Boundary Layer Flow and Heat Transfer over a. Continuous Surface in the Presence of. Hydromagnetic Field

Transcription

1 International Jornal of Mathematical Analsis Vol. 8, 4, no. 38, HIKARI Ltd, Bondar Laer Flow and Heat Transfer over a Continos Srface in the Presence of Hdromagnetic Field M. Dakshinamoorth, P. Geetha and M. B. K. Moorth 3 Department of Phsics, Institte of Road and Transport, Erode , India Department of Mathematics, Bannari Amman Institte of Technolog Sathamangalam-638 4, India Corresponding athor 3 Department of Mathematics, Institte of Road and Transport Erode , India Copright 4 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth. This is an open access article distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in an medim, provided the original work is properl cited. Abstract Two dimensional stead flow of an electricall condcting, viscos incompressible flid past a continosl moving srface is considered in the presence of niform transverse magnetic field. The non-linear partial differential eqations, governing the problem nder consideration have been transformed b a similarit transformation into a sstem of ordinar differential eqations, which is solved nmericall b appling Rnge-Ktta Gill method along with shooting techniqe. The effects of the parameters sch as reference temperatre, exponent, magnetic, Prandtl nmber and Eckert nmber are discssed graphicall on velocit and temperatre distribtions. Kewords: Bondar laer, Hdro magnetic Flow, Heat transfer, Moving srface

2 86 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth INTRODUCTION In recent ears, the problems of free convective heat transfer flows throgh a poros medim nder the inflence of a magnetic field have been attracted the attention of a nmber of researchers becase of their possible applications in man branches of science and technolog, sch as its applications in transportation cooling of re-entr vehicles and rocket boosters, cross-hatching on ablative srfaces and film vaporization in combstion chambers. Flow throgh a poros medim have nmeros engineering and geophsical applications, for example, in chemical engineering for filtration and prification process; in agricltre engineering to std the ndergrond water resorces; in petrolem technolog to std the movement of natral gas, oil and water throgh the oil reservoirs. In view of these applications, man researchers have stdied MHD free convective heat and mass transfer flow in a poros medim. Rahman and Sattar [] have investigated the effect of heat generation or absorption on convective flow of a micropolar flid past a continosl moving vertical poros plate in presence of a magnetic field. The chemical reaction effect on heat and mass transfer flow along a semi-infinite horizontal plate has been stdied b Anjalidevi and Kandaswam [] and later it was extended for Hiemenz flow b Seddeek et al. [5] and for polar flid b Patil and Klkarni []. Salem and Abd El-Aziz [4] have reported the effect of hall crrents and chemical reaction on hdromagnetic flow of a stretching vertical srface with internal heat generation or absorption. Ibrahim et al. [7] stdied the effect of chemical reaction and radiation absorption on the nstead MHD free convection flow past a semi-infinite vertical permeable moving plate with heat sorce and sction. A detailed nmerical std has been carried ot for nstead hdromagnetic natral convection heat and mass transfer with chemical reaction over a vertical plate in rotating sstem with periodic sction b Parida et al. []. Rajeswari et al. [3] have investigated chemical reaction, heat and mass transfer on nonlinear MHD bondar laer flow throgh a vertical poros srface in presence of sction. Mahd [] has stdied the effect of chemical reaction and heat generation or absorption on doble diffssive convection from vertical trncated cone in a poros media with variable viscosit. Pal and Talkdar [9] have stdied pertrbation analsis of nstead magnetohdrodnamic convective heat mass transfer in bondar laer slip flow past a vertical permeable plate with a thermal radiation and chemical reaction. Frther the effect of thermal radiation, heat and mass transfer flow of a variable viscosit flid past a vertical poros plate in presence of transverse magnetic field was investigated b Makinde and Ogl [5]. The analsis of MHD

