field(B) Magnetic Induced 0.6 M=0.20, Ec=0.005 Temperature( M=0.20, Ec=0.006 -3 0.4

-3.5 0 1 2 3 4 5 6 7 8 9 10 0.2 y

Fig. 4 Induced magnetic field against coordinate y for different 0 values of M and Ec with Sc=0.6, Pr=0.71, Prm=2.0, R=0.5, k=5, 0 1 2 3 4 5 6 7 8 9 10 y Gr=2 and Gm=2 Fig. 7 Temperature profiles for different M and Ec

56 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

1.4 1

0.9 1.2 0.8 Sc=0.22, K=5.0 1 0.7 Sc=0.60, K=5.0 )  0.6 Sc=0.78, K=5.0 0.8 R=0.9, Prm=2.0 Sc=0.60, K=10.0 R=0.95,Prm=2.0 0.5 Sc=0.60, K=15.0 R=0.98,Prm=2.0 0.6 Sc=0.60, K=20.0 R=0.5, Prm=2.0 0.4 Concentration Temperature( R=0.5, Prm=3.5 0.3 0.4 R=0.5, Prm=5.0

0.2 0.2 0.1

0 0 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 y y Fig. 8 Temperature profiles for different R and Prm Fig. 9 Concentration profiles for different Sc

TABLE I THE NUMERICAL VALUES OF VELOCITY (U), INDUCED MAGNETIC FIELD (B) AND FLUID TEMPERATURE WHEN PRM  M

U B  Y M=0.50 M=0.75 M=0.50 M=0.75 R=0.5 1.0 0.0 0.0000 0.0000 0.0000 -0.0000 1.0000 1.0000 2.0 3.4360 3.2607 -0.2078 -0.2916 0.1961 0.1644 4.0 1.8516 1.7719 -0.3093 -0.4305 0.0384 0.0270 6.0 1.2335 1.2084 -0.2961 -0.4103 0.0075 0.0044 8.0 1.0591 1.0524 -0.2537 -0.3510 0.0015 0.0007 10.0 1.0146 1.0131 -0.2105 -0.2911 0.0003 0.0001

TABLE II COMPARISON OF VALUES OF THE FLOW VELOCITY WITH OBTAINED BY RAPTIS ET AL. (2003) AND SAHIN AHMED (2010) FOR PRM=0.1

RAPTIS ET AL. (2003) SAHIN AHMED (2010) PRESENT WORK Y R=0.1 R=0.8 R=1.0 R=0.1 R=0.8 R=1.0 R=0.1 R=0.8 R=1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 3.39344 2.83302 2.77543 3.35839 2.70542 2.68540 3.2584 2.6054 2.4854 4.0 1.88766 1.57742 1.53862 1.85232 1.51786 1.46272 1.8523 1.5179 1.4627 6.0 1.24757 1.13381 1.11986 1.24063 1.12212 1.10496 1.2406 1.1221 1.1050 8.0 1.06255 1.02759 1.02367 1.06134 1.02555 1.02107 1.0613 1.0256 1.0211 10.0 1.01511 1.00536 1.00439 1.01492 1.00503 1.00396 1.0149 1.0050 1.0040

TABLE III COMPARISON OF VALUES OF THE INDUCED MAGNETIC FIELD WITH OBTAINED BY RAPTIS ET AL. (2003) AND SAHIN AHMED (2010) FOR PRM=0.1

RAPTIS ET AL. (2003) SAHIN AHMED (2010) PRESENT WORK

Y R=0.1 R=0.8 R=1.0 R=0.1 R=0.8 R=1.0 R=0.1 R=0.8 R=1.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

2.0 -0.09437 -0.07526 -0.07392 -0.08750 -0.06367 -0.05916 -0.0875 -0.0637 -0.0592

4.0 -0.14786 -0.11214 -0.10870 -0.13880 -0.09687 -0.08914 -0.1388 -0.0969 -0.0892

6.0 -0.14358 -0.10543 -0.10150 -0.13538 -0.09161 -0.08389 -0.1354 -0.0916 -0.0839 8.0 -0.12354 -0.08931 -0.08594 -0.11669 -0.07775 -0.07101 -0.1167 -0.0777 -0.0710 10.0 -0.10263 -0.07472 -0.07070 -0.09699 -0.06421 -0.05859 -0.0970 -0.0642 -0.0586

