Study of MHD Oblique Stagnation Point Assisting Flow on Vertical Plate with Uniform Surface Heat Flux

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1 World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: 011 Std of MHD Obliqe Stagnation Point Assisting Flow on Vertical Plate with Uniform Srface Heat Fl Phool Singh Ashok Jangid N.S. omer and Deepa Sinha International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101 Abstract he aim of this paper is to std the obliqe stagnation point flow on ertical plate with niform srface heat fl in presence of magnetic field. Ug Stream fnction partial differential eqations corresponding to the momentm and energ eqations are conerted into non-linear ordinar differential eqations. Nmerical soltions of these eqations are obtained g Rnge-Ktta Fehlberg method with the help of shooting techniqe. In the present work the effects of striking angle magnetic field parameter Grashoff nmber the Prandtl nmber on elocit and heat transfer characteristics hae been discssed. Effect of aboe mentioned parameter on the position of stagnation point are also stdied. Kewords Heat fl Obliqe stagnation point Mied conection Magneto hdrodnamics I. INRODUCION HE stagnation flow of an incompressible iscos flid oer ertical srface is important in arios processes. he technical process concerning etrsion of polmer sheets rolling and manfactring artificial fibers inoles the stagnation point flow. In stagnation point flow a rigid wall or a stretching srface occpies the entire horizontal ais. he flid domain is normal to that ais and the flid strikes on that srface either orthogonal or at some angle of incidence. his simple model of obliqe stagnation point wold enable s to nderstand how a bondar laer begins to deelop. Mied (combined forced and free) conection in stagnation flows is er important. he boanc forces take place when the temperatre difference between the two walls and the free stream is becomes high. hese forces modif the flow and the thermal fields significantl. In sch flows the local heat transfer rate and local skin friction can be significantl enhanced or diminished in comparison to the pre forced conection case. Depending on the forced flow direction the boanc forces ma assist or oppose the forced flow. his reslts in increase or decrease in heat transfer rate. hese reslts hae been discssed b Chamkha [1] Chamkha and Phool Singh is with the Department of Mathematics itm Uniersit Sector -A Grgaon Harana India (phone: ; fa: ; phool.gh4@gmail.com). Ashok Jangid and N.S. omer is with the Department of Applied Sciences itm Uniersit Sector -A Grgaon Harana India ( ashjangid@gmail.com and tomar_ns@rediffmail.com). Deepa Sinha is with the Centre for Mathematical Sciences (CMS) Banasthali Uniersit Rajasthan-040 India ( deepaha001@gmail.com). Issa [] Kmari [] Adin and Kaa [4] and Prasad et al. [5].Present research field has attracted man researchers in recent ears de to its applications. he std of magneto hdrodnamic flow of an electricall condcting flid cased b the deformation of the walls of a essel containing a flid is of considerable interest in modern metallrgical and metalworking processes. Magnetic field enhances the elocit gradient and heat transfer rate at srfaces de to the increase in the Lorentz force. Attia [6] inestigated effect of increag magnetic field on elocit bondar laer thickness. Kmari and Nath [7] [8] inestigated the MHD effects on Mawell flids for stead and nstead cases. he fond that elocit gradient at the srface and heat transfer increase with magnetic parameter. Singh et al. [9] [10] reported effect of magnetic parameter and radiations on stretching sheet for stead and nstead flow. Zia et al. [11] [1] inestigated Magnetohdrodnamic free conection flow of a iscos flid throgh inclined poros plate in the presence of high temperatre. All the aboe mentioned stdies confined their discssions b assming orthogonal flow. Howeer it is known that the flid ma incident at some angle on the srface. he angle of incidence of flid leads to a local increase/ decrease in the transport phenomena. herefore to predict the flow behaior accratel it is necessar to take into accont non-orthogonal flows for incompressible flids. Grosan et al. [1] and Singh et al. [14] analzed the magneto-hdrodnamic obliqe stagnation-point flow on a flat plate and stretching sheet respectiel. It was fond that the magnetic parameter cases a shift in the position of the stagnation-point. Lok et al. [15] [16] inestigated the nonorthogonal stagnation-point for Newtonian and non- Newtonian flows towards a sheet. he fond that the position of stagnation point depend on stretching sheet parameter and angle of incidence. Mahapatra et al. [17] and Labropl et al. [18] [19] analzed obliqe stagnation point flow of incompressible isco-elastic flid towards a stretching sheet. ille and Weidman [0] stdied interaction between two planar obliqe stagnation point flow of different immiscible flid. he std of heat transfer and flow field is necessar for determining the qalit of the final prodct. Amaoche and Bokari [1] stdied the inflence of thermal conection on non-orthogonal stagnation point flow. he athors in the present paper stdied the stead twodimensional boanc indced mied conection oppog flow of a iscos incompressible flid striking at different International Scholarl and Scientific Research & Innoation 5() scholar.waset.org/ /101

