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1.Shaik ABZAL, 2. G. V. RAMANA REDDY, 3. S. VIJAYAKUMAR VARMAUNSTEADY MHD FREE CONVECTION FLOW AND MASSTRANSFER NEAR A MOVING VERTICAL PLATE IN THEPRESENCE OF THERMAL RADIATION1.AUDISANKARA COLLEGE OF ENGINEERING AND TECHNOLOGY, GUDUR, A.P, INDIAUSHARAMA COLLEGE OF ENGINEERING,TELAPROLU,A.P, INDIA3.S.V.UNIVERSITY, TIRUPATI, A.P, INDIA2.ABSTRACT: The problem of unsteady MHD free convection flow and mass transfer near a moving verticalplate in the presence of thermal radiation has been examined in detail in this paper. The governingboundary layer equations of the flow field are solved by a closed analytical form. A parametric study isperformed to illustrate the influence of radiation parameter, magnetic parameter, Grashof number, Prandtlnumber on the velocity, temperature, concentration and skin-friction. The results are discussed graphicallyand qualitatively. The numerical results reveal that the radiation induces a rise in both the velocity andtemperature, and a decrease in the concentration. The model finds applications in solar energy collectionsystems, geophysics and astrophysics, aero space and also in the design of high temperature chemicalprocess systems.KEYWORDS: MHD, Radiation, unsteady, concentration and skin-friction INTRODUCTIONThe phenomenon of magnetohydrodynamic flow with heat transfer has been a subject ofgrowing interest in view of its possible applications in many branches of science and technology andalso industry. Free convection flow involving heat transfer occurs frequently in an environment wheredifference between land and air temperature can give rise to complicated flow patterns. The study ofeffects of magnetic field on free convection flow is often found importance in liquid metals,electrolytes and ionized gasses. At extremely high temperatures in some engineering devices, gas, forexample, can be ionized and so becomes an electrical conductor. The subject of magnetohydrodynamics has attracted the attention of a large number of scholars due to its diverseapplications in several problems of technological importance. The ionized gas or plasma can be madeto interact with the magnetic field and can frequently alter heat transfer and friction characteristicson the bounding surface. Heat transfer by thermal radiation is becoming of greater importance whenwe are concerned with space applications, higher operating temperatures and also power engineering.In astrophysics and geophysics, it is mainly applied to study the stellar and solar structures,interstellar matter, radio propagation through the ionosphere etc. In engineering, the problemassumes greater significance in MHD pumps, MHD journal bearings etc. Recently, it is of great interestto study the effects of magnetic field and other participating parameters on the temperaturedistribution and heat transfer when the fluid is not only an electrical conductor but also when it iscapable of emitting and absorbing thermal radiation.Viskanta (1963) had initiated the problem by examining the effects of transverse magnetic fieldon heat transfer of an electrically conducting and thermal radiating fluid flow in a parallel channel.Later, Grief et al. (1971) had investigated for an exact solution for the problem of laminar convectiveflow in a vertical heated channel in the optically thin film. Subsequently, Gupta et al. (1974) studiedthe effect of radiation on the combined free and forced convection of an electrically conducting fluidflowing inside an open ended vertical channel in the presence of uniform magnetic field.Soundalgekar (1979) had studied free convection effects on the flow past a vertical oscillating plate.Transformations of the boundary layer equation for free convection effects on flow past a verticalsurface studied by Vedhanayagam et al. (1980). Kolar et al. (1988) had analyzed a free convectiontranspiration of radiation effects over a vertical plate while, Soundalgekar (1993) worked inhydromagnetic natural convection flow past a vertical surface and the problem of heat transfer byconsidering radiation as an important application of space and temperature related problems. Later,Takhar (1996) and Hossian et al. (1996) analyzed the effects of radiation using the Rosseland diffusionapproximation for mixed convection of an optically dense viscous incompressible fluid in presence of copyright FACULTY of ENGINEERING ‐ HUNEDOARA, ROMANIA29