III. RESULTS AND DISCUSSION To get a physical insight into the problem, the numerical section was performed and a set of results is reported evaluation of the analytical results reported in the previous graphically in Figs. 1-8 for the cases cooling Gr  0 of the

57 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

plate i.e. free convection currents convey heat away from the the induced magnetic field B negatively increasing, when plate into the boundary layer. During the numerical radiation parameter R increases from 0.0 (non-radiating) calculations the physical parameters are considered as Pr = through 0.4 and 0.7 (thermal conduction is dominant over 0.71 (air), Gr = 2 (thermal buoyancy forces are dominant over radiation) and magnetic Prandtl number Prm increases from the viscous hydrodynamic forces in the boundary layer), R = 2.5 through 3.5 and 5.0. The influence of thermal Grashof 0.5 < 1 (thermal conduction is dominant over the thermal number Gr, mass Grashof number Gm and Schmidt number radiation), Ec = 0.001  1 (Enthalpy difference is dominant Sc on the induced magnetic field B has been presented in over the kinetic energy). These results are obtained to Figure 6. It is observed that for all combinations of Gr, Gm illustrate the effects of various physical parameters like and Sc, the values of B remains negative i.e. there is a clear magnetic parameter M, radiation parameter R, chemical absolute increase in B when Gr or Gm or Sc increases. reaction parameter k, heat source parameter Q, Schmidt The behavior of fluid temperature by the influence of number Sc, Prandtl number Pr, Ec and magnetic parameter (M), radiation parameter (R) and magnetic field parameter Prm on the velocity, temperature magnetic Prandtl number (Prm) are shown in Figures 7 and and concentration fields. 8. It is seen that magnetic parameter M, radiation parameter Fig.1 describes the velocity profiles due to variation in R and magnetic Prandtl number Prm increase the fluid magnetic field parameter M and viscous heating Ec due to temperature. Finally, from figure 9, it is observed that cooling of the plate ( G  0 ). It is observed that with an concentration decreases with increase of Schmidt number Sc r or chemical reaction parameter k. The numerical values of increase in M=0.20 through 0.24 and 0.26 (all are weak fluid velocity (u) induced magnetic field (B) and temperature magneto hydrodynamic flow) there is an increase in velocity ( ) for different values of R, i.e. the flow is accelerated. Physically it meets the logic that Prm, M, Ec, k, Sc, Ec, Gr and Gm for the case Prm << M. the applied magnetic field B0 , is therefore moving effectively In this situation, it is found that when M increases from with the free stream. The resulting Lorentzian body force will M=0.50 to 0.75, there is a significant deceleration in the flow therefore not act as a drag force as in conventional MHD is noticed. It is due to the fact that, the presence of magnetic flows, but as an aiding body force. This result has also been field in an electrically conducting fluid introduces Lorentz found by Chamkha and Camille [23] and Zueco et al. [27]. force, which acts against the flow if the applied magnetic The maximum velocity is achieved at intermediate distances field is normal in direction as considered in the present study. from the plate into the boundary layer transverse to the wall. This type of resistive force tends to slow down in the flow And moving away from the wall, the free stream velocity field. These results concurred with many studies on magneto profiles converge in accordance with the boundary conditions hydrodynamic free convection boundary layer flows. On the imposed here. Fig. 2 reveals the influence of radiation other hand, with an increase in values of M from 0.50 to 0.75, parameter R on the flow transport for Pr = 0.71. It is found the induced magnetic field elevates absolutely, while the fluid that the velocity increases as radiation parameter R increases temperature decelerates when R increases from 0.5 to 1.0. i.e. the flow is accelerated. Physically it meets the logic that Finally, in Tables 1 and 2, we made a comparison of our when R increases, fluid performs a less interaction of results for flow velocity and induced magnetic field with radiation with the moment boundary layer. And with an those Raptis et al. [24] and Ahmed and Liu [19]. It is seen increase in Prm = 2.0 through 2.5 and 2.8 causes a noticeable that the results are in good agreement with their study. increase in the fluid velocity, in particular at short distance from the wall. Fig. 3 depicts the variation in velocity as Schmidt number Sc changes from 0.22 through 0.60 and 0.78 when chemical reaction parameter k = 5 and 10. It is observed that an increase in Sc and k, there is a clear increase in velocity i.e. the flow is accelerated. The effect of magnetic parameter M and viscous dissipative heat Ec on the induced magnetic field B has been presented in Fig. 4. And it is found that values of B remain negative i.e. induced magnetic flux reversal arises for all distances into the boundary layer, transverse to the plate. In all the combinations of M and Ec, values of B are at peak at short distance from the wall. And moving away from the wall it tends to zero. Therefore, when M increases from 0.20 through 0.24 and 0.26, and Ec increases from 0 through 0.005 and 0.009, the induced magnetic field found to be negatively increasing. Fig. 5 describes the effect of radiation parameter R and magnetic Prandtl number Prm on the induced magnetic field B for conducting air (Pr = 0.71) and weak magneto hydrodynamic flow (M=0.20). It is found that

