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    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278 Radiation Effect On Unsteady MHD Free Convective Flow Past An Exponentially Accelerated Vertical Plate With Viscous And Joule Dissipations Maitree Jana1, Sanatan Das2 and Rabindra Nath Jana3 1,3 (Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India) 2 (Department of Mathematics, University of Gour Banga, Malda 732 103, India)ABSTRACT The effects of radiation on unsteady MHD propulsion in astronautics, nuclear reactor thermalfree convection flow of a viscous incompressible dynamics and ionized-geothermal energy systems.electrically conducting fluid past an exponentiallyaccelerated vertical plate in the presence of a An excellent summary of applications can be founduniform transverse magnetic field on taking in Hughes and Young [1]. Abd-El-Naby et al.[2]viscous and Joule dissipations into account have have studied the finite difference solution of radiationbeen studied. The governing equations have been effects on MHD free convection flow over a verticalsolved numerically by the implicit finite difference plate with variable surface temperature. The transientmethod of Crank- Nicolsons type. The variations radiative hydromagnetic free convection flow past anof velocity and fluid temperature are presented impulsively started vertical plate with uniform heatgraphically. It is found that both the magnetic and mass flux have been investigated byfield and radiation decelerate the fluid velocity. Ramachandra Prasad et al. [3]. Takhar et al.[4] haveAn increase in Eckert number leads to rise in fluid analyzed the radiation effects on MHD freevelocity and temperature. Further, it is found that convection flow of a radiating fluid past a semi-the magnitude of the shear stress at the plate infinite vertical plate using Runge-Kutta-Mersonincreases with an increase in either radiation quadrature. Samria et al. [5] studied theparameter or magnetic parameter. The the rate of hydromagnetic free convection laminar flow of anheat transfer at the plate decreases with an elasto-viscous fluid past an infinite plate. Recently,increase in radiation parameter. the natural convection flow of a conducting visco- elastic liquid between two heated vertical platesKeywords: MHD free convection, radiative heat under the influence of transverse magnetic field hastransfer, Prandtl number and Grashof number, been studied by Sreehari Reddy et al.[6]. In all theseviscous and Joule dissipations. investigations, the viscous dissipation is neglected. The viscous dissipation heat in the natural convectiveI. INTRODUCTION flow is important, when the flow field is of extreme The most common type of body force on a size or at low temperature or in high gravitationalfluid is gravity defined in magnitude and direction by field. Such effects are also important in geophysicalthe corresponding acceleration vector. Sometimes, flows and also in certain industrial operations and areelectromagnetic effects are important. The electric usually characterized by the Eckert number. Aand magnetic fields themselves obey a set of physical number of authors have considered viscous heatinglaws, which are expressed by Maxwells equations. effects on Newtonian flows. Maharajan and GebhartThe solution of such problems requires the [7] reported the influence of viscous dissipationsimultaneous solution of the equations of fluid effects in natural convective flows, showing that themechanics and of electromagnetism. One special case heat transfer rates are reduced by an increase in theof this type of coupling is the field known as dissipation parameter. Israel-Cookey et al. [8] havemagnetohydrodynamics (MHD). The hydromagnetic investigated the influence of viscous dissipation andflows and heat transfer have become more important radiation on unsteady MHD free convection flow pastin recent years because of its varied applications in an infinite heated vertical plate in a porous mediumagricultural engineering and petroleum industries. with time dependent suction. Zueco Jordan [9] hasRecently, considerable attention has also been used network simulation method (NSM) to study thefocused on new applications of effects of viscous dissipation and radiation onmagnetohydrodynamics (MHD) and heat transfer unsteady MHD free convection flow past a verticalsuch as metallurgical processing. Melt refining porous plate. Suneetha et al. [10] have analyzed theinvolves magnetic field applications to control effects of viscous dissipation and thermal radiationexcessive heat transfer rate. Other applications of on hydromagnetic free convection flow past anMHD heat transfer include MHD generators, plasma impulsively started vertical plate. Suneetha et al.