3 Bondar laer flow and heat transfer 86 mixed-convection interaction with thermal radiation and higher order chemical reaction is carried ot b Makinde [6]. Aziz [] theoreticall examined a similarit soltion for a laminar thermal bondar laer over a flat plate with a convective srface bondar condition. He fond an interesting reslt that a similarit soltion is possible if the convective heat transfer along with the hot flid on the lower srface of the plate is inversel proportional to the sqare root of the axial distance. Recentl, the combined effects of an exponentiall decaing internal heat generation and a convective bondar condition on the thermal bondar laer over a flat plate are investigated b Olanrewaj et al. [8]. In their std athors have neglected the Sherwood effect. Similar analsis has been carried ot b Makinde [, 3] withot heat sorce and with heat sorce [4], neglecting chemical reaction effect. There has been considerable interest in stding the effect of chemical reaction [6] and heat sorce effect on the bondar laer flow problem with heat and mass transfer of an electricall condcting flid in different geometr [7,4 and 3]. The problem of flow and heat transfer over a moving srface has drawn considerable attention and a ver good amont of literatre has been generated on this problem (Fang [5] and Ishaket al. [8]). Kmari and Nath [] discssed the problem of MHD bondar laer flow of a non-newtonian flid over a continosl moving srface with a parallel free stream. Recentl, Jat and Chadhar [9] stdied the flow of incompressible viscos condcting flid past a continos moving srface in the presence of transverse magnetic field. Sondalgekar et al. [6] stdied the flow of incompressible viscos flid past a continosl moving semi-infinite plate b considering variable viscosit and variable temperatre. The objective of the present paper is to std the effects on both momentm and heat transfer problem with viscos dissipation and Jole heat transfer for an electricall condcting flid past a continosl moving plate in the presence of a niform transverse magnetic field. MATHEMATICAL FORMULATION Consider the two-dimensional stead flow of an electricall condcting, viscos, incompressible flid past a continosl moving srface with niform velocit U in the presence of niform transverse magnetic field of strength B. The magnetic Renolds nmber is assmed to be small so that the indced magnetic field can be neglected. The x-axis is taken along the srface and -axis normal to it as shown in fig.. The flid properties are assmed to be isotropic and constant. Therefore, nder the sal bondar laer approximations, the governing eqations of motion are

4 86 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth v x () B v x e () p e p p c B c T c k T v x T..(3) where is the densit, be the coefficient of viscosit, e the electrical condctivit, k thermal condctivit of the flid and cp the specific heat at constant pressre. The other smbols have their sal meanings. The corresponding bondar conditions are: = ; = U; v = ; T= Tw(x) = ; = ; T = T ;..(4) The continit eqation () is satisfied b introdcing the stream fnction (x, ), sch that x v,..(5) The momentm and energ eqations ()and (3) can be transformed to the corresponding ordinar differential eqations b introdcing the following similarit transformations (Blasis (98)): () f Ux..(6) x U..(7)

5 Bondar laer flow and heat transfer 863 T T T T w....(8) where is a reference kinematic viscosit. It will be assmed that the temperatre difference between the moving srface and the free stream varies as Ax n. i.e.tw(x)-t =Ax n.....(9) where A is a constant, n is exponent, and x is measred from the leading edge of the srface. The momentm and energ eqations () (3) after some simplifications, redce to the following forms: f ''' ' f r '' ' n Pr f Pr '' r f r Re m ' r f Pr Ec f r The corresponding bondar conditions are f '' r f r '' Rem ' Pr Ecf '.()....() = ; f = ; f = ; = = ; f = ; =..() Where the prime () denotes differentiation with respect to and e x Re m B (Magnetic parameter) U

6 864 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth Ec c p U ( T T w ) (Eckert nmber)..(3) and r T T r w T T (Dimensionless reference temperatre, constant) It is important to note that r is negative for liqids (Pr >.) and positive for gases (Pr <.). The eqations () and () constitte a non-linear copled bondar vale problem prescribed at two bondaries, the analtical soltion of which is not feasible. Therefore, these eqations have been solved nmericall on compter sing Newton s shooting techniqes with the Rnge-ktta Gill method with a step size of.. The corresponding velocit and temperatre profiles are shown in fig. to fig.9. RESULTS AND DISCUSSION The profiles for velocit and temperatre are shown in fig. to fig.9. It is seen from fig. that the velocit decreases as n and Rem, the power-law index of the srface temperatre variation (exponent) and the magnetic parameter increases with Prandtl nmber Pr =. and the variable viscosit parameter r = respectivel. From fig.3 and fig. 4, it is nderstood that the velocit profiles are almost identical for different vales of n and Rem. From fig.5 the velocit decreases as Magnetic parameter Rem increases. Figs. 6,7, 8 and 9 show the temperatre profiles for different vales of n and Rem. It is seen from fig.6 that as n increases the temperatre decreases for fixed vale of Rem. Fig.7 and fig.8 reveal that the temperatre decreases as n and Rem increases. Fig. 9 shows that the temperatre decreases as the magnetic parameter increases with Pr=., n =.3 and Ec =. Fig. gives the vales of - ' (), the heat transfer rate for varios vales of n and Rem. From the figre it is observed that as Rem increases the heat transfer rate - ' () decreases bt as n increases the heat transfer rate increases. So the parameters n and Rem have considerable inflence on the heat transfer rate - ' ().

