THE EFFECTS OF RADIATION ON UNSTEADY MHD CONVECTIVE HEAT TRANSFER PAST A SEMI-INFINITE VERTICAL POROUS MOVING SURFACE WITH VARIABLE SUCTION

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1 Latin merican pplied Research 8:7-4 (8 THE EFFECTS OF RDITION ON UNSTEDY MHD CONVECTIVE HET TRNSFER PST SEMI-INFINITE VERTICL POROUS MOVING SURFCE WITH VRIBLE SUCTION. MHDY Math. Department Science, Soth Valle Universit, Qena, EGYPT * sci_eng@ahoo.com bstract Nmerical soltions for the effects of radiation on a MHD convective heat transfer past a semi-infinite poros plate with a magnetic field are obtained. It is assmed that the poros plate moves with a constant velocit in the direction of flid flow, and the free stream velocit follows the exponentiall increasing small pertrbation law. The magnetic field acts perpendiclar to the poros srface which absorbs the flid with a sction velocit varing with time. The governing eqations for the flow are transformed into a sstem of nonlinear ordinar differential eqations b pertrbation techniqe and then are solved nmericall b sing the shooting method. The effects of the varios parameters on the velocit, temperatre profiles as well as the srface skinfriction and srface heat transfer are illstrated graphicall. Kewords Unstead state, Radiation, Moving srface, Magnetic field, Sction. I. INTRODUCTION It is known that the effects of radiation on MHD flow and heat transfer problem have become more important indstriall. t high operating temperatre, radiation effects can be qite significant. Man processes in engineering areas occr at high temperatre and a knowledge of radiation heat transfer becomes ver important for the design of the pertinent eqipment. Nclear power plants, gas trbines and the varios proplsion devices for aircraft, missiles, satellites and space vehicles are examples of sch engineering areas. Raptis (1998 investigate the stead flow of a viscos flid throgh a ver poros medim bonded b a poros plate sbjected to a constant sction velocit b the presence of thermal radiation. Bestman (199 examined the natral convection bondar laer with sction and mass transfer in a poros medim. His reslts confirmed the hpothesis that sction stabilizes the bondar laer and allows the most efficient method in bondar laer control et known. bds Sattar and Hamid Kalim (1996 investigated the nstead free convection interaction with thermal radiation in a bondar laer flow past a vertical poros plate. Makinde (5 examined the transient free convection interaction with thermal radiation of an absorbing emitting flid along moving vertical permeable plate. Chamkha (4 assmed that the plate is embedded in a niform poros medim and moves with a constant velocit in the flow direction in the presence of a transverse magnetic field. Raptis and Perdikis ( stdied the nstead free convection flow of water near 4 o C in the laminar bondar laer over a vertical moving poros plate. Sondalgekar and Patti (198 stdied the problem of the flow past an implsivel started isothermal infinite vertical plate with mass transfer effects. Takhar et al. (1996 stdied the radiation effects on MHD free-convection flow of a gas past a semi-infinite vertical plate. Seddeek (, 1 stdied thermal radiation and boanc effects on MHD free convective heat generating flow over an accelerating permeable srface with temperatre-dependent viscosit in the case of stead state. The radiation effects on heat transfer over a stretching srface have been stdied b Elbashbesh (. In spite of all these stdies, the nstead MHD convective heat transfer in the presence of radiation has received little attention. Hence, the main objective of the present investigation is to consider the case of a semi-infinite moving poros plate in a poros medim with the presence of radiation and constant velocit in the flow direction when the magnetic field is imposed transverse to the plate and sbjected to variable sction. It is also assmed that an exponential variation with time is imposed to temperatre. II. MTHEMTICL FORMULTION We consider, nstead flow of a laminar, incompressible flid past a semi-infinite vertical poros moving plate embedded in a poros medim in the presence of radiation and sbjected to a transverse magnetic field (Fig. 1. In or problem we assmed that there is no applied voltage which implies the absence of an electric field. The transversel applied magnetic field and magnetic Renolds nmber are ver small and therefore the indced magnetic field is negligible (Cowling, Viscos and Darc s resistance terms are taken into accont with constant permeabilit of the poros medim. The governing eqations for or problem nder the sal bondar laer approximation can be written as follows: ˆ υ =, (1 ˆ 7

