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International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of ...

International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.

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    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101197 | P a g eEffects of Hall Current on MHD Free and Forced Convectionflow of Newtonian fluid through a Porous medium in an Infinitevertical plate in presence of Thermal radiation heat transfer andsurface temperature oscillation1E.Neeraja and 2M.Veera Krishna1Department of Mathematics, M.S.Thakur college of sciences, Seawoods, Nerul, Navi Mumbai (Maharastra) -400706 (INDIA)2Department of Mathematics, Rayalaseema University, KURNOOL (A.P) - 518002 (INDIA)ABSTRACTIn this paper, we study the steady andunsteady magneto hydro dynamic (MHD)viscous, incompressible free and forcedconvective flow of an electrically conductingNewtonian fluid through a porous medium in thepresence of appreciable thermal radiation heattransfer and surface temperature oscillationtaking hall current into account. The fluid isassumed to be optically-thin and magneticReynolds number small enough to neglectinduced hydro magnetic effects. Secondary(cross-flow) effects are incorporated. Thegoverning equations are solved analytically usingcomplex variables. Detailed computations of theinfluence of governing parameters on theunsteady mean flow velocity (u1) and unsteadymean cross flow velocity (w1), the plate shearstresses for the unsteady main and the secondaryflow and also temperature gradients due to theunsteady main flow and the unsteady cross flow,are presented graphically and tabulated. Theclosed-form solutions reveal that the shear stresscomponent due to a steady mean flowexperiences a non-periodic oscillation whichvaries as a function of the Hartmann number (M2) and radiation parameter (K1). However theshear stress components due to main and crossflows for an unsteady mean flow are subjected toperiodic oscillation which depends on Hartmannnumber, inverse Darcy parameter, radiationparameter but also on the Prandtl number andfrequency of oscillation. Applications of themodel include fundamental magneto-fluiddynamics, MHD energy systems and magneto-metallurgical processing for aircraft materials.Keywords: steady and unsteady flows, thermalradiation heat transfer, hall current effects, free andforced convective flows, surface temperatureoscillation and porous medium.Keywords - About five key words in alphabeticalorder, separated by commaI. INTRODUCTIONSeveral authors have considered thermalradiaition effects on convection flows with andwithout magnetic fields. A seminal study wascommunicated by Audunson and Gebhart [5] whoalso presented rare experimental data for radiation-convection boundary layer flows of air, argon andammonia, showing that thermal radiation increasesconvective heat transfer by up to 40 %. Larson andViskanta [15] investigated experimentally theunsteady natural convection-radiation in arectangular enclosure for the case of fire-generatedthermal radiative flux, showing that thermalradiation dominates the heat transfer in theenclosure and alters the convective flow patternssubstantially. Helliwell and Mosa [12] reported onthermal radiation effects in buoyancy-driven hydromagnetic flow in a horizontal channel flow with anaxial temperature gradient in the presence of Jouleand viscous heating. Bestman [6] studied magnetohydro dynamic rarefied oscillatory heat transferfrom a plate with significant thermal radiation usinga general differential approximation for radiationflux and perturbation methods for small amplitudeoscillations. Yasar and Moses [23] developed a one-dimensional adaptive-grid finite-differencingcomputer code for thermal radiation magneto hydrodynamic (RMHD) simulations of fusion plasmas.Alagoa et al. [2] studied magneto hydro dynamicoptically-transparent free-convection flow, withradiative heat transfer in porous media with time-dependent suction using an asymptoticapproximation, showing that thermal radiationexerts a significant effect on the flow dynamics. El-Hakiem [10] analyzed thermal radiation effects ontransient, two dimensional hydro magnetic freeconvection along a vertical surface in a highlyporous medium using the Rosseland diffusionapproximation for the radiative heat flux in theenergy equation, for the case where free-streamvelocity of the fluid vibrates about a mean constantvalue and the surface absorbs the fluid with constantvelocity. Israel-Cookey et al. [14] described theeffects of viscous dissipation and thermal radiationon transient magneto hydro dynamic free-