58 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

IV. APPENDIX

11 m Sc  Sc22 4 kSc , m  Pr  Pr  R Pr 1222    1 2 2 mM1  Pr  1  Pr   4 Pr 3 2 m m m Gm m Pr  Gr m Pr  AA12mm, 123 2 2 3 2 2 m1 1  Prm  m1  Prm  M  1  m1  m2 1  Prm  m2  Prm  M  1  m2

MPrm A1 m 1 MPrm A2 m 2 MPrm A3 m 3 AAAA3  1 1  2  , 4  2 , A5  2 , A6  2 , AAAA7  4  5  6  m1 m 1 Prm m2 m 2 Prm m3 m 3 Prm

2 2 2 2 2 2 2 2 2 Pr Pr A PrA1 m 1   Pr A4 m 1 / Prmm  PrA2 m 2   Pr A4 m 1 / Pr  A  m 7 ,,A  A  8 R Pr 9 R Pr 10 R Pr 4Pr2  2Pr Pr  4m2  2 m Pr  4m2  2 m Pr  mm4 1 1 4 2 2 4 PrA2 m 2 Pr A2 m 2 / Pr 3 3  6 3 m  2Pr A1 A 2 m 1 m 2 2Pr A4 A 5 m 1 m 2 / Prm  AA11 , 12 RRPr 2 Pr 4m2  2 m Pr  m m    m  m Pr  3 3 441 2 1 2

2Pr A2 A 3 m 2 m 3 2Pr A5 A 6 m 2 m 3 / Prmm  2Pr A1 A 3 m 1 m 3  2Pr A4 A 6 m 1 m 3 / Pr  AA13 , 14 22RRPr Pr m m    m  m Pr  m m    m  m Pr  2 3 2 3 441 3 1 3

2Pr A4 A 7 m 1 2Pr A5 A 7 m 2 AA15 , 16 22RRPr Pr mPr    m  Pr  Pr  mPr    m  Pr  Pr  1 m 1 m 442 m 2 m

2Pr A6 A 7 m 3 A17  , AAAAAAAAAAA18 8 9 10 11 12 13 14 15 16 17  2 R Pr mmPr     Pr  Pr  33mm4 GrA m  Pr  GrA Pr AA18 2 m , 8 m 19 3 3 2 20 3 2 2 2 m2 1  Prm  m2  Prm  M  1  m2 8Prm  4 1  Prm  Pr m  2M  1  Prm 2Gr m A GrPr A 2Gr m A GrPr A AA1 9 mm9 , 2 10 10 21 3 2 2 22 3 2 2 8m1  41Pr m m1  2Prm  M  1  m1  8 m2  41Pr m m2  2Prm  M  1  m2 2Gr m A GrPr A A  3 11 m 11 23 3 2 2 8m3  4 1  Prmm m3  2Pr M  1  m3 Gr m m  A GrPr A A  1 2 12 m 12 24 322 m1  m 2   1  Prmm m1  m 2  Pr M  1  m1  m 2  Gr m m  A GrPr A A  2 3 13 m 13 25 322 m2  m 3   1  Prmm m2  m 3  Pr M  1  m2  m 3  Gr m m  A GrPr A A  1 3 14 m 14 26 322 m1  m 3   1  Prmm m1  m 3  Pr M  1  m1  m 3 