2 . MHDY Fig. 1. The sketch of the phsical model ˆ ˆ 1 pˆ ˆ + ˆ υ = + v + g β ( T T ˆ ˆ ˆ t ρ x ˆ v σ B ˆ ˆ, ( Κˆ ρ T T T 1 q ˆ r + υ = α, ( ˆ ˆ t ˆ ρ c p ˆ where û and υˆ are the components of dimensional velocities in the xˆ and ŷ directions, respectivel xˆ and ŷ are the dimensional distances along and perpendiclar to the plate, respectivel q r is the radiative heat flx and T, pˆ are the temperatre and pressre. Properties v, ρ, Κˆ,α and β are the kinematic viscosit, densit, permeabilit, the flid thermal diffsivit, and thermal expansion coefficients of the flid, respectivel, assmed constant. Frthermore, σ, B, and c p are the flid electrical condctivit, magnetic indction and the specific heat at constant pressre, respectivel. Moreover, the appropriate bondar conditions for the velocit and temperatre fields are: nt ˆˆ ˆ = : ˆ = ˆ p, T = T + ε ( T T e, ˆ nt ˆˆ ˆ : ˆ = U = U ( 1+ ε e, T T, (4 where û p is the wall dimensional velocit, nˆ and ε are constants, T is a reference temperatre. The radiative heat flx q r nder the Rosseland approximation has the form 4 4δ T q r =, χ ˆ where δ is the stefan-boltzman s constant and χ is the mean absorption coefficient. From the continit eqation Eq. (1, it is obvios that the sction velocit normal to the plate is a fnction of time onl and it can be taken in the form: nt ˆˆ υ = V ( 1+ ε e, (5 where is a real positive constant vale, ε and ε are small real nmbers less than nit, V is a scale of sction velocit which has non-zero positive constant. Otside the bondar laer, Eq. ( gives ˆ 1 pˆ d U v ˆ σ B = + U ˆ + U, (6 ρ xˆ d tˆ Κˆ ρ On introdcing the dimensionless qantities, (the partial differential eqations are converted into dimensionless form ˆ ˆ ˆ υ V ˆ t V =, υ =, =, t =, U V v v Uˆ ˆ p nˆ v T T U =, =, n =, =. (7 U U V T T In view of Eqs. (5-(7, the governing Eqs. ( and ( redce to the dimensionless form: nt d U ( 1+ ε e = + t d t (8 + Gr + N( U, 4 nt 1 R ( + w ( 1+ e = Pr ε +, (9 t ˆ Κ V v ρ c p σ B v where Κ =, Pr =, Mn =, k v ρ V g v β Tw T Gr =, N = Mn + Κ U V ( 1 4δ ( Tw T T R =, =. χ v ρ c p Tw T The previos bondar conditions take the following dimensionless form: nt = : =, = 1+ ε e, : = U = 1+ ε e,, (1 where Gr is the Grashof nmber, Mn is the magnetic field parameter, K is the permeabilit parameter, R is the radiation parameter, Pr is the Prandtl nmber, and is the temperatre parameter. w III. SOLUTION PROCEDURE Eqations (8 and (9 represent a set of partial differential eqations that can not be solved in closed form. However, it can be redced to a set of ordinar differential eqations that can be solved nmericall. This can be done b representing the velocit and temperatre as follows: nt = f ( + ε e f ( + O(, 1 ε nt ( + ε e g1( + O( ε = g. (11 Sbstitting Eq. (11 into Eqs. (8 and (9 and neglecting the coefficients of terms of O ( ε, we obtain the following pairs of eqations for ( f, g and f. ( 1, g1 nt, 8