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101198 | P a g econvection flow past an infinite vertical heated platein an optically thin environment with time-dependent suction showing that increased cooling(positive Grashof number) of the plate andincreasing Eckert number boost velocity profile andtemperature, a rise in magnetic field, thermalradiation and Darcian drag force decelerate the flowand increasing thermal radiation and magnetic fieldcool the flow in the porous medium. Other excellentstudies of thermal radiation-convection magnetohydro dynamics include Duwairi and Damseh [8],Raptis et al. [18] who considered axi-symmetricflow and Duwairi and Duwairi [9] who studiedthermal radiation heat transfer effects on the hydromagnetic Rayleigh flow of a gray viscous fluid.Vasil’ev and Nesterov [22] who presented a twodimensional numerical model for radiative-convective heat transfer in the channel of an MHDgenerator with a self-sustaining current layer.Duwairi [7] considered Ohmic and viscousdissipation effects on thermal radiating hydromagnetic convection. Ouaf [17] has consideredthermal radiation effects on hydro magneticstretching porous sheet flow. Aboeldahab andAzzam [1] have described the effects of magneticfield on hydro magnetic mixed free-forced heat andmass convection of a gray, optically-thick,electrically-conducting viscous fluid along a semi-infinite inclined plate for high temperature andconcentration using the Rosseland approximation.Zueco [24] has modeled using the networksimulation technique, the collective effects of walltranspiration, thermal radiation and viscous heatingeffects on hydro magnetic unsteady free convectionflow over a semi-infinite vertical porous plate for anon-gray fluid (absorption coefficient dependent onwave length). Alam et al. [3] have very recentlyinvestigated the influence of thermal radiation,variable suction and thermo phoretic particledeposition on steady hydro magnetic free-forcedconvective heat and mass transfer flow over aninfinite permeable inclined plate using theNachtsheim–Swigert shooting iteration techniqueand a sixth-order Runge-Kutta integration scheme.Ghosh and Pop [11] have studied thermal radiationof an optically-thick gray gas in the presence ofindirect natural convection showing that thepressure rise region leads to increase in the velocitywith an increase of radiation parameter. RecentlyAnwerbeg. O and S.K, Ghosh [4] investigated hydromagnetic free and forced convection of an optically-thin gray gas from vertical flat plate subject to asurface temperature oscillation with significantthermal radiation. In this paper, we study the steadyand unsteady magneto hydro dynamic (MHD)viscous, incompressible free and forced convectiveflow of an electrically conducting Newtonian fluidthrough a porous medium in the presence ofappreciable thermal radiation heat transfer andsurface temperature oscillation taking hall currentinto account.II. FORMULATION AND SOLUTION OF THEPROBLEMWe consider a two dimensional unsteadyMHD flow of a viscous incompressible electricallyconducting fluid occupying a semi infinite region ofspace bounded by porous medium through aninfinite vertical plate moving with the constantvelocity U, in the presence of a transverse magneticfield. The surface temperature of the plate oscillateswith small amplitude about a non-uniform meantemperature. The co-ordinate system is such that thex-axis is taken along the plate and y-axis is normalto the plate. A uniform transverse magnetic field Bois imposed parallel to y-direction. All the fluidproperties are considered constant except theinfluence of the density variation in the buoyancyterm, according to the classical Boussinesqapproximation. The radiation heat flux in the x-direction is considered negligible in comparison tothe y-direction. The unsteady MHD equationgoverning the fluid through a porous medium underthe influence of transverse magnetic field withbuoyancy force, then takes the vectorial form,  )(1. 2TTgBJqvqqtq(2.1)The equation of continuity is0. q (2.2)Ohm’s law for a moving conductor states BqEJ  (2.3)Maxwell’s electromagnetic field equations areJB e (Ampere’s Law) (2.4)tBE (Faraday’s Law) (2.5)0B. (Solenoidal relation i.e., magnetic fieldcontinuity) (2.6)0J . (Gauss’s Law i.e., Conservation of electriccharge) (2.7)In which EB,q, and J are, respectively,the velocity vector, magnetic field vector, electricfield vector and current density vector, T is thetemperature of the fluid, T is the temperature faraway the plate , g is the gravitational acceleration , is the coefficient of volume expansion,  is thedensity of fluid,  is the electrical conductivity, eis the magnetic permeability of the fluid, t is time, vis dynamic viscosity and Bo is the magnetic fluxdensity component normal to the plate surface .According to Shercliff [20] and Hughes and young[13], the following assumptions are compatible withthe fundamental equations (2.1) to (2.7) of magnetohydro dynamics. zx BBBwu ,,),,0,( 0 Bq (2.8)
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101199 | P a g e zxzyxJJEEE ,0,),,,(  JE (2.9)Where, u and w are the velocitycomponents along the x-direction and z-directionrespectively. Since magnetic Reynolds number isvery small for metallic liquid or partially ionizedfluid the induced magnetic field produced by theelectrically conducting fluid is negligible. Also asno external electric field is applied, the polarizationvoltage is negligible so that following Meyer [16],E=0. Ohmic and viscous heating effects are alsoneglected. The appropriate boundary conditions tobe satisfied by equations (2.1) and (2.3) are;0,0,0;0),1)((,0,  yatwuyatexTTwUu tiw(2.10)Where Φ designated wall-free streamtemperature difference,vU i.e., dimensionlessvelocity ratio and  is the frequency of oscillationin the surface temperature of the plate. Theconditions (2.10) suggest solutions to equations(2.1) to (2.3) for the variables , vu and  of theform,, 10ueuu ti (2.11), 1weww tio (2.12)))(( 1  tiowex  (2.13)Since )0,,0( 0BB  and q = (u, 0, w), When thestrength of the magnetic field is very large, thegeneralized Ohm’s law is modified to include theHall current, so thatH)qμ(EσHJHτωJ e0ee (2.14)Where, q is the velocity vector, H is themagnetic field intensity vector, E is the electricfield, J is the current density vector, e is thecyclotron frequency, e is the electron collisiontime,  is the fluid conductivity and, eμ is themagnetic permeability. In the above equation theelectron pressure gradient, the ion-slip and thermo-electric effects are neglected. We also assume thatthe electric field E=0 under assumptions reduces towHσμJmJ 0ezx  (2.15)uHσμJmJ 0exz  (2.16)On solving these equations (2.15) and (2.16), wehave,)(,0),( umwm1HσμJJmuwm1HσμJ 20ezy20ex  (2.17)Where eeωτm  is the hall parameter.For the case of an optically-thin gray gas, thethermal radiation flux gradient may be expressed asfollows (Siegel and Howell [21]))(*444TTayqr  (2.18)and rq is the radiative heat flux, a isabsorption coefficient of the fluid and * is theStefan-Boltzmann constant. We assume that thetemperature differences within the flow aresufficiently small such that 4T may be expressed asa linear function of the temperature. This isaccomplished by expanding 4T in a Taylor seriesabout T and neglecting higher order terms, leadingto:43434  TTTT (2.19)Making use of the equation (2.17) thecomponents 0u , 0w and 0 represent the steady meanflow and temperature fields, and satisfy thefollowing equations:000ywxu(2.20)0000202)()(0 ukumwm1Hσμxgyuv 20ew (2.21)000202)(0 wkmuwm1Hσμywv 20e  (2.22)20210yqcycK rpp(2.23)Where K designates thermal conductivityand cp is the specific heat capacity under constantpressure. The corresponding boundary conditionsarewooTTwUu  ,0, at 0y(2.24) TTwu oo,0,0 at y(2.25)Again making use of the equation (3.17),the components 1u , 1w and 1θ represent the steadymean flow and temperature fields, and satisfying thefollowing equations:0ywxu 11(2.26)111112121)()( ukumwm1Hσμxgyuvui 20ew (2.27)111121211)(wkmuwm1Hσμywvwi 20e   (2.28)rp212p yqρc1yθρcKtΦ(2.29)The corresponding boundary conditions arewTTwUu  ,0, 11 at 0y(2.30) TTwu ,0,0 11 at y(2.31)
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101200 | P a g eProceeding with the analysis we introducedimensionless quantities to normalize the flowmodel:vUyyUewwUeuuUwwUuu,,,,11110000  ,,ULevθθ,ULvθθ,vUtt 11002 ,)(42LUxvgGr w,2Uv 23212212202 16,,KUTvaKkDUvBM TTTTθθw10Where Gr is Grashof number, 2M is theHartmann (magneto hydro dynamic number), K1 isthe thermal radiation-conduction number, K isthermal conductivity and 1 is dimensionlesstemperature D-1is the inverse Darcy parameter.Using equation (2.29) together with the equations(2.18) and (2.19) the dimensionless form of equation(2.23) becomes:001202 Kdyd(2.32)Making use of non-dimensional variables,together with equations (2.18) and (2.19) thedimensionless form of equation (2.29) becomes:0Pr)( 11212 iKdyd(2.33)We are introducing complex variables,0 Fiwu o  (2.34)Hiwu  11(2.35)where 1i .Combining equations (2.21) and (2.22)with the help of (2.34), the differential equation forsteady mean flow in dimensionless form becomes:oGrFDmMdyFd )1( 12222(2.36)Combing equations (2.27) and (2.28) withthe help of (2.35), the differential equation forunsteady mean flow in dimensionless form reducesto:112222)1(  GrHiDmMyH (2.37)The corresponding boundary conditions for steadymean flow (non-dimensional) are,1ou ,0ow 1o at 0y (2.38),0ou ,0ow 0o at y (2.39)The corresponding boundary conditions forunsteady mean flow (non-dimensional) are,11 u 01 w , 11  at 0y (2.40),01 u 01 w , 01  at y (2.41)The boundary conditions (2.38), (2.39), (2.40) and(2.41) can be written subject to equation (2.34 and2.35) as follows:1F , 10  at 0y (2.42)0F , 00  at y (2.43)and,1H 11  at 0y(2.44),0H 01  at y (2.45)Equations (2.36) and (2.32) subjects to theboundary conditions (2.42) and (2.43) can be solvedand the solution for the steady mean flow can beexpressed as:ykyDmMyDmMeeDmMKGreyiwyuF1122122)1(1221)1(00)1()()((2.46)in which yke 1 = )(y0 .Equations (2.36) and (2.33) subjects to theboundary conditions (2.44) and (2.45) may also besolved yielding the following solution for unsteadymean flow: yiDCyiDCyiDCeeGrietyiwtyuH)()(22)(11221111),(),((2.47)andPriK1ety  1),( .Where, the functions 0 and 1 denote thetemperature fields due to the main flow and crossflows, respectively. Of interest in practical MHDplasma energy generator design are thedimensionless shear stresses at the plate, which maybe defined for steady and unsteady mean flow,respectively as follows:1122122112201)1(1|KDmMDmMKGrDmMdydFy(2.48)    221122110()(|iDCiDCGriiDCyHy(2.49)It is evident from equations (2.48) and(2.49) that the shear stress component due to themain flow for the steady mean flow equations (2.48)and the shear stress components due to main andcross flows given by equation (2.49) do not vanishat the plate. Inspection of these expressions alsoreveals that the shear stress component as definedby equation (2.48) due to a steady mean flow issubjected to a non-periodic oscillation that dependson Hartmann number, inverse Darcy parameter andradiation- conduction parameter. In contrast to this,