2 angles of incidence on a ertical semi-infinite flat plate. Flid flow is considered in presence of magnetic field with niform srface heat fl. II. FORMULAION OF PROBLEM he phsical model considered here consist of a iscos incompressible stead two-dimensional mied conection flow of an electricall condcting flid striking at some angle of incidence on a ertical semi-infinite flat plate. A constant magnetic field o B is applied in - direction as shown in Figre 1. Fig. 1 Phsical Model of the Problem he coordinate sstem in Figre 1 is sch that represents the ertical distance or the distance along the srface and represents the horizontal distance or the distance normal to the srface. Far awa from the plate the elocit and the temperatre of the niform main stream are cos ) ( i b j i a V and respectiel where i and j are nit ectors along the and aes a and b are the positie constant with dimension (time) -1. he entire srface of the plate is maintained at a constant heat fl w q. he iscos dissipation Jole heating and indced magnetic field are neglected. he goerning eqations of continit momentm and energ nder aboe assmptions with Bosesq approimation are 0 (1) ) ( 1 g Bo p β σ () 1 p () and α (4) where and are elocit components along and aes respectiel is kinematic iscosit σ is electrical condctiit p is the pressre is temperatre of the plate is densit of the flid β is coefficient of thermal epansion g is acceleration de to boanc and α is the thermal diffsiit. Bondar conditions are: 0 0 at k qw as a b a cos (5) where k is coefficient of thermal condctiit. his flow model is called faorable flow for / 0 π and nfaorable flow for π π < /. Case / π is known as orthogonal flow. Eliminating pressre terms from () and () we get the following eqations. 0 B Zg σ β (6) Let is Stream fnction defined as / and /. Introdcing the dimensionless ariables defined as k qwl l l and where ( ) ( ) 4 1/ cos 1/ b a l. Eqation of Continit (1) is also satisfied b ) ( and ) ( defined as aboe and (6) and (4) redces to 0 M Gr (7) Pr 1 (8) World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: International Scholarl and Scientific Research & Innoation 5() 011 scholar.waset.org/ /101 International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101

3 World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: 011 International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101 respectiel. Where Pr / α is the Prandtl nmber Gr g βq 4 / k w l is Grashoff nmber based on the heat fl q w > 0 and M σ B l o / is magnetic field parameter. We seek soltion of (7) in the form of f ( g( where the fnction f ( ) and g( ) are referring to the normal and tangential component of the flow. herefore elocit components are gien b ( f '( ( and ( f (. From (7) and (8) we hae f '''( f ( f ''( f '( Mf '( 0 (9) '' ( f ( '( f ' ( ( M[ ( cos ] Gr '( 0 (10) ''( Pr F ( '( 0 (11) and bondar conditions becomes f ( 0) 0 f '(0) 0 g(0) 0 (0) 0 '(0) 1 at 0 f '( ) ' ( ) cos ( ) 0 as. (1) A phsical qantit of interest is the skin friction or shear stress τ w at the wall which is defined as τ w μ or 0 τ w μ( f ''(0) '(0)) (1) he diiding streamlines 0 and the cre 0 intersect the wall at the stagnation point where τ w 0 hence the location of stagnation point s is gien b '(0) s (14) f ''(0) III. NUMERICAL SIMULAION he set of non-linear copled differential eqations (9) (10) and (11) sbject to the bondar conditions (1) constitte a two-point bondar ale problem. In order to sole these eqations nmericall we follow nmerical shooting techniqe with Rnge Ktta Fehlberg integration scheme. In this method it is most important to choose the appropriate finite ales of. he soltion process is repeated with another large ale of ntil two sccessie ales of f ''(0) g '' (0) and (0) differ onl after a desired digit signifing the limit of the bondar along. he last ale of is chosen as appropriate ale of the limit for that particlar set of parameters. he three ordinar differential eqation (9) (10) and (11) were first conerted into a set of eight first-order