[11] 270 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278have studied the effects of thermal radiation on the the y - direction. As the plate are infinite long, thenatural convective heat and mass transfer of a viscous velocity and temperature distribution are functions ofincompressible gray absorbing-emitting fluid flowing y and t only. We assume that the magneticpast an impulsively started moving vertical plate with Reynolds number is much less than unity and henceviscous dissipation. Ahmed and Batin [12] have the induced magnetic field can be neglected inobtained an analytical model of MHD mixed comparison with the applied magnetic field in theconvective radiating fluid with viscous dissipative absence of any input electric field.heat.The objective of the present work is to study theeffects of radiation on unsteady MHD free convectiveflow of a viscous incompressible electricallyconducting fluid past an exponentially acceleratedvertical plate in the presence of a uniform transversemagnetic field on taking viscous and Jouledissipations into account. At time t  0 , both thefluid and plate are at rest with constant temperature T . At time t > 0 , the plate at y = 0 starts to movein its own plane with a velocity u0 ea t , a  being aconstant and the plate temperature is raised to Tw . Auniform magnetic field B0 is applied perpendicularto the plate. The governing equations have beensolved numerically using Crank- Nicolsons method.It is found that the fluid velocity u decreases with an Fig.1: Geometry of the problemincrease in either magnetic parameter M 2 orradiation parameter Ra . It is observed that the fluid Under the above assumptions and using the usualvelocity u increases with an increase in either Boussinesqs approximation, the equations of momentum and energy can be written asmagnetic parameter M 2 or Eckert number Ec . It isalso found that the fluid temperature  decreases u   2u   B0 2   2  g  (T  T )  u , (1)with an increase in radiation parameter Ra . Further, t y it is found that the absolute value of the shear stress 2 T  2T  u   qr x at the plate ( = 0) increases with an increase in  cp  k 2      B0 u  2 2 , (2) t y  y  yeither Ra or M 2 or Ec . The rate of heat transfer where u is the velocity in the x -direction, T the  d   at the plate ( = 0) decreases with an temperature of the fluid, g the acceleration due to  d  =0 gravity,  the coefficient of thermal expansion, increase in Ra . the kinematic coefficient of viscosity,  the fluid density, k the thermal conductivity, c p the specificII. FORMULATION OF THE PROBLEMAND ITS SOLUTION heat at constant pressure and qr the radiative heat Consider the unsteady hydrodynamic flow flux.of a viscous incompressible radiative fluid past an The initial and boundary conditions areexponentially accelerated vertical plate in the u  0, T  T for all y  0 and t  0,presence of a uniform transverse magnetic field on u   u0 eat , T  T at y  0 at t > 0, (3)taking into account viscous and Joule dissipations.The x -axis is taken along the vertical plate in an u  0, T  T as y   for t > 0.upward direction and y -axis is taken normal to the It has been shown by Cogley et al.