3 Latin merican pplied Research 8:7-4 (8 For zero order of ε f + f + N 1 f + Gr g, (1 ( = / ( g + g + 4 R ( g + g =, 1 ( Pr + (4 R + g (1 sbject to the bondar conditions = : f = U, g 1, p = : f 1, g, (14 For O(1 of ε f + N + n (1 f + f + f + Gr g, (15 1 ( = 1 + (4 R / ( g + g R ( g + g 1 g 4R( g + w ( g1g + g1 g + g1 n g1 + = ( Pr + g, (16 sbject to the bondar conditions = : f =, g 1, 1 1 = : f1 1, g1, (17 The phsical qantities of interest are the wall shear stress τ w and the local srface heat transfer rate q w. These are defined as ˆ τ w = μ = U V ( ˆ ρ, (18 w therefore, the local friction factor C f is given b C w f U V = ρ τ = (. (19 The local heat transfer coefficient in terms of Nsselt nmber can be expressed as x T N x = Tw T. ( ˆ w or it can be also qantitativel evalated b N Re 1 x x = ( (1 where Re x = V x / v is the local Renolds nmber. IV. RESULTS ND DISCUSSION In the previos sections, we have formlated and solved the problem of the inflence of radiation on an nstead MHD convection heat transfer past a semi-infinite vertical poros moving plate. In the nmerical comptation, varios vales of the material parameters are sed. In addition, the bondar condition is approximated b max =6, which is sfficientl large for the velocit to approach the relevant stream velocit. representative set of reslts is reported graphicall in Figs These reslts are obtained to illstrate the inflence of the radiation parameter R, the magnetic field parameter Mn, permeabilit parameter K, velocit of the plate, the dimensionless exponential index n, temperatre parameter, Grashof nmber Gr, and Prandtl nmber Pr, on the velocit, temperatre profiles, srface skin friction and srface heat transfer in terms of Nsselt nmber. Figres and show that the dimensionless velocit and the dimensionless temperatre increase as the radiation parameter R increases. lso, we observe that the magnitde of the stream wise velocit increases and the inflection point for the velocit distribtion moves frther awa from the srface. In Fig. 4, the velocit is plotted for several vales of Mn while R = 5.. s seen in this figre, for a given vale of magnetic field, the velocit profiles decrease monotonicall with an increase in. For small and large vales of, the effect of Mn is rather insignificant. Onl when.5 < < 4 the vale of the velocit is redced significantl depending on Mn Mn =. Gr =. K Pr =.7 n =.1 ε =. t = 1. R = 1. R =. R =. R = 4. R = Fig.. Effect of radiation parameter on velocit Mn =. Gr =. K Pr =.7 n =.1 ε =. t = 1. R = 1. R =. R =. R = 4. R = Fig.. Effect of radiation parameter on temperatre The velocit profiles for different vales of plate moving velocit in the direction of flid flow are described in Fig. 5 with R = 5.. lthogh we have different initial plate moving velocities, the velocit decas to the constant vale for the given material parameter. Figres 6 and 7 show that the dimensionless velocit and the dimensionless temperatre increase as the dimensionless exponential index n increases for the two cases R=. and R = 5.. The velocit profiles for different vales of Grashof nmber Gr while R = 1. are described in Fig. 8. It is observed that an increase in Gr leads to an increase in the vale of velocit. In addition, 9

4 . MHDY Pr =.7 Gr =. K R = 5. ε =. n =.1 t = 1. Mn =. Mn = 1. Mn =. Mn = 5. Mn = the crves show that the peak vale of the velocit increases rapidl near the wall of the plate as Grashof nmber increases, and then decas to the relevant free stream velocit. Figre 9 depicts the temperatre profiles as a fnction of the transversal coordinate for varios vales of the Prandtl nmber, Pr. The nmerical reslts show that the effect of increasing vales of Prandtl nmber reslts in a decreasing thermal bondar laer thickness and more niform temperatre distribtion across the bondar laer, while the vales of the phsical parameters are fixed at real constants,,, ε=., the dimensionless exponential index n =., scale of free stream velocit, Prandtl nmber Pr =.7 and t = 1. and for the two cases and R =.. Fig. 4. Effect of magnetic field on velocit 1. Pr =.7 Gr =. Mn =. ε =. K t = 1. Mn =. Pr =.7 =. Gr =. K = 5. R = 5. p = 1. ε =. p =. 15 n =.1 t = 1. n =.1,.,.5,.7, 1. n =.1,.,.5,.7, R = Fig. 7. Effect of n on temperatre Fig. 5. Effect of velocit of the plate on velocit. Gr =.1,.5, 1.,., 4.. n =.1,.,.5,.7, Pr =.7 Gr =. Mn =. ε =. K t = Fig. 6. Effect of n on velocit R = 5..4 Pr =.7 R = 1. Mn =. ε =. K n =. t = Fig. 8. Effect of Grashof nmber on velocit Figre 1 shows the velocit profiles for different vales of the permeabilit parameter K. Clearl as K increases the peak vale of velocit tends to increase with and R =.. The inflence of the tempera- 4

5 Latin merican pplied Research 8:7-4 (8 tre parameter is plotted in Fig. 11. It is obviosl that as increases leads to increase the thermal bondar laer thickness.. K =.1,.5, 1., R =. Pr = 5.,.7,.1 Mn =. Gr ε =. K t = 1. n = Pr = 5. Pr =.7 Pr = Fig. 9. Effect of Prandtl nmber on temperatre The effect of time on velocit and temperatre profiles is considered in Figs. 1 and 1. s it is clear from the figres that the peak vale of velocit is increasing with increasing of time, the temperatre increases in the range. 614 and decreases when In addition, some graphs of the srface skin friction and srface heat transfer against the sction velocit parameter are reported in Figs The effect of radiation parameter on the srface skin friction and srface heat transfer illstrated in Figs. 14 and 15 respectivel, and it is clear from these figres that both of skin friction and heat transfer increase with increasing the radiation parameter R. It has been observed from Fig. 16 that for a constant plate moving velocit with given material parameters, Mn, K, Pr, Gr,, and,. the effect of increasing vales of sction velocit parameter reslts in a slightl increasing srface skin friction for the lower vales of. It is also evident that the srface skin friction decreases b increasing the plate moving velocit. s shown in Fig. 17, the skin friction on the poros plate increases b increasing the strength of sction velocit. It has been shown that for small vales of the dimensionless exponential index n, the increment of srface skin friction has a gentle slope. Figre 18 illstrates the variation of srface heat transfer verss the sction velocit parameter for several vales of Prandtl nmber. Nmerical reslts show that for given material parameters which are listed in figre, the srface heat transfer tends to decrease b increasing the magnitde of sction velocit for and less decrease b increasing the magnitde of sction velocit when R =...4 Pr =.7 Gr =. Mn =. ε =. n =. t = Fig. 1. Effect of K on velocit =. =.1 =. = = 1. R =. Mn =. Gr =. K Pr =.7 R =. n =. ε =. t = Fig. 11. Effect of on temperatre Pr =.7 Mn =. ε =. n =.1 R =. K Gr =. = 6. t =., 1.,.,.,4.,5. 6.,7.,8.,9., Fig. 1. Effect of time on velocit 41