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101201 | P a g ethe shear stress components as computed in equation(2.49) due to the main and cross flows for anunsteady mean flow are subjected to periodicoscillation which is a function of not only Hartmannnumber and radiation- conduction parameter, butalso the Prandtl number and the frequency ofoscillation. The shear stress for equation (2.48) willvanish at the plate (y=0) at a critical value of thefree convection parameter i.e. Grashof number,defined by the condition:  122112211DmMKDmMGrCrit(2.50)The shear stress for equation (2.48) willvanish at the plate (y=0) when    212111 )( DDiCCiDCGrCrit  (2.51)Also of interest in plasma MHD generatordesign is the dimensionless temperature gradient atthe plate. This can be shown to take the form, for theunsteady main flow, as follows:100| Kdydy (2.52)For the unsteady cross flow thedimensionless temperature gradient at the plate(y=0) isPr| 101iKdydy (2.53)Comparing equations (2.51) and (2.52) it isimmediately deduced that in the absence of anoscillating surface i.e., for  =0, the dimensionlesstemperature gradient due to a steady and unsteadymean follows are identical.III. RESULTS AND DISCUSSIONThe flow governed by the non-dimensionalparameters namely viz., Hartmann number M,inverse Darcy parameter D-1,hall parameter m, K1 isthe thermal radiation-conduction number, surfacetemperature oscillation i.e.,  and Grashof numberGr. Selected computations for the velocity andtemperature fields have been provided in figures (1-17) & (18-24) respectively. Default values of thedimensionless thermo-physical parameters werespecified as follows, unless otherwise indicated: M= 2, m=1, K1 =1,  =2, Gr = 2 and Pr = 0.025which correspond to weak free convection currentsin liquid metal flow under strong magnetic fieldwith equal thermal radiation and thermal conductioncontribution, with surface temperature oscillation.Computations for the shear stresses at the plate areprovided in tables (1-3) and for temperature gradientat the plate in tables (4-7).We note that steady mean flow is simulatedfor which there will be no surface temperatureoscillation i.e.  = 0. The magnitude of the velocityreduces with increase in the intensity of themagnetic field. i.e., Mean flow velocity, (u0) iscontinuously reduced with increasing M. Thetransverse magnetic field generates a retarding bodyforce in the opposite direction to the flow whichserves to decelerate the flow. As such magnetic fieldis an effective regulatory mechanism for the regime(Fig. 1). The magnitude of the velocity (u0) reduceswith increase in the inverse Darcy parameter D-1.Lower the permeability of the porous medium lesserthe fluid speed in the entire fluid region (Fig. 2). Anincrease in radiation-conduction number, K1 has anadverse effect on the velocity due to steady meanflow (u0) for all values of y. K1 represents therelative contribution of thermal radiation heattransfer to thermal conduction heat transfer. ForK1 < 1 thermal conduction exceeds thermal radiationand for K1 > 1 this situation is reversed. For K1 = 1the contribution from both modes is equal. In allcases steady mean flow velocity is a maximum atthe plate (y = 0) and decays smoothly to the lowestvalue far from the plate (Fig. 3). Conversely anincrease in free convection parameter, Grashofnumber Gr, boosts the steady mean flow velocity,(u0). Increasing thermal buoyancy (Fig. 4) thereforeaccelerates the mean steady flow in particular at andnear the plate surface. Fig.5 exhibits the magnitudeof the velocity (u0) increases with increase in thehall parameter m. Figures (6 to 17) correspond to theunsteady mean flow distributions due to the mainflow (u1) and also cross flow (w1). The frequency ofoscillation is prescribed as 2 unless otherwiseindicated. Magnetic field effects on the unsteadymean main flow velocity component (u1) and theunsteady mean cross flow velocity component (w1),respectively, are presented in figures 6 and 12. Themagnitude of the velocity (u1) is strongly reducedwith increase in the intensity of the magnetic fieldparameter M, with the maximum effect sustained atthe plate surface where peak (u1) value plummetsfrom 0.037 for M = 5 to 0.0035 for M = 10 (Fig. 6).Unsteady mean cross flow velocity component (w1)is also reduced in magnitude with a rise in M. For M= 10 cross flow velocity is almost totally suppressedat all locations transverse to the wall. With weakermagnetic field the backflow presence is substantial.As such very strong magnetic field may be appliedin operations to successfully inhibit backflownormal to the plate surface (Fig. 12). The magnitudeof the velocities (u1 & w1) reduces with increase inthe inverse Darcy parameter D-1. Lower thepermeability of the porous medium the fluid speedretards in the entire fluid region (Fig. 7 & 13).Increasing radiation-conduction number, K1, as withthe steady mean flow (u0) discussed in figure 3,again has an opposing influence on unsteady mainflow velocity (u1). The profiles are similar to thosefor steady mean flow. Conversely cross flowvelocity (w1) is positively affected by an increase inK1 as depicted in figure (5). Profiles become lessnegative as K1 rises from 1 through 2 to 3. As suchbackflow is inhibited considerably with increasing