[13] that in theplate (see Fig.1). At time t  0 , both the fluid and optically thin limit for a non-gray gas near equilibrium, the following relation holdsplate are at rest with constant temperature T . At  qr  e p time t > 0 , the plate at y = 0 starts to move in itsown plane with a velocity u0 eat , a  being a y 0   4(T  T ) K    d , 0  T  0 (4)constant and Tw is the plate temperature . A uniform where K is the absorption coefficient,  is themagnetic field B0 is applied perpendicular to the wave length, e p is the Planks function andplate. It is also assumed that the radiative heat flux in subscript 0 indicates that all quantities have beenthe x - direction is negligible as compared to that in 271 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278evaluated at the temperature T which is the of initial conditions. In all numerical solutions thetemperature of the plate at time t  0 . Thus, our continuous partial differential equation is replacedstudy is limited to small difference of plate with a discrete approximation. In this context thetemperature to the fluid temperature. word discrete means that the numerical solution isOn the use of the equation (4), equation (2) becomes known only at a finite number of points in the physical domain. The number of those points can be T  2T cp  k 2  4 T  T  I selected by the user of the numerical method. In t y general, increasing the number of points not only 2 increases the resolution but also the accuracy of the  u        B0 u , 2 2 (5) numerical solution. The discrete approximation  y  results in a set of algebraic equations that arewhere evaluated (or solved) for the values of the discrete  unknowns. The mesh is the set of locations where the  e p  0  I  K  0  T    d . 0 (6) discrete solution is computed. These points are called nodes and if one were to draw lines between adjacent nodes in the domain the resulting image wouldIntroducing non-dimensional variables resemble a net or mesh. yu tu 2 u T  T When time dependent solutions are   0 ,   0 , u  ,  , (7)   u0 Tw  T important, the Crank-Nicolson scheme has significantequations (1) and (5) become advantages. The Crank-Nicolson scheme is not significantly more difficult to implement and it has a u  2u  Gr  2  M 2u, (8) temporal truncation error that is O( 2 ) as   explained by Recktenwald [14]. The Crank-Nicolson   2  u 2   2 2 scheme is implicit, it is also unconditionally stable Pr   PrEc    M u   Ra , (9) [15- 17]. In order to solve the equations (8) and (9)   2       under the initial and boundary conditions (10), an  B0 2 implicit finite difference scheme of Crank-Nicolsonswhere M 2  is the magnetic parameter, type has been employed. The right hand side of the  u2 equations (8) and (9) is approximated with the 4IT  c p average of the central difference scheme evaluated atRa  the radiation parameter, Pr  the the current and the previous time step. The finite k k difference equation corresponding to equations (8) g  (Tw  T )Prandtl number, Gr  3 the Grashof and (9) are as follows: u0 ui, j 1  ui, j  2 u0number and Ec  the Eckert number. c p (Tw  T )  ui 1, j  2ui , j  ui 1, j  ui 1, j 1  2ui , j 1  ui 1, j 1     The corresponding boundary conditions for u and   2( ) 2 are M2 u  0,   0 for   0 and   0,  Gr 2  i, j 1  i, j 2  ui, j 1  ui, j ,  u  ea ,   1 at   0 for  > 0, (10) (11) u  0,   0 as    for  > 0, i, j 1  i , j Pr  a where a  2 is the accelerated parameter. u0  i 1, j  2i , j  i 1, j  i 1, j 1  2i , j 1  i 1, j 1       2(  ) 2 III. NUMERICAL SOLUTION  u 2  ROne of the most commonly used numerical methods  i 1, j  ui , j  is the finite difference technique, which has better  Pr Ec     2    M ui , j    i, j 1   i , j , stability characteristics, and is relatively simple,     2 accurate and efficient. Another essential feature of (12)this technique is that it is based on an iterative The boundary conditions (10) becomeprocedure and a tridiagonal matrix manipulation. ui,0  0, i,0  0 for all i  0,This method provides satisfactory results but it mayfail when applied to problems in which the u0, j  e a j  ,  0, j  1, (13)differential equations are very sensitive to the choice u N , j  0,  N , j  0, 272 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278where N corresponds to  . Here the suffix i number Gr , Eckert number Ec , acceleratedcorresponds to y and j corresponds to  . Also parameter a and time  in Figs.3-13. It is seen from   j 1   j and   i 1 i . Knowing the Fig.3 that the velocity u decreases with an increasevalues of  , u at a time  we can calculate the in magnetic parameter M 2 . This implies that the magnetic field has retarding influence on the velocityvalues at a time    as follows . We substitute field. The presence of a magnetic field in an i  1,2,..., N 1 , in equation (12) which constitute a electrically-conducting fluid introduces a force calledtri-diagonal system of equations, the system can be Lorentz force which acts against the flow if thesolved by Thomas algorithm as discussed in magnetic field is applied in the normal direction asCarnahan et al.