7 Bondar laer flow and heat transfer 865 CONCLUSION The effects of power law index of the srface temperatre variation (exponent), magnetic parameter and variable parameter on a hdromagnetic flow and heat transfer on a continosl moving srface have been stdied nmericall sing the RK Gill method together with the shooting techniqe. From the previos reslts and discssion, we conclde the following () The velocit decreases with the increase of power law index of the srface temperatre variation (exponent) and the magnetic parameter () The temperatre decreases with the increase of the power law index of the srface temperatre variation (exponent) and the magnetic parameter. (3) The heat transfer rate increases rapidl with the increase of power law index of the srface temperatre variation (exponent) whereas when the magnetic parameter increases the heat transfer rate decreases. ACKNOWLEDGEMENT The athors wish to thank the Chairman, Trstee, Chief Exective, Principal, Bannari Amman Institte of Technolog for their spport to do this work. The athors wish to thank the Director IRT Chennai, and the Principal IRTT, Erode for their generos help. Bo Bo O x U Fig.. Coordinate sstem for continosl moving srface Continos moving srface

8 866 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth f ' (),,8,6,4, n=.3,rem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem=.3 5 Fig. velocit profile against for different vales of n and Rem with Pr=. and r =. f ' (),,8,6,4, n=.3,rem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem= Fig.3 velocit profile against for different vales of n and Rem with Pr=., Ec= and r =.

9 Bondar laer flow and heat transfer 867 f ' (),,8,6,4, n=.3,rem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem= Fig.4 velocit profile against for different vales of n and Rem with Pr=.,Ec=.5 & r =.,,8,6 f' (),4, Rem= Rem= Fig.5 velocit profile against for different vales of n and Rem with Pr=., n =.3 & Ec =.

10 868 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth,,8,6,4 n=.3,,rem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem=.3, 5 Fig.6 Temperatre profile against for different vales of n and Rem with Pr=. and r =.,,8,6,4 n=.3,rem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem=.3, 5 5 Fig.7 Temperatre profile against for different vales of n & Rem with Pr=.,Ec =. & r =.

11 Bondar laer flow and heat transfer 869,,8,6 n=.3,rrem= n=.3,rem=.3 n=,rem= n=,rem=.3 n=,rem= n=,rem=.3,4, n=.3 Fig.. 8 Temperatre profile against for different vales of n and Rem with Pr=., Ec=.5& r =.,,8,6,4, 5 5 Fig. 9 Temperatre profile against for different vales of n & Rem with Pr=.,n =.3&Ec =

12 87 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth,,8,6,4,,3 Fig. Heat transfer rate References [] S. P. Anjalidevi and R. Kandasam, Effects of chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate, Heat and Mass Transfer, vol. 35, no. 6, pp , 999. [] A. Aziz, A similarit soltion for laminar thermal bondar laer over a flat plate with a convective srface bondar condition, Commnications in Nonlinear Science and Nmerical Simlation, vol. 4, no. 4, pp , 9. [3] A. A. Bakr, Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar flid with oscillator plate velocit and constant heat sorce in a rotating frame of reference, Commnications in Nonlinear Science and Nmerical Simlation, vol. 6, no., pp ,. [4] V. Bisht, M. Kmar, and Z. Uddin, Effect of variable thermal condctivit and chemical reaction on stead mixed convection bondar laer flow with heat and mass transfer inside a cone de to a point, Jornal of Applied Flid Mechanics, vol. 4, no. 4, pp ,. [5] Fang T, Similarit soltions for a moving flat plate thermal bondar laer, Acta Mech., 63; 6-7, 3. [6] K. Gangadhar, N. B. Redd, and P. K. Kameswaran, Similarit soltion of hdromagnetic heat and mass transfer over a vertical plate with convective srface bondar condition and chemical reaction, International Jornal of Nonlinear Science, vol. 3, no. 3, pp ,.