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101202 | P a g ethermal radiation. We observe that an unsteadymean cross flow velocity trough occurs near theplate surface at y ~1. At the plate cross flow velocityvanishes and at large distance from the plate againvanishes (Fig. 8 & 14). Figures 9 and 15 show theeffect of Grashof number on the unsteady meanmain flow velocity component (u1) and the unsteadymean cross flow velocity component (w1),respectively. In both cases increasing buoyancy hasan adverse effect on both velocity fields. Mean flowvelocity remains however positive always i.e., thereis no presence of backflow. Cross flow velocity isconsistently negative throughout the regimeindicating that backflow is always present. Lowerbuoyancy forces i.e. smaller Grashof number servesto reduce the backflow. Very little effect ofoscillation frequency ( ) is computed, in figure 10,on the unsteady mean main flow velocitycomponent, (u1) which decreases very slightly as rises from 2 through 4 to 6. However frequencyexerts a marked influence on the unsteady meancross flow velocity component (w1), as shown infigure 16 which is decreased substantially with arise in  . Backflow is therefore augmented withincreasing oscillation frequency, with the maximumeffect at close proximity to the plate. In engineeringdesign applications, therefore this region (y ~ 1)would be of particular interest in controllingbackflow during MHD generator operations. Themagnitudes of the velocities (u1 and w1) increasewith increase in the hall parameter m (Fig. 11 & 17).The magnitude of the temperature 0 reduces withincrease in radiation-conduction number K1 due tosteady mean flow (Fig. 18). The magnitude of thetemperature 1 reduces with increase in K1 and Prdue to unsteady mean flow (Fig. 19 & 20). Similarlythe magnitude of the temperature 1 reduces withincrease in K1 and enhances with increase in Pr dueto unsteady cross flow (Fig. 21 &22). Themagnitude of the temperature 1 reduces in5.20  y and y=4.5 and enhances within thedomain 2.5<y<4.5 with increase in surfacetemperature oscillation  due to unsteady meanflow (Fig. 23). Likewise the magnitude of thetemperature 1 enhances in 5.15.0  y andreduces within the domain 5.45.2  y withincrease in surface temperature oscillation  due tounsteady cross flow (Fig. 24).Tables (1-3) also show the combinedinfluence of several of the dimensionless parameterson the shear stress at the plate. An increase in M andinverse Darcy parameter D-1causes increase in theshear stress at the plate y=0 due to a steady meanflow (u0) i.e.  values become increasinglynegative. Increasing radiation-conduction parameter,K1 , hall parameter m and Grashof number Gr alsodecreases the shear stress i.e. decelerates the flow, inconsistency with the velocity distribution shown infigure (1-5) (Table. 1). Table (2) indicates that shearstress at the plate with unsteady mean flow due tomain flow (u1) reduces with increase in Gr,  andm, and is greatly increased with an increase in thesquare root of the Hartmann magneto hydrodynamic number (M), inverse Darcy parameter D-1and radiation-conduction parameter, K1. Unsteadymean flow is therefore enhanced strongly with anincrease in magnetic field strength. An increase inthermal radiation-conduction number (K1) alsodecreases values but very slightly indicating thatthermal radiation has a very weak effect on flow atthe plate surface. On the other hand, the shear stressat the plate with unsteady mean flow due to crossflow direction (w1) as shown in table (3) isconsiderably increased with square root of theHartmann magneto hydro dynamic number (M)since the magnetic retarding force acts in the samedirection as this flow component and boostssecondary (cross) flow. A rise in thermal radiation-conduction number (K1) also increases the slightlysince values become less negative indicating thatbackflow is resisted with greater thermal radiationcontribution. The shear stress enhances due to crossflow with increasing M, D-1and , and reduces withincreasing m, K1 and Gr. An increase in frequencyof oscillation ( ) causes the temperature gradient atthe plate due to unsteady mean flow i.e. to decreaseslightly, as indicated in table 4; converselyincreasing thermal radiation-conduction parameter(K1) has a strong positive effect inducing a majorincrease in Temperature gradient at the plate due tounsteady cross flow affected more strongly (Table.5) with an increase in frequency of oscillation ( ),being reduced from -0.03531 for  = 2, to -0.07036for  = 4 and to the lowest value of -0.10491 for = 6 (all at K1 = 1). In contrast to this thetemperature gradient, is increased somewhat with anincrease in thermal radiation-conduction parameter(K1), for any value of  . The combined influenceof thermal radiation-conduction parameter (K1) andPrandtl number (Pr) on the temperature gradient atthe plate due to unsteady main flow is shown in(table. 6). An increase in thermal radiation-conduction parameter (K1) substantially decreasesvalues for all Prandtl numbers. An increase inPrandtl number also decreases values. Lower Prvalues imply a higher thermal conductivity andcorrespond to liquid metals (Pr<<1). Pr = 1 impliesthat energy and momentum are diffused at the samerate and the lowest value of occurs for Pr =1, at agiven value of K1. Finally in table 7, temperaturegradient at the plate due to unsteady cross flow, isobserved to be increased with an elevation inthermal radiation-conduction parameter (K1) at anyPrandtl number (Pr). Conversely increasing Prandtlnumber markedly reduces the value of at a fixedvalue of K1. A similar trend was observed in the