simltaneos eqations. o sole this sstem we reqire eight initial conditions bt we hae onl fie initial conditions f (0) and f '(0) on f ( ) g (0) and g '(0) on g ( ) and one initial condition '(0) on ( ). Still there are three initial conditions f ''(0) g ''(0) and (0) are not prescribed. Howeer the ales of f '( g ''( and ( ) are known at. Now we emplo the nmerical shooting techniqe where these three ending bondar conditions were sed to prodce three nknown initial conditions at 0. Finall the problem has been soled nmericall g Rnge-Ktta Fehlberg integration scheme. IV. RESULS AND DISCUSSIONS In the absence of an analtical soltion of a problem nmerical soltion is indeed an obios and natral choice. hs the goerning bondar laer and thermal bondar laer eqations (9) (10) and (11) with bondar conditions (1) are soled g Rnge-Ktta Fehlberg method with shooting techniqe. Different ales of M Pr and Gr taking step size sed for nmerical simlation. While nmerical simlation step size 0.00 and 0.00 were also checked and ales of f ''(0) '(0) and (0) were fond in each case correct p to si decimal places. Hence the scheme sed in this paper is stable and consistent. It is worth mentioning that small ales of Pr ( << 1) phsicall corresponds to liqid metals while Pr 1 corresponding to diatomic gases inclding air. On the other hand Pr >> 1corresponds to high iscosit oils and Pr 7 corresponds to water at room temperatre. De to decopled bondar laer eqation (9) it is fond that there is onl a niqe ale of redced skin friction for orthogonal flow f ''(0) which is er good agreement with the ale f ''(0) fond b Hiemenz as described b [15]. ABLE I VALUES OF f ''(0) FOR DIFFEREN VALUES OF ANGLE OF INCIDENCE AND MAGNEIC FIELD PARAMEER M. Vale of f ''(0) M 0 M M M 5 π / π / π / π / Effect of striking angle on flow has been shown in Figre and able I. It is fond that bondar laer thickness increase as striking angles increases for faorable flow whereas thermal bondar laer thickness decrease as striking angle increases as shown in Figre. Vales of bondar laer separation part s for assisting flow are gien in able V and VI for Pr 0. 71and Gr 0. able V has been calclated with the help of able[i-iii] and (14). It has been obsered from able V that stagnation point shifts towards origin as striking angle increased from π / 5 to π / for an ale of Gr. Stagnation point moe awa from origin as magnetic parameter increased. Grashoff nmber Gr gies the effect of boanc. hs it is the dominant parameter in free conection. Effect of Gr can be seen from able V and able International Scholarl and Scientific Research & Innoation 5() scholar.waset.org/ /101

4 World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: 011 VI that in presence of boanc parameter ( Gr ) shifting of stagnation point is more prononced than absence of boanc parameter ( Gr 0 ). ABLE II VALUES OF g ''(0) FOR DIFFEREN VALUES OF ANGLE OF INCIDENCE AND MAGNEIC FIELD PARAMEER M WHEN Gr 0 AND Pr Vale of '(0) when Gr 0 and Pr M 0 M M M 5 π / π / π / π / International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101 ABLE III VALUES OF g ''(0) FOR DIFFEREN VALUES OF ANGLE OF INCIDENCE AND MAGNEIC FIELD PARAMEER M WHEN Gr AND Pr Vale of '(0) when Gr and Pr M 0 M M M 5 π / π / π / π / ABLE IV VALUES OF (0) FOR DIFFEREN VALUES OF ANGLE OF INCIDENCE AND MAGNEIC FIELD PARAMEER M WHEN Gr 0 AND Pr Vale of (0) when Gr 0 and Pr M 0 M M 5 π / π / π / π / ABLE V VALUE OF SAGNAION POIN s FOR DIFFEREN MAGNEIC FIELD PARAMEER M AND ANGLE OF INCIDENCE WHEN Gr 0 Pr Vale of s when Gr 0 Pr M 0 M M M 5 π / π / π / ABLE VI VALUE OF SAGNAION POIN s FOR DIFFEREN MAGNEIC FIELD PARAMEER M AND ANGLE OF INCIDENCE WHEN Gr Pr Vale of s when Gr Pr M 0 M M M 5 π / π / π / π / he magnetic parameter M represents the importance of magnetic field on the flow. he presence of transerse magnetic field sets in Lorentz force which reslts in retarding Fig. Velocit profile f '( erss for different ales when M 0. Fig. emperatre profile ( ) erss for different ales when Pr and Gr 0. Fig. 4 Velocit profile f '( erss for different ales M when π /. force on the elocit field and therefore as magnetic parameter increases so does the retarding force and hence the elocit decrease. his is shown in Figre 4. Same effect is on temperatre that can be seen from Figre 5. Vale of International Scholarl and Scientific Research & Innoation 5() scholar.waset.org/ /101