59 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

Gr mPr  A Gr Pr A A  1 mm15 15 27 322 m1 Prm    1  Prm m1  Prm   Pr m M  1  m1  Prm  Gr mPr  A Gr Pr A A  2 mm16 16 28 322 m2 Prm    1  Prm m2  Prm   Pr m M  1  m2  Prm  Gr mPr  A Gr Pr A A  3 mm17 17 29 322 m3 Prm    1  Prm m3  Prm   Pr m M  1  m3  Prm 

AAAAAAAAAAAA30 19  20 21 22 23 24 25 26 27 28 29 

MAMAPrm 30 Prm 19 MAMAPrm 21 Prm 22 A31  , A32  ,,A33 MA 20 A 34  , A35  m3  Prm m2  Prm 2m1  Prm 2m2  Prm

MAMAMAMAPrm 23 Prm 24 Prm 25 Prm 26 A36  , A37  , A38  , A39  2m3  Prm m1 m 2  Prm m2 m 3  Prm m1 m 3  Prm

MAMAMAPrm 27 Prm 28 Prm 29 A40  ,,A41  A42  m1 m2 m3 AAAAAAAAAAAAA  43 31 32 33 34 35 36 37 38 39 40 41 42 

REFERENCES [15] A. A. Raptis and V. M. Soundalgekar, “MHD flow past a steadily moving infinite vertical porous plate with constant heat flux,” Nuclear [1] V. M. Soundalgekar, “Hydromagnetic flow near an accelerated plate in the Engineering and Design, vol. 72, No. 3, 1982, pp. 373-379. presence of a magnetic field,” Applied Scientific Research, vol. 12, No.1, [16] M. M. Alom, I. M. Rafiqul and F. Rahman, “Study heat and mass transfer 1965, pp. 151-156 by mixed convection flow from a vertical porous plate with induced [2] V. M. Soundalgekar, S. K. Gupta, and N. S. Birajdar, “Effects of mass magnetic field, constant heat and mass fluxes,” Thammasat Int. J. Sci. transfer and free convection currents on MHD Stokes problem for a Tech, vol. 13, No. 4, 2008, pp. 1-13. vertical plate,” Nuclear Eng. Des., vol. 53, 1979, pp. 339-346. [17] O. A. Beg, A. Y. Bakier, V. R. Prasad, J. Zueco, and S. K. Ghosh, “Non [3] V. M. Soundalgekar, M. R. Patil, and M. D. Jahagirdar, “MHD Stokes similar, laminar study, electrically-conducting forced convection liquid problem for a vertical plate with variable temperature,” Nuclear Eng. metal boundary layer flow with induced magnetic field effects,” Int. J. Des., vol. 64, 1981, pp. 39-42. Thermal Sciences, vol. 48, 2009, pp. 1596-1606. [4] M. Kumari, and G. Nath, “Development of two-dimensional boundary [18] S. Ahmed, “The study of heat and mass transfer on free convective three- layer with an applied magnetic field due to an impulsive motion,” Indian dimensional unsteady flows over a porous vertical plate,” Journal of J. pure Appl. Math., vol. 30, 1999, pp. 695-708. Energy, Heat and Mass Transfer, vol.31, 2009, pp. 89-110. [5] M. M. Alam. and M. A. Sattar, “ Transient MHD heat and mass transfer [19] S. Ahmed, and I-Chung Liu, “Mixed convective three-dimensional heat and flow with thermal diffusion in a rotating system,” J. Energy Heat Mass mass transfer flow with transversely periodic suction velocity,” Int. J. trans, vol. 22m, 1999, pp. 9-21. Applied Mathematics and Mechanics, vol.6, 2010, pp. 58-73. [6] R. Muthucumaraswamy and Janakiraman, “MHD and Radiation effects on [20] W. G. England and A. F. Emery, “Thermal radiation effects on the laminar moving isothermal vertical plate with variable mass diffusion,” Theoret. free convection boundary layer of an absorbing gas,” Journal of Heat Appl. Mech., vol.33, No.1, 2006, pp.17-29. Transfer, vol. 91, 1969, pp. 37-44. [7] V. Rajesh and S. V. K. Varma, “Thermal diffusion and radiation effects on [21] A. Raptis and C. V. Massalas, “Magnetohydrodynamic flow past a plate by MHD flow past an impulsively started infinite vertical plate with variable the presence of radiation, Heat and Mass Transfer, vol. 34, 1998, pp. 107- temperature and mass diffusion,” JP