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101203 | P a g eearlier studies of for example Duwairi and Damseh[8], Raptis et al. [18] and more recently by Samad etal. [19]. The analytical solutions presented thereforeprovide a succinct confirmation of earlier results andalso reveal some new interesting phenomena in theinteraction of radiation, magnetic field and effect ofhall current, porous medium and periodicity ofsurface temperature.IV. FIGURES AND TABLES3.1. Velocity Distributions due to steady meanflowFig. 1: Velocity distribution due to a steady meanflow for various M with 0 , Gr=2, D-1=1000,m=1, K1=1, Pr=0.025Fig. 2: Velocity distribution due to a steady meanflow for various D-1with 0 , Gr=2, m=1, M=2,K1=1, Pr=0.025Fig. 3: Velocity distribution due to a steady meanflow for various K1 with 0 , Gr=2, m=1, M=2,D-1=1000, Pr=0.025Fig. 4: Velocity distribution due to a steady meanflow for various Gr with 0 , M=2, m=1 , D-1=1000, K1=1, Pr=0.025Fig. 5: Velocity distribution due to a steady meanflow for various m with 0 , M=2, Gr=12, D-1=1000, K1=1, Pr=0.0253.2. Velocity Distributions due to unsteady meanand cross flow
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101204 | P a g eFig. 6: Unsteady mean flow distribution due to meanflow for various M with 2 , Gr=2, m=1, D-1=1000, K1=1, Pr=0.025Fig. 7: Unsteady mean flow distribution due to meanflow for various D-1with 2 , Gr=2, M=2, m=1,K1=1, Pr=0.025Fig. 8: Unsteady mean flow distribution due to meanflow for various K1 with 2 , Gr=2, m=1, D-1=1000, M=2, Pr=0.025Fig. 9: Unsteady mean flow distribution due to meanflow for various Gr with 2 , M=2, D-1=1000,m=1, K1=1, Pr=0.025Fig. 10: Unsteady mean flow distribution due tomean flow for various  with 2Gr , M=2, m=1,D-1=1000, K1=1, Pr=0.025Fig. 11: Unsteady mean flow distribution due tocross flow for various m with 2Gr , M=2,  =2,D-1=1000, K1=1, Pr=0.025
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101205 | P a g eFig. 12: Unsteady mean flow distribution due tocross flow for various M with 2 , Gr=2, m=1,D-1=1000, K1=1, Pr=0.025Fig. 13: Unsteady mean flow distribution due tocross flow for various D-1with 2 , Gr=2, m=1,M=2, K1=1, Pr=0.025Fig. 14: Unsteady mean flow distribution due tocross flow for various K1 with 2 , Gr=2, m=1,D-1=1000, M=2, Pr=0.025Fig. 15: Unsteady mean flow distribution due tocross flow for various Gr with 2 , M=2, m=1, D-1=1000, K1=1, Pr=0.025Fig. 16: Unsteady mean flow distribution due tocross flow for various  with Gr=2, M=2, m=1, D-1=1000, K1=1, Pr=0.025Fig. 17: Unsteady mean flow distribution due tocross flow for various m with Gr=2, M=2,  =2, D-1=1000, K1=1, Pr=0.0253.3. Temperature Distributions due to steadyflow
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101206 | P a g eFig. 18: Temperature distribution due to a steadymean flow for various K1 with  =03.4. Temperature Distributions due to unsteadymean and cross flowFig. 19: Temperature distribution due to a unsteadymean flow for various K1 with  =2, Pr=0.025Fig. 20: Temperature distribution due to a unsteadymean flow for various K1 with  =2, Pr=0.025Fig. 21: Temperature distribution due to a unsteadymean flow for various Pr with  =2, K1=1Fig. 22: Temperature distribution due to a unsteadymean flow for various Pr with  =2, K1=1Fig. 23: Temperature distribution due to a unsteadymean flow for various  with Pr=0.025, K1=1
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101207 | P a g eFig. 24: Temperature distribution due to a unsteadymean flow for various  with Pr=0.025, K1=1V. CONCLUSIONSExact solutions have been derived usingcomplex variables for the transient Magneto hydrodynamic convection flow of an electrically-conducting, Newtonian, optically-thin fluid from aflat plate with thermal radiation and surfacetemperature oscillation effects. Our analysis hasshown that:1. Steady mean flow velocity, (u0), is decreasedwith increasing thermal radiation (K1), inverseDarcy parameter D-1and magnetic hydrodynamic parameter (M), increased withGrashof number (Gr) and hall parameter m.2. Unsteady mean flow velocity (u1) is reducedwith increasing radiation-conduction number,K1 , slightly decreases with increasingfrequency of oscillation (  ) and also falls witha rise in Grashof number, inverse Darcyparameter D-1and magnetic hydro dynamicparameter, M. Mean flow velocity (u1) isenhanced with increasing hall parameter m.3. Conversely cross flow velocity (w1) isincreased with a rise in K1 but decreased with arise in Gr, D-1and  . Strong magnetic fieldalso practically eliminates backflow.4. The magnitude of the temperature 0 reduceswith increase in radiation-conduction numberK1 due to steady mean flow.5. The magnitude of the temperature 1 reduceswith increase in K1 and Pr due to unsteadymean flow. The magnitude of the temperature1 reduces with increase in K1 and enhanceswith increase in Pr due to unsteady cross flow.6. The magnitude of the temperature 1 reducesin 5.20  y and y=4.5 and enhances withinthe domain 2.5<y<4.5 with increase in surfacetemperature oscillation  due to unsteadymean flow. Likewise the magnitude of thetemperature 1 enhances in 5.15.0  y andreduces within the domain 