5 World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: 011 International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101 f ''(0) for faorable flow is decreases as magnetic parameter increase as shown in able I. It is obsered from able IV that (0) increases with increase in magnetic parameter. Fig. 5 emperatre profile ( ) erss for different ales M when Pr 0.71 Gr 0 and π /. Fig. 6 emperatre profile ( ) erss for different ales Pr when Gr 0 and π /. ABLE VII VALUES OF (0) FOR DIFFEREN VALUES OF PRANDAL NUMBER Pr GRASHOF NUMBER Gr AND MAGNEIC FIELD PARAMEER M WHEN π /. Pr Vale of (0) when π /. Gr 0 Gr M 0 M M 0 M Pr Pr Pr Pr It is obsered from Figres (6) that the temperatre profile decreases with an increase in the Prandtl nmber Pr for striking angle π /. his is in agreement with the phsical fact that at higher Prandtl nmber flid has a thinner thermal bondar laer and this increases the gradient of temperatre. Vale of (0) decreases as Prandtl nmber increase as shown in able VII. V. CONCLUSION he two dimensional obliqe stagnation point flow on ertical plate with niform srface heat fl in presence of magnetic field is stdied. A nmerical soltion for the goerning eqations is obtained which allows the comptation of the flow and heat transfer characteristics for arios ales of the magnetic parameter striking angle Grashoff nmber and the Prandtl nmber. he main reslts of the paper can be smmarized as follows: 1) he elocit bondar laer thickness decreases as striking angle decreases. ) Stagnation point shifts towards origin as striking angle increases. ) hermal bondar laer thickness increases as striking angle decreases. 4) Velocit and temperatre decrease as magnetic field increases. 5) emperatre decreases with increase in the Prandtl nmber. 6) In absence of bonac forces stagnation point moes to left of origin while in presence it moes right of origin as striking angle decreases. 7) Stagnation point moe awa from origin as magnetic parameter increases. hese reslts hae potential technological applications in liqid-based sstems. REFERENCES [1] Chamkha A.J. Hdromagnetic mied conection stagnation flow with sction and blowing International Commnications in Heat and Mass ransfer ol. 5 pp [] Chamkha A.J. and C. Issa Mied conection effects on nstead flow and heat transfer oer a stretched srface International Commnications in Heat and Mass ransfer ol. 6 pp [] Kmari M. Variable iscosit effects on free and Mied conection bondar-laer flow from a horizontal srface in a satrated poros medim-ariable heat fl Mechanics Research Commnications ol. 8 pp [4] Adin O. and A. Kaa Mied conection of a iscos dissipating flid abot a ertical flat plate Applied Mathematical Modelling ol. 1 pp [5] Prasad K.V. K. Vajrael and P.S. Datti Mied conection heat transfer oer a non-linear stretching srface with ariable flid properties International Jornal of non-linear Mechanics ol. 45 pp [6] Attia H. A. Hdromagnetic stagnation point flow with heat transfer oer a permeable srface he Arabian Jornal for Science and Engineering ol [7] Kmari M. and G. Nath Unstead MHD mied conection flow oer an implsiel stretched permeable ertical srface in a qiescent flid. International Jornal of non-linear Mechanic ol. 45 pp [8] Kmari M. and G. Nath Stead mied conection stagnation-point flow of pper conected Mawell flids with magnetic field International Jornal of non-linear Mechanic ol. 44 pp [9] Singh P. N.S. omer and D. Sinha Nmerical std of heat transfer oer stretching srface in poros media with transerse magnetic field Proceeding of International Conference on Challenges and application of Mathematics in Sciences and echnolog ISBN X International Scholarl and Scientific Research & Innoation 5() scholar.waset.org/ /101

6 World Academ of Science Engineering and echnolog International Jornal of Mechanical and Mechatronics Engineering Vol:5 No: 011 International Science Inde Mechanical and Mechatronics Engineering Vol:5 No: 011 waset.org/pblication/101 [10] Singh P. A. Jangid N.S. omer and D. Sinha Effects of hermal Radiation and Magnetic Field on Unstead Stretching Permeable Sheet in Presence of Free Stream Velocit International Jornal of Engineering and Mathematical Sciences ol. 6 no. pp [11] Zia U. M. Kmar and V. Bisht Radiation heat transfer effect on a moing semi-infinite tilted poros heated plate with niform sction in the presence of transerse magnetic field Ganita ol. 60 no. 1 pp [1] Zia U. and M. Kmar ransient MHD free conection flow of an incompressible iscos dissipatie flid throgh inclined poros plate. International J Essential Sciences ol. no [1] Grosan. I. Pop C. Renic and D.B. Ingham Magnetohdrodnamic obliqe stagnation-point flow Meccanica ol. 44 pp [14] Singh P. N.S. omer S. Kmar and D. Sinha MHD obliqe stagnation-point flow towards a stretching sheet with heat transfer International Jornal of Applied Mathematics and Mechanics ol. 6 no.1 pp [15] Lok Y.Y. N. Amin and I. Pop Mied conection near a nonorthogonal stagnation point flow on a ertical plate with niform srface heat fl Acta Mechanica ol. 186 pp