[18]. Thus  is known for all values considered in the present problem. This type ofof  at time  . Then knowing the values of  and resistive force tends to slow down the motion ofapplying the same procedure with the boundary electrically conducting fluid. It is revealed from Fig.4conditions, we calculate, u from equation (11). This that the fluid velocity u decreases with an increase inprocedure is continued to obtain the solution till radiation parameter R . The radiation parameterdesired time  . The Crank-Nicolson scheme has a arises only in the energy equation in the thermal    truncation error of O  2  O  2 , i.e. the diffusion term and via coupling of the temperature field with the buoyancy terms in the momentumtemporal truncation error is significantly smaller. equation, the velocity is indirectly influenced by thermal radiation effects. An increase in R clearly reduces the fluid velocity. Fig.5 displays that the velocity u increases with an increase in Prandtl number Pr . Fig.6 shows that the velocity u increases with an increase in Grashof number Gr . It is due to the fact that an increase of Grashof number has a tendency to increase the thermal effect. This gives rise to an increase in the induced flow. Fig.7 reveals that the fluid velocity u increases with an increase in Eckert number Ec . It is seen from Figs.8 and 9 that the velocity u increases with an increase in either accelerated parameter a or time  . It is illustrated from Fig.10 that the fluid temperature  increases with an increase in magnetic parameter M 2 . Fig.11 display that the fluid temperature  Fig.2: Finite difference grids decreases with an increase in radiation parameter R . This is due to the fact that the radiation provides an The implicit method gives stable solutions additional means to diffuse energy. Fig.12 shows thatand requires matrix inversions which we have done at the fluid temperature  increases with an increase instep forward in time because this problem is an Prandtl number Pr . It is seen from Fig.13 that theinitial-boundary value problem with a finite number fluid temperature  increases with an increase inof spatial grid points. Though, the corresponding Eckert number Ec . This is due to the fact that Eckertdifference equations do not automatically guarantee number is the ratio of the kinetic energy of the flowthe convergence of the mesh   0 . To achieve to the boundary layer enthalpy difference. The effectmaximum numerical efficiency, we used the of viscous dissipation on flow field is to increase thetridiagonal procedure to solve the two- point energy, yielding a greater fluid temperature.conditions governing the main coupled governingequations of momentum and energy. Theconvergence (consistency) of the process is quitesatisfactory and the numerical stability of the methodis guaranteed by the implicit nature of the numericalscheme. Hence, the scheme is consistent. Thestability and consistency ensure convergence.IV. RESULTS AND DISCUSSION We have presented the non-dimensionalfluid velocity u and the fluid temperature  forseveral values of the magnetic parameter M 2 ,radiation parameter R , Prandtl number Pr , Grashof 273 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278 Fig.6: Velocity profiles for different Gr whenFig.3: Velocity profiles for different M 2 whenGr  5 , Ra  2 , a  0.5 , Ec  0.5 , Pr  0.25 and M 2  5 , R  2 , a  0.5 , Ec  0.5 , Pr  0.25 and  0.2   0.2Fig.4: Velocity profiles for different R when Fig.7: Velocity profiles for different Ec whenM 2  5 , Gr  5 , a  0.5 , Ec  0.5 , Pr  0.25 and M 2  5 , Gr  5 , a  0.5 , R  2 , Pr  0.25 and  0.2   0.2 Fig.8: Velocity profiles for different a whenFig.5: Velocity profiles for different Pr when M 2  5 , Gr  5 , R  2 , Ec  0.5 , Pr  0.25 andM 2  5 , Gr  5 , a  0.5 , Ec  0.5 and   