5.45.2  y withincrease in surface temperature oscillation due to unsteady cross flow.7. An increase in M and inverse Darcy parameterD-1causes increase in the shear stress at theplate y=0 due to a steady mean flow (u0).Increasing radiation-conduction parameter, K1,hall parameter m and Grashof number Gr alsodecreases the shear stress.8. The shear stress at the plate with unsteadymean flow due to main flow (u1) reduces withincrease in Gr, m and  , and is greatlyincreased with an increase in the square root ofthe Hartmann magneto hydro dynamic number(M), inverse Darcy parameter D-1andradiation-conduction parameter, K1.9. The shear stress enhances due to cross flowwith increasing M, D-1and , and reduceswith increasing K1, m and Gr.10. Temperature gradient at the plate due tounsteady cross flow is reduced substantiallywith an increase in frequency of oscillation butelevated with an increase in thermal radiation-conduction parameter (K1), for any value of .11. An increase in thermal radiation-conductionparameter (K1) reduces strongly as does anincrease in Prandtl number (Pr).12. Temperature gradient at the plate due tounsteady cross flow, increases with thermalradiation-conduction parameter (K1) butreduces with a rise in Prandtl number.Table 1: Shear stress( ) at the plate y=0 due to a steady mean flow (u0) for various values of thermal radiation-conduction number (K1), square root of the Hartmann magneto hydro dynamic number (M), Gr and m or  = 0M I II III IV V VI VII VIII IX2 -2.45655 -2.84415 -2.99445 -2.74445 -2.85856 -1.52498 -1.11489 -2.15526 -2.001455 -4.64789 -4.83145 -5.26688 -4.73365 -4.75698 -2.44565 -2.00534 -4.15585 -3.888598 -6.25566 -6.33475 -6.33248 -6.25446 -6.53266 -4.45244 -3.21156 -5.62281 -4.6652510 -9.66655 -9.81498 -10.1156 -9.75278 -9.84422 -6.22415 -5.00012 -7.99859 -5.23011I II III IV V VI VII VIII IXD-11000 2000 3000 1000 1000 1000 1000 1000 1000K1 1 1 1 2 3 1 1 2 1
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101208 | P a g eGr 2 2 2 2 2 4 6 2 2m 1 1 1 1 1 1 1 2 3Table 2: Shear stress ( ) at the plate y=0 with unsteady mean flow due to main flow (u1) for various values offrequency of oscillation ( ) and square root of the Hartmann magneto hydro dynamic number (M), K1, Gr, mand Pr = 0.025M I II III IV V VI VII VIII IX X XI2 -22.714 -31.971 -39.088 -22.714 -22.714 -22.665 -23.284 -22.714 -22.714 -22.612 -22.4525 -22.946 -32.141 -39.218 -22.945 -22.947 -22.904 -22.626 -22.945 -22.945 -22.854 -22.6658 -23.374 -32.442 -39.464 -23.374 -23.375 -23.331 -22.861 -23.374 -23.372 -23.356 -23.15210 -23.762 -32.722 -39.694 -23.762 -23.762 -23.718 -23.676 -23.762 -23.764 -23.511 -23.354I II III IV V VI VII VIII IX X XID-11000 2000 3000 1000 1000 1000 1000 1000 1000 1000 1000K1 1 1 1 2 3 1 1 1 1 1 1Gr 2 2 2 2 2 4 6 2 2 2 2 2 2 2 2 2 2 2 4 6 2 2m 1 1 1 1 1 1 1 1 1 2 3Table 3: Shear stress ( ) at the plate y=0 with unsteady mean flow due to cross flow (u1) for various values offrequency of oscillation ( ), square root of the Hartmann magneto hydro dynamic number (M), K1, Gr , m andPr = 0.025M I II III IV V VI VII VIII IX X XI2 -22.008 -31.268 -38.377 -22.005 -22.004 -21.945 -21.045 -22.004 -22.004 -15.486 -13.8955 -22.234 -31.435 -38.511 -22.238 -22.238 -22.125 -21.945 -22.215 -22.245 -18.225 -15.4458 -22.665 -31.734 -38.764 -22.666 -22.665 -22.625 -22.169 -22.666 -22.665 -20.562 -18.45210 -23.0585 -31.017 -38.988 -23.053 -23.053 -23.049 -22.982 -23.052 -23.054 -21.468 -19.256I II III IV V VI VII VIII IX X XID-11000 2000 3000 1000 1000 1000 1000 1000 1000 1000 1000K1 1 1 1 2 3 1 1 1 1 1 1Gr 2 2 2 2 2 4 6 2 2 2 2 2 2 2 2 2 2 2 4 6 2 2m 1 1 1 1 1 1 1 1 1 2 3Table 4: Temperature gradient at the plate due to unsteady main flow for various values of frequency ofoscillation ( ) and thermal radiation-conduction parameter (K1) for Pr = 0.025. I II III2 -1.00000 -1.41432 -1.732144 -1.00125 -1.41466 -1.732286 -1.00279 -1.41521 -1.732468 -1.00424 -1.41732 -1.73284K1 1 2 3Table 5: Temperature gradient at the plate due to unsteady cross flow for various values of frequency ofoscillation () and thermal radiation-conduction parameter (K1) with Pr = 0.025 I II III2 -0.02499 -0.01768 -0.015324 -0.04993 -0.03534 -0.028946 -0.07479 -0.05299 -0.035668 -0.09222 -0.06735 -0.04225K1 1 2 3Table 6: Temperature gradient at the plate due to unsteady main flow for various values of thermal radiation-conduction parameter (K1) and Prandtl number (Pr) with  =2
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101209 | P a g eK1 I II III IV V1 -1.28082 -1.00125 -1.00045 -1.32049 -1.466352 -1.55949 -1.41462 -1.32645 -1.84538 -1.945223 -1.82125 -1.73229 -1.63322 -2.00536 -2.144554 -2.06095 -2.00019 -1.98882 -2.11432 -2.16654I II III IV VPr 0.025 0.05 0.075 0.025 0.025 2 2 2 4 6Table 7: Temperature gradient at the plate due to unsteady cross flow for various values of thermal radiation-conduction parameter (K1) and Prandtl number (Pr) with  =2K1 I II III IV V1 -0.80022 -0.04992 -0.02452 -0.95263 -1.143622 -0.65725 -0.03534 -0.02965 -0.83452 -0.931633 -0.56289 -0.02885 -0.00145 -0.67425 -0.844354 -0.49734 -0.02496 -0.00035 -0.05921 -0.62432I II III IV VPr 0.025 0.05 0.075 0.025 0.025 2 2 2 4 6ACKNOWLEDGEMENTSWe kindly acknowledge Late Prof.D.V.Krishna, Department of Mathematics, SriKrishnadevaraya University, Anantapur (AndhraPradesh), India for their useful remarks on themanuscript and Prof. R. Siva Prasad, Departmentof Mathematics, Sri Krishnadevaraya University,Anantapur (AP), India for providing the materialwhich was used to validate our computationalwork. Also, part of the computational facilities wasprovided by Department of Mathematics,Rayalaseema University, Kurnool (AP), India.REFERENCES[1]. Aboeldahab E.M. and El-Din. A.A.G.