Journal of Heat and Mass Transfer, 109. vol. 3, No. 1, 2009, pp. 17-39. [22] M. A. Hossain, M. A., Alim and D. A. S. Rees, “The effects of radiation on [8] A. G. Vijaya Kumar and S. V. K. Varma, “Thermal radiation and mass free convection from a porous vertical plate,” Int. J. Heat and Mass transfer effects on MHD flow past an impulsively started exponentially transfer, vol. 42, No. 1, 1999, pp. 181-191. accelerated vertical plate with variable temperature and mass diffusion, [23] A. J. Chamkha. and I. Camille I, “Effects of heat generation/absorption and Far East Journal of Applied Mathematics, vol.55, No.2, 2011, pp. 93- thermophoresis on hydromagnetic flow with heat and mass transfer over a 115. flat surface,” International Journal of Numerical Methods for Heat [9] E. L. Cussler, “Diffusion mass transfer in fluid systems,” 1988, Cambridge Fluid Flow,10, 2000, pp. 432-448. University Press. [24] J. Zueco, O. Anwar Bég, H. S. Takhar and V. R. Prasad, “Thermophoretic [10] P. L. Chambre, J. D. Young, “On the diffusion of chemically reactive hydromagnetic dissipative heat and mass transfer with lateral mass flux, species in alaminar boundary layer flow,” The physics of fluids, vol.1, heat source, Ohmic heating and thermal conductivity effects. Network 1958, pp. 48-54. Simulation Numerical Study,” Applied Thermal Engineering, 29, 2009, [11] A. Apleblat, “Mass transfer with a chemical reaction of the first order: pp. 2808-2815. Analytical solutions,” The Chemical Engineering Journal, vol.19, 1980, [25] A. A. Raptis, C. Perdikis and A. Leontitsis,“ Effects of radiation in an pp. 19-37. optically thin gray gas flowing past a vertical infinite plate in the presence [12] U. N. Das, R. K. Deka. and V. M. Soundalgekar, “Effects on mass of a magnetic field,” Heat and Mass Transfer, vol. 39, 2003, pp. 771-773. transfer on flow past an impulsively started infinite vertical plate with [26] M. A. Emad and A. A. Gamel El-Din, “Thermal radiation effects on constant heat flux and chemical reaction,” Foushung im ingenieurwesen, magnetohydrodynamic flow past a semi-infinite vertical plate in the vol.60, 1994, pp.284-287. presence of mass diffusion, Can. J. Phys., vol. 83, No. 3, 2005, pp. 243- [13] U. N. Das, R. K. Deka. and V. M. Soundalgekar, “Effects on mass 256. transfer on flow past an impulsively started infinite vertical plate with a [27] A. Orhan and K. Ahmet, “Radiation effect on MHD mixed convection flow chemical reaction,” The Bulliten GUMA, vol.5, 1999, pp. 13-20. about a permeable vertical plate, Heat and Mass Trnasfer, vol. 45, 2008, [14] M. B. Glauert, “The boundary layer on a magnetized plate,” Journal of pp. 239-246. Fluid Mechanics, vol. 12, No. 4, 1962, pp. 625-638.

60 International Conference on Advances in Marine, Industrial and Mechanical Engineering (ICAMIME'2014) April 15-16, 2014 Johannesburg (South Africa)

[28] S. K. Ghosh, O. A. Beg and J. Zueco, “Hydromagnetic free convection flow with induced magnetic field effects,” Physics and Astronomy, vol. 45, No.2, 2009, pp. 175-185. [29] Sahin Ahmed, “Induced magnetic field with radiating fluid over a porous vertical plate: Analytical study, Journal of Navel of Architecture and Marine Engineering,” vol. 7, No.2, 2010, pp. 61-72. [30] S. Ahmed, “Mathematical model on induced magnetic field with viscous/magnetic dissipation bounded by a porous vertical plate in presence of radiation,” Int. J. of Appl. Math and Mech, 8(1), 2012, pp: 86-104.

61