13 Bondar laer flow and heat transfer 87 [7] F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr, Effect of the chemical reaction and radiation absorption on the nstead MHD free convection flow past a semi infinite vertical permeable moving plate with heat sorce and sction, Commnications in Nonlinear Science and Nmerical Simlation, vol. 3, no. 6, pp , 8. [8] Ishak A, Nazar R and Pop I, Flow and heat transfer characteristics on a moving flat plate in a parallel stream with constant heat flx, Heat & Mass Transfer, 45; , 8. [9] Jat R N and Chadhar S, Hdromagnetic flow and heat transfer on a continos moving srface, Applied Mathematical Sciences, 4; 65-78,. [] Kmari M and Nath G, MHD bondar-laer flow of a non-newtonian flid over a continosl moving srface with a parallel free stream, Acta Mech., 46; 39-5,. [] A. Mahd, Effect of chemical reaction and heat generation or absorption on doble-diffsive convection from a vertical trncated cone in poros media with variable viscosit, International Commnications in Heat and Mass Transfer, vol. 37, no. 5, pp ,. [] D. Makinde, On MHD heat and mass transfer over a moving vertical plate with a convective srface bondar condition, Canadian Jornal of Chemical Engineering, vol. 88, no. 6, pp ,. [3] O. D. Makinde, Similarit soltion of hdromagnetic heat and mass transfer over a vertical plate with a convective srface bondar condition, International Jornal of Phsical Sciences, vol. 5, no. 6, pp. 7 7,. [4] O. D. Makinde, Similarit soltion for natral convection from a moving vertical plate with internal heat generation and a convective bondar condition, Thermal Science, vol. 5, spplement, pp. S37 S43,. [5] O. D. Makinde and A. Ogl, The effect of thermal radiation on the heat and mass transfer flow of a variable viscosit flid past a vertical poros plate permeated b a transverse magnetic field, Chemical Engineering Commnications, vol. 95, no., pp , 8. [6] O. D. Makinde, MHD mixed-convection interaction with thermal radiation and nth order chemical reaction past a vertical poros plate embedded in a poros medim, Chemical Engineering Commnications, vol. 98, no. 4, pp ,. [7] N.F.M. Noor, S. Abbaband, and I. Hasim, Heat and Mass Transfer of thermophoretic MHD flow over an inclined radiate isothermal permeable srface in presence of heat sorce/sink, International Jornal of Heat and Mass Transfer, vol. 55, no. 7, pp. 8,.

14 87 M. Dakshinamoorth, P. Geetha and M. B. K. Moorth [8] P. O. Olanrewaj, O. T. Arlogn, and K. Adebimpe, Internal heat generation effect on thermal bondar laer with a convective srface bondar condition, American Jornal of Flid Dnamics, vol., no., pp. 4,. [9] D. Pal and B. Talkdar, Pertrbation analsis of nstead magnetohdrodnamic convective heat and mass transfer in a bondar laer slip flow past a vertical permeable plate with thermal radiation and chemical reaction, Commnications in Nonlinear Science and Nmerical Simlation, vol. 5, no. 7, pp ,. [] K. Parida, M. Achara, G. C. Dash, and S. Panda, MHD heat and mass transfer in a rotating sstem with periodic sction, Arabian Jornal for Science and Engineering, vol. 36, no. 6, pp. 39 5,. [] P. M. Patil and P. S. Klkarni, Effects of chemical reaction on free convective flow of a polar flid throgh a poros medim in the presence of internal heat generation, International Jornal of Thermal Sciences, vol. 47, no. 8, pp , 8. [] M. M. Rahman and M. A. Sattar, Magnetohdrodnamic convective flow of a micropolar flid past a continosl moving vertical poros plate in the presence of heat generation/absorption, Jornal of Heat Transfer, vol. 8, no., pp. 4 5, 6. [3] R. Rajeswari, B. Jothiram, and V. K. Nelson, Chemical reaction, heat and mass transfer on nonlinear MHD bondar laer flow throgh a vertical poros srface in the presence of sction, Applied Mathematical Sciences, vol. 3, no. 49-5, pp , 9. [4] A. M. Salem and M. Abd El-Aziz, Effect of Hall crrents and chemical reaction on hdromagnetic flow of a stretching vertical srface with internal heat generation/absorption, Applied Mathematical Modelling, vol. 3, no. 7, pp , 8. [5] M. A. Seddeek, A. A. Darwish, and M. S. Abdelmegid, Effects of chemical reaction and variable viscosit on hdromagnetic mixed convection heat and mass transfer for Hiemenz flow throgh poros media with radiation, Commnications in Nonlinear Science and Nmerical Simlation, vol., no., pp. 95 3, 7. [6] V.M. Sondalgekar, H.S. Takhar, U.N. Das, R.K. Deka and A. Sarmah, Effect of variable viscosit on bondar laer flow along a continosl moving plate with variable srface temperatre, Heat and Mass Transfer, 4, 4-44, 4. Received: Jl 5, 4