(2005): “Thermal radiation effects onMHD flow past a semi-infinite inclinedplate in the presence of mass diffusion”,Heat & Mass Transfer, 41(12), pp. 1056-1065.[2]. Alagoa. K.D., Tay. G. and Abbey. T.M.(1998): “Radiative and free convectiveeffects of MHD flow through a porousmedium between infinite parallel plateswith time-dependent suction”.Astrophysics and Space Science, 260(4),pp. 455-468.[3]. Alam. M.S., M.M. Rahman, and M.A.Sattar (2008): “Effects of variable suctionand thermophoresis on steady MHDcombined free-forced convective heat andmass transfer flow over a semi-infinitepermeable inclined plate in the presence ofthermal radiation”. Int. J. ThermalSciences, 47(6), pp. 758-765.[4]. Anwerbeg. O and S.K. Ghosh (2010):“Analytical study of magneto hydrodynamic radiation-convection with surfacetemperature oscillation and secondaryflow effects”, Int. J. of Appl. Math andMech. 6 (6), pp. 1-22.[5]. Audunson.T. and Gebhart.B. (1972): “Anexperimental and analytical study ofnatural convection with appreciablethermal radiation effects”, J. FluidMechanics, 52(1), pp. 57-95.[6]. Bestman. A.R. (1989): “Radiative transferto oscillatory hydro magnetic rotatingflow of a rarefied gas past a horizontal flatplate”. Int. J. Numerical Methods inFluids, 9(4), pp. 375-384.[7]. Duwairi.H.M. (2005): “Viscous and Jouleheating effects on forced convection flowfrom radiate isothermal porous surfaces”,Int. J. Numerical Methods Heat FluidFlow, 15(5), pp. 429-440.[8]. Duwairi.H.M. and Damseh.R.A), pp. 57-95.[9]. Bestman. A.R. (1989): “Radiative transferto oscillatory hydro magnetic rotatingflow of a rarefied gas past a horizontal flatplate”. Int. J. Numerical Methods inFluids, 9(4), pp. 375-384.[10]. Duwairi.H.M. (2005): “Viscous and Jouleheating effects on forced convection flowfrom radiate isothermal porous surfaces.(2004): “Magneto hydro dynamic naturalconvection heat transfer from radiatevertical porous surfaces”, Heat MassTransfer, 40(10), pp. 787-792.[11]. Duwairi.H.M. and Duwairi.R.M. (2005):“Thermal radiation effects on MHD-Rayleigh flow with constant surface heatflux”, Heat and Mass Transfer, 41(1), pp.51-57.[12]. El-Hakiem.M.A. (2000): “MHDoscillatory flow on free convectionradiation through a porous medium with
    • E.Neeraja and M.Veera Krishna/ International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1197-12101210 | P a g econstant suction velocity”, J. MagnetismMagnetic Materials, 220(2-3), pp. 271-276.[13]. Ghosh.S.K. and Pop.I. (2007): “Thermalradiation of an optically-thick gray gas inthe presence of indirect naturalconvection”, Int. J. Fluid MechanicsResearch, 34(6), pp. 515-520.[14]. Helliwell. J.B. and Mosa. M.F. (1979):“Radiative heat transfer in horizontalmagneto hydro dynamic channel flowwith buoyancy effects and an axialtemperature gradient”, Int. J. Heat MassTransfer, 22, pp. 657-668.[15]. Hughes. W.F. and Young. F.J. (1966): TheElectro magneto dynamics of Fluids, JohnWiley and Sons, New York, USA.[16]. Israel-Cookey.C, A. Ogulu, and V.B.Omubo-Pepple. (2003): “Influence ofviscous dissipation and radiation onunsteady MHD free-convection flow pastan infinite heated vertical plate in a porousmedium with time-dependent suction”,Int. J. Heat and Mass Transfer, 46(13),pp. 2305-2311.[17]. Larson.D.W. and R.Viskanta. (1976):“Transient combined laminar freeconvection and radiation in a rectangularenclosure”, J. Fluid Mechanics, 78(1), pp.65-85.[18]. Meyer. R.C. (1958): “On reducingaerodynamic heat-transfer rates bymagneto hydro dynamic techniques”, J.Aerospace Sciences, 25, pp. 561- 566.[19]. Ouaf. M.E.M. (2005): “Exact solution ofthermal radiation on MHD flow over astretching porous sheet”, AppliedMathematics and Computation, 170(2),pp. 1117-1125.[20]. Raptis.A, Perdikis.C, and Takhar.H.S.(2004): “Effect of thermal radiation onMHD flow”. Applied Mathematics andComputation, 153(3), pp. 645-649.[21]. Samad. M.A. and Rahman. M.M. (2006):“Thermal radiation interaction withunsteady MHD flow past a vertical porousplate immersed in a porous medium”, J.Naval Architecture and MarineEngineering, 3, pp. 7-14.[22]. Shercliff. J.A. (1965): A Textbook ofMagneto hydro dynamics, CambridgeUniversity Press, UK.[23]. Siegel. R. and Howell. J.R. (1993):Thermal Radiation Heat Transfer,Hemisphere, USA.[24]. Vasil’ev. E.N. and D.A. Nesterov. (2005):“The effect of radiative-convective heattransfer on the formation of current layer”,High Temperature, 43(3), pp. 396-403.[25]. Yasar. O. and Moses. G.A. (1992): “R-MHD: an adaptive-grid radiation-magnetohydro dynamics computer code”.Computer Physics Communications, 69(2-3), pp. 439-458.[26]. Zueco. J. (2007): “Network simulationmethod applied to radiation and viscousdissipation effects on MHD unsteady freeconvection over vertical porous plate”,Applied Mathematical Modelling, 31(9),pp. 2019-2033.Appendix:),1(Pr),1( 1221  DmMK21212121)1()1121)1()11211222222112222221DmMDmMDDmMDmMC   ,21,2121121222122112122212KrPKDKrPKC