0.2   0.2 274 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278Fig.9: Velocity profiles for different time  when Fig.12: Temperature profiles for different Pr whenM 2  5 , Gr  5 , a  0.5 , Ec  0.5 , Pr = 0.25 andR2 M 2 = 5 , Ec = 0.5 , R = 2 and  = 0.2Fig.10: Temperature profiles for different M 2 when Fig.13: Temperature profiles for different Ec whenR  2 , Ec  0.5 , Pr  0.25 and   0.2 M 2  5 , R  2 , Pr  0.25 and   0.2 Numerical values of the rate of heat transfer    0  at the plate   0 are presented in Tables 1 and 2 for several values of radiation parameter Ra , Prandtl number Pr , time  , magnetic parameter M 2 and Eckert number Ec . It is seen from Table 1 that the rate of heat transfer    0  at the plate   0 decreases with an increase in either Prandtl numbe Pr or time  . It is also seen that the rate of heat transfer    0  at the plate   0 increases with an increase in radiation parameter R . Further, it is seen from Table 2 that the rate of heat transfer    0  increases with an increase in magnetic parameterFig.11: Temperature profiles for different R when M 2 while it decreases with an increase in time  forM 2  5 , Ec  0.5 , Pr  0.25 and   0.2 fixed values of radiation parameter Ra . 275 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278 Table 1. Rate of heat transfer    0  at the plate   0 Pr  R 0.5 0.71 1.25 1.5 0.2 0.4 0.6 0.8 2 1.45461 1.44657 1.38121 1.28118 1.40435 1.39435 1.37435 1.35435 3 1.52588 1.47031 1.34243 1.23809 1.64988 1.64132 1.63123 1.62598 4 1.64211 1.57474 1.40540 1.29520 1.71077 1.70500 1.69877 1.69085 5 1.75135 1.69003 1.51410 1.40561 1.82159 1.81524 1.80802 1.80165 Table 2. Rate of heat transfer    0  at the plate   0 M2 Ec R 6 8 10 12 0.1 0.2 0.3 0.4 2 1.39768 1.41948 1.43180 1.43987 1.47126 1.46680 1.45627 1.44450 3 1.60759 1.61302 1.61810 1.62216 1.66308 1.64587 1.63179 1.61808 4 1.72446 1.73223 1.73737 1.74145 1.76149 1.75448 1.74503 1.73485 5 1.81566 1.81893 1.82207 1.82467 1.84589 1.83635 1.82810 1.82001The non-dimensional shear stress  x at the plate  0  due to the flow is given by  du  x    . (14)  d  0Numerical values of the non-dimensional shear stress x due to the flow at the plate (  0) are presentedin Figs.14-19 against radiation parameter R forseveral values of magnetic parameter M 2 , Grashofnumber Gr , Prandtl number Pr , Eckert number Ec , accelerated parameter a and time  . Fig.14shows that for the fixed values of the radiationparameter R , the absolute value of the shear stress x increases with an increase in magnetic parameterM 2 . On other hand, it is observed that the absolute Fig.14: Shear stress  x for different M 2 whenvalue of the shear stress  x decreases for R < 1 and Pr  0.71 , Gr  5 , a  0.5 , Ec  0.5 and   0.2it increases for R  1 for fixed values of M 2 . Fig.15reveals that the absolute value of the shear stress  xdecreases for R < 0.75 and it increases for R  0.75with an increase in Prandtl number Pr . Fig.16 showsthat the absolute value of the shear stress  xdecreases with an increase in Grashof number Gr . Itis illustrated from Fig.17 that the absolute value ofthe shear stress  x increases with an increase inaccelerated parameter a . It is revealed from Fig.18that the absolute value of the shear stress  xdecreases with an increase in Eckert number Ec .Fig.19 displays that the absolute value of the shearstresses  x increases with an increase in time  . Fig.15: Shear stress  x for different Pr when M 2  5 , Gr  5 , a  0.5 , Ec  0.5 and   0.2 276 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278Fig.16: Shear stress  x for different  when Fig.19: Shear stress  x for different  whenM  5 , Pr  0.71 , a  0.5 , Ec  0.5 and   0.2 2 M  5 , Pr  0.71 , Gr  5 , a  0.5 and Ec  0.5 2 V. CONCLUSION The radiation effects on unsteady MHD free convection flow of a viscous incompressible electrically conducting fluid past an exponentially accelerated vertical plate in the presence of a uniform transverse magnetic field by taking into account viscous and Joule dissipations have been studied. The governing equations have been solved numerically by the implicit finite difference method of Crank- Nicolsons type. It is found that the fluid velocity u decreases with an