6 . MHDY Pr =.7 Mn =. ε =. n =.1 R =. K Gr =. = 6. t =., 1.,.,.,4.,5. 6.,7.,8.,9.,1. Srface Skin Friction =. Pr =.7 Mn =. ε =. n =.1 t = 1. K =. =. =. R = Fig. 1. Effect of time on temperatre Fig. 16. Effect of on the srface skin-friction Srface Skin Friction 4.5 Pr =.7 Mn =. ε =. K 4. n =.1 t = R =. R = 1. R = Fig. 14. Effect of R on the srface skin-friction Srface Skin Friction Pr =.7 Mn =. ε =. t = 1. K n = 1. n = 1. n =.1 n =.1 n =.1 R = Fig. 17. Effect of n on the srface skin-friction n =.1 Srface Heat Transfer Pr =.7 Mn =. ε =. K n =.1 t = 1. R =. R =. R = Fig. 15. Effect of R on the srface heat transfer Srface Heat Transfer Mn =. ε =. n =.1 t = 1. K Pr =.7 Pr = 5. Pr =.7, 1., 5. Pr = 1. R = Fig. 18. Effect of Pr on the srface heat transfer 4

7 Latin merican pplied Research 8:7-4 (8 V. CONCLUSIONS The plate velocit was maintained at constant vale and the flow was sbjected to a transverse magnetic field in the presence of radiation effect. The governing eqations were developed and transformed into a sstem of nonlinear ordinar differential eqations b pertrbation techniqe and are solved nmericall b emploing the shooting method. The srface skin friction coefficient, srface heat transfer as well as the details of velocit and temperatre profiles are presented for varios vales of parameters of the problem. The nmerical reslts indicate that the temperatre, skin friction and heat transfer increase as the radiation parameter increases. The velocit increases with the increase in radiation parameter, the exponential index, poros medim, Grashof nmber and plate moving velocit, while it decreases as the magnetic field parameter increases. REFERENCES bds Sattar, M.D., and M.D. Hamid Kalim, Unstead free-convection interaction with thermal radiation in a bondar laer flow past a vertical poros plate, J Math Phs Sci., 5-7 (1996. Bestman,.R., Natral convection bondar laer with sction and mass transfer in a poros medim, Int J Energ Res, 14, (199. Chamkha,.J., Unstead MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption, Int J Eng Sci., 4, 17- (4. Cowling, T.G., Magnetohdrodnamics, Interscience, New York, (1957. Elbashbesh, E.M.., Radiation effect on heat transfer over a stretching srface, Can. J. Phs., 78, (. Makinde, O.D., Free convection flow with thermal radiation and mass transfer past a moving vertical poros plate, Int Comm Heat Mass Transfer,, (5. Raptis,., Radiation and free convection flow throgh a poros medim, Int Comm Heat Mass Transfer, 5, (1998. Raptis,., and C. Perdikis, Free convection flow of water near 4 o C past a moving plate, Forschng im Ingenierwesen, 67, 6-8 (. Seddeek, M.., The effect of variable viscosit on hdromagnetic flow and heat transfer past a continosl moving poros bondar with radiation, Int. Commn. Heat Mass Transfer, 7, (. Seddeek, M.., Thermal radiation and boanc effects on MHD free convective heat generating flow over an accelerating permeable srface with temperatre-dependent viscosit, Can. J. Phs., 79, 75-7 (1. Sondalgekar, V.M., and M.R Patti, Stokes problem for a vertical plate with constant heat flx, strophs Space Sci., 7, (198. Takhar, H.S., R.S.R. Gorla and V.M. Sondalgekar, Radiation effects on MHD free convection flow of a gas past a semi-infinite vertical plate, Int. J. Nmer. Meth. Heat Flid Flow, 6, 77-8 (1996. Received: gst 14, 7. ccepted: Febrar 14, 8. Recommended b Sbject Editor: Walter mbrosini 4

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