ANNALS OF FACULTY ENGINEERING HUNEDOARA – International Journal Of Engineeringmagnetic field. Thereafter, Soundalgakar et al. (1997) generalized the problem by considering theeffect of radiation on the natural convection flow of a gas past a semi infinite plate. The effects ofcritical parameters influencing the mass transfer on the MHD flow past an impulsively started infinitevertical plate with variable temperature or constant heat flux was discussed by Shankar et al. (1997).It has been reported that, in the optically thin film, the fluid does not absorb its own emittedradiation which means that there is no self absorption, but the fluid does absorb radiation emitted bythe boundary. Hussain et al. (1999) reported interesting observations in the problem of naturalconvection boundary layer flow, induced by the combined buoyancy forces from mass and thermaldiffusion from a permeable vertical flat surface with non uniform surface temperature andconcentration but a uniform rate suction of fluid through the permeable surface. Revankar (2000)studied the free convection effects on the flow past an impulsively started or oscillating infinitevertical plate with different boundary conditions and thereafter, Hussain et al. (2000) discussed theeffect of radiation on free convection from a porous vertical plate. Several investigators like Sahoo etal. (2003) Muthcumaraswamy et al. (2004) reported their observations on the heat and mass transfereffects on moving vertical plate in presence of thermal radiation. Recently, Shateyi et al. (2007)studied magnetohydrodynamic flow past a vertical plate with radiative heat transfer and Majumder etal. (1968) gave an exact solution for MHD flow past an impulsively started infinite vertical plate inpresence of thermal radiation.The purpose of the present paper is to solve analytically the problem of the unsteady freeconvection flow and mass transfer of an optically thin viscous, electrically conducting incompressiblefluid near an infinite vertical plate which moves with time dependent velocity in presence oftransverse uniform magnetic field and thermal radiation. The flow phenomena has been characterizedwith the help of flow parameter and the effect of these parameters on the velocity field, temperature,concentration and skin friction have been analyzed and the results with respect to various flow entitieshave been presented graphically and discussed qualitatively. MATHEMATICAL FORMATION OF PROBLEMWe consider unsteady free convection flow and mass transfer of a viscous incompressible,electrically conducting and radiating fluid along an infinite non-conducting vertical flat plate (orsurface) in presence of a uniform transverse magnetic field B0 applied in the direction of the flow. Onthis plate an arbitrary point has been chosen as the origin of a coordinate system with x ' - axis isalong the plate in the upward direction and the y ' -axis normal to plate. Initially for time t ' 0 , the'plate and the fluid are at same constant temperature T in a stationary condition, with the same'species concentration C at all points. Subsequently ( t ' 0 ) , the plate is assumed to be acceleratingwith velocity U 0 f (t ') in its own plane along the x ' - axis; instantaneously the temperature of the''plate and the concentration are raised to Tw and Cw respectively which are hereafter regarded as aconstant. For free convection flows, here we also assume that all the physical properties of the fluid isassumed to be in the direction of the x ' -axis, so the physical variables are functions of the space coordinate y ' and time t ' only. Under the assumptions, the fully developed flow of a radiating fluid isgoverned by the following set of equations are: u ' 2u ' σ B02υ g β (T ' T ' ) g β ( C ' C ' ) ν u ' u '2ρ t ' y 'k'ρ CP(1) T ' 2T ' qr κ t ' y '2 y '(2) C ' 2C ' D 2 t ' y '(3)In view if the physics af the problem, following are the initial and boundary conditions:For t 0 : u ' 0, T ' T ' , C ' C ' , y '''' (4)For t 0, u ' U 0 f ( t ) T ' Tw C ' C w ,y ' 0and u ' 0, T ' T ' , C ' C ' , as y ' where u ' is the velocity in the x ' -direction, υ the kinematics viscosity, k ' is the thermal diffusivity,β is the volumetric coefficient of thermal expansion , β* is the volumetric coefficient of expansion forconcentration, ρ is the density, σ is the electrical conductivity , κ the thermal conductivity, g is theacceleration due to gravity, T ' is the temperature of the fluid near the plate, T is the temperature ofthe fluid far away from the plate, C ' is the species concentration, Cp is the specific heat at constantpressure, D is the chemical molecular diffusivity, qr is the radiative flux.In the situation of optically thick film, the fluid does not absorb its own emitted radiation, wherethere is no self absorption but it does absorb radiation emitted by the boundaries. It has been shown byCogly et al [19] that in the optically thick limit for a non gray gas near equilibrium is: qr de 4 (T ' T ' ) K λζ bλ d λ 4 I1 (T ' T ' )0 y ' dT ' ζ30(5)Tome IX (Year 2011). Fascicule 3. ISSN 1584 – 2673

ANNALS OF FACULTY ENGINEERING HUNEDOARA – International Journal Of Engineeringwhere K λζ is the absorption coefficient, ebλ is Planck function and the subscriptthe wall.Introducing the following dimensionless variables and parameters as:y' Gr U0 yυ, u' ζrefers to values attU 02k 'U 02T ' T 'C ' C 'u K ,t ' ,θ ',φ,υυ2Tw T 'Cw' C 'U0g βυ (Tw' T ' )g β υ (Cw' C ' )4I1υ 2μ Cpυ,,,,F Gm ScPr DκU 02U 03U 03κ(6)where Pr is the Prandtl number, Gr is the thermal Grashof number, Gm is the mass Grashof number, Mis the magnetic parameter, F is the radiation parameter and Sc is the Schmidt number.With the help of Eqn (6), the governing Eqns (1) - (3) reduce to: u 2u1 2 Grθ Gcφ M uK t y (7) θ1 2θ F θ t Pr y 2 Pr(8) φ 1 2φ t Sc y 2(9)The corresponding initial and boundary conditions in non-dimension form are:u 0, θ 0, ϕ 0 y 0, t 0u f (t ), θ 1, φ 1 at y 0 , t 0u 0 , θ 0 , φ 0 as y (10)The system Eqns (7) - (9) subject to the boundary conditions (10), includes the effect of freeconvection and mass transfer on the flows near a moving isothermal vertical plate. SOLUTION OF PROBLEMIn order to reduce the above system of partial differential equations to a system of ordinarydifferential equations in dimensionless form, we assume the trial solution for the velocity, temperatureand concentration as:u ( y, t ) u0 ( y )eiωt(11)θ ( y, t ) θ 0 ( y )eiωt(12)iωtφ ( y, t ) φ0 ( y )e(13)In view of the above, the corresponding boundary conditions can be re written asu0 f (t )e iωt , θ0 e iωt , φ0 e iωt asu0 0, θ0 0, φ0 0 as y y 0(14)The solutions of Eqs.(11) – (13) satisfying the boundary conditions (14) are given byu ( y, t ) e Ny f (t ) GrGme Ny e m2 y ) 2e Ny e m2 y )((m Nm1 N22θ ( y, t ) e m y2φ ( y, t ) e m y1(15)(16)(17)Knowing the velocity field, the skin-friction at the plate can be obtained, which in nondimensional form is given by u GrGm Nf (t ) 2( m2 N ) 2 ( m1 N )m2 Nm1 N y y 0(18)m1 iω Scm2 F iω PrN M iω 1/ K RESULTS AND DISCUSSIONIn order to get a physical insight in to the problem the effects of various governing parameterson the physical quantities are computed and represented in Figures 1-17 and discussed in detail. copyright FACULTY of ENGINEERING ‐ HUNEDOARA, ROMANIA31

ANNALS OF FACULTY ENGINEERING HUNEDOARA – International Journal Of EngineeringThe effect of Prandtl number is noticed in Fig .1. For a constant value of Schmidt number, as thePrandtl number increases, the velocity field is found to be decreasing. Further, it is observed that aswe move away from the plate, the velocity increases and thereafter it is found to be decreasing. Also,far away from the plate, it is noticed that the variation in the velocity is not significant even if thePrandtl number increases. Therefore, it can be concluded that the effect of Prandtl number isincreasing the velocity field is only up to some level and thereafter, its contribution is not thatsignificant. For a fixed value of Prandtl number, the contribution of Schmidt number is seen in Fig.2.The increase in the value of Schmidt number, contributes to the decreasing in the velocity field. In theboundary layer region the fluid velocity observed to be decreasing and thereafter, as we move awayfrom the plate, the velocity is found to be decreasing.Fig.1: Effect of Pr on theFig.2. Effect of Sc on theFig.3: Effect of F on thevelocity fieldvelocity fieldvelocity fieldThe effect of radiation parameter on the velocity field is illustrated in Fig.3. Increase in theradiation parameter contributes to the decrease in the velocity field. However, the trend seems tohave been reversed as we move away from the plate. Therefore, the velocity field seems to behavedifferently in each of these situations. The decrease in the velocity in boundary layer region can beattributed to the fact that the intra molecular forces within the fluid decreases which would havecontributed to the increase in the velocity. But the presences of magnetic field suppress such anincrease as a result of which the velocity reduces. However, in the later stage, it is observed that aswe move far away from the plate, the influence of the magnetic field is not felt resulting in theincrease of fluid velocity. The contribution of the Magnetic field on the velocity profiles is noticed inFig .4. It is observed that as the magnetic intensity increases, the velocity field decreases throughoutthe analysis as long as the radiation parameter is held constant. Further, it is also noticed that thevelocity of the fluid medium raises within the boundary layer region and thereafter, it decreases whichclearly indicates that the radiation parameter has not that much of significant effect as was in theinitial stage. The contribution of the porosity factor of the fluid bed on the velocity field is illustratedin Fig.5. In general, it is noticed that, as the porosity decreases, the velocity also decreases for aconstant Gr. Further, as we move far away from the fluid bed, the effect of both velocity and Gr onthe velocity is found to be almost zero. The influence of frequency of excitation for a constant Prandtlnumber (Pr) is shown in Fig.6. In general it is noticed that increase in the frequency of excitation,contributes to the decrease in the velocity of the fluid medium. Further, as was seen in all otherearlier situations, as we move away from the plate the velocity decreases.Fig.4: Effect of M on the velocity Fig.5: Effect of k on the velocity Fig.6: Effect of ω on the velocityfieldfieldfieldThe influence of thermal Grashof number on the velocity field is illustrated graphically in Fig.7.When Gm is held constant, and as the thermal Grashof number is increased, in general the fluidvelocity increases. However, as we move away from the bounding surface of the fluid, it is noticed thatirrespective of the nature of thermal Grashof number, the velocity remains to be zero and hence theinfluence of thermal Grashof number do not qualitatively contributes to the velocity field. The effectof mass Grashof number for a constant value of thermal Grashof Number is illustrated in Fig.8. For afixed thermal Grashof Number, the increase in the mass Grashof number, in general contributes to theincrease in the velocity field. However, it does not have any influence as we move away from thebounding surface.32Tome IX (Year 2011). Fascicule 3. ISSN 1584 – 2673

ANNALS OF FACULTY ENGINEERING HUNEDOARA – International Journal Of EngineeringFig. 7: Effect of Gr on theFig. 8: Effect of Gm on theFig. 9: Effect of ω on thevelocity fieldvelocity fieldtemperature fieldThe influence of frequency of excitation for a constant radiation parameter on the temperatureis studied in Fig.9. When the radiation parameter is held constant and as the frequency of excitation isincreased, it is noticed that, the temperature decreases. Relatively when the frequency of excitation isvery small, the profiles for the temperature are found to be linear of course with a negative slope. Butwhen it is increased, the profiles for the temperature are found to be parabolic in nature. In tune withall earlier observations, it is noticed that, in this situation, the effect of frequency of excitation is notsignificant as we move away from the bounding surface. For a constant value of the Prandtl Number,the influence of radiation parameter on the temperature field is studied in Fig.10. Increase, in theradiation parameter contributes in general to decrease in the temperature. The effect of suchradiation parameter is not significant as we move away from the boundary. As the radiation parameterincreases, the profiles for the temperature field are found to be more parabolic in nature. Theinfluence of Prandtl number for a fixed radiation parameter is illustrated graphically in Fig.11. It isnoticed that, as the Prandtl number increases, in general the temperature falls down. Also, theincrease in Prandtl number contributes to the parabolic nature of temperature profiles. Further, theeffect is found to be significant in the initial stages and not that predominant as we move away fromthe plate. The influence of the frequency of excitation on the temperature profiles when the Prandtlnumber is held constant is illustrated in Fig.12. As the frequency of excitation is increased, in generalit is seen the temperature decreases. Such as effect is found to be more dispersive and pre dominantwithin the boundary layer region. However, the contribution of both participating parameters as wemove away from the plate is not significant.Fig.10: Effect of F on theFig. 11: Effect of Pr on theFig.12: Effect of ω on thetemperature fieldtemperature fieldtemperature fieldThe influence of Schmidt number on the concentration of fluid medium is shown graphically inFig.13. The relation for the Schmidt number on the concentration is perfectly found to linear and ofcourse inversely. As the Schmidt number increases, the concentration decreases as long as thefrequency of excitation is held constant. The influence of frequency of excitation on the concentrationfield is studied graphically in Fig.14. It is observed that, as the frequency of excitation increases, asignificant drop in the temperature is noticed. Also, it is seen that as the frequency of excitation isincreased, the temperature profiles are found to be more parabolic in nature.Fig.13: Effect of Sc on theconcentration fieldFig.14: Effect of ω on theconcentration field copyright FACULTY of ENGINEERING ‐ HUNEDOARA, ROMANIAFig.15: Effect of Sc on Skinfriction33

ANNALS OF FACULTY ENGINEERING HUNEDOARA – International Journal Of EngineeringFig.16: Effect of Pr on the SkinfrictionThe influence of Schmidt number on Skin friction withrespect to the frequency of excitation is shown graphically inFig.15. When the magnetic intensity is held constant and theSchmidt number is increased, the skin friction reduces quitesignificantly. Though not much of variation is seen on theboundary, its effect is found to be highly dispersive.The consolidated contribution of the frequency ofexcitation and Prandtl number on the skin friction is illustratedgraphically in Fig.16. In general, it is seen that as the frequencyof excitation is increased, the skin friction decreases when themagnetic intensity is held constant throughout theinvestigation.The influence of magnetic fiel