increase in either magnetic parameter M 2 or radiation parameter Ra . It is observed that the fluid velocity increases with an increase in either Grashof number Gr or time  . ItFig.17: Shear stress  x for different a when is also found that the fluid temperature  decreasesM 2  5 , Pr  0.71 , Gr  5 , Ec  0.5 and   0.2 with an increase in radiation parameter Ra . Further, it is found that the absolute value of the shear stress  x at the plate (  0) increases with an increase in either Ra or M 2 or  . The rate of heat transfer    0  at the plate (  0) increases with an increase in Ra or  . It is also found that the rate of heat transfer falls with increasing Eckert number Ec . REFERENCES [1] W.F. Hughes, and F. J. Young, The electro- Magneto-Dynamics of fluids, John Wiley & Sons, New York, USA (1966). [2] M. A. Abd-El-Naby, M.E. Elasayed, Elbarbary, Y. Nader, and Abdelzem, Finite difference solution of radiation effects on MHD free convection flow over a verticalFig.18: Shear stress  x for different Ec when plate with variable surface temperature. J. Appl. Math. 2, 2003, 65 - 86.M 2  5 , Pr  0.71 , Gr  5 , a  0.5 and   0.2 [3] V. Ramachandra Prasad, N. Bhaskar Reddy, and R. Muthucumaraswamy, Transient radiative hydromagnetic free convection flow past an impulsively started vertical plate with uniform heat and mass flux. J. App. Theo. Mech. 33(1), 2006, 31- 63. 277 | P a g e
    • Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.270-278[4] H.S. Takhar, R. S. R. Gorla, and V. M . Ramachandra Prasad, Effects of viscous Soundalgekar, Radiation effects on MHD dissipation and thermal radiation on free convection flow of a radiating fluid past hydromagnetic free convection flow past an a semi-infinite vertical plate. Int. J. impulsively started vertical plate, J. Naval Numerical Methods for Heat & Fluid Flow, Archit. Marine Eng., 2, 2008, 57-70. 6(2), 1996, 77 - 83. [11] S. Suneetha, N. Bhaskar Reddy, and V.[5] N.K. Samaria, M.U.S. Reddy, R. Prasad, Ramachandra Prasad, Effects of thermal H.N. Gupta, Hydromagnetic free convection radiation on the natural convective heat and laminar flow of an elasto-viscous fluid past mass transfer of a viscous incompressible an infinite plate, Springer Link, 179 (1), gray absorbing-emitting fluid flowing past 2004, 39-43. an impulsively started moving vertical plate[6] P. Sreehari Reddy, A. S. Nagarajan, with viscous dissipation, Thermal Sci., 13 M.Sivaiah, Natural convection flow of a (2), 2009, 71-181. conducting visco-elastic liquid between two [12] S. Ahmed, and A. Batin, Analytical model heated vertical plates under the influence of of MHD mixed convective radiating fluid transverse magnetic field, J. Naval Archit. with viscous dissipative heat, Int. J. Engin. Marine Eng., 2, 2008, 47–56. Sci. Tech., 2(9), 2010, 4902- 4911.[7] R.L. Maharajan, and B. B. Gebhart, [13] A.C. Cogley, W.C. Vincentine, and S.E. Influence of viscous heating dissipation Gilles, A Differential approximation for effects in natural convective flows, Int. J. radiative transfer in a non-gray gas near Heat Mass Trans., 32 (7), 1989, 1380–1382. equilibrium, AIAA Journal, 6 (1968) 551-[8] Israel – C. Cookey, A. Ogulu, and V.M. 555. Omubo-Pepple, Influence of viscous [14] G.W. Recktenwald, Finite-difference dissipation and radiation on unsteady MHD approximations to the heat equation, 2011. free convection flow past an infinite heated [15] W.F. Ames, Numerical Methods for Partial vertical plate in a porous medium with time Differential Equations, Academic Press, dependent suction, Int. J. Heat Mass Trans., Inc., Boston, Third edition, 1992. 46 (13), 2003, 2305-2311. [16] E. Isaacson, and H. B. Keller, Analysis of[9] J. Zueco Jordan, Network Simulation Numerical Methods, Dover, New York, Method Applied to Radiation and 1994. Dissipation Effects on MHD Unsteady Free [17] R. L. Burden, and J. D. Faires, Numerical Convection over Vertical Porous Plate, Analysis, Brooks/ Cole Publishing Co., New Appl.Math. Modeling, 31 (20) , 2007, 2019 – York, Sixth edition, 1997. 2033. [18] B. Carnahan, H.A. Luther, and J.O. Wilkes,[10] S. Suneetha, N. Bhaskar Reddy, and V. Applied Numerical Methods, John Wiley & Sons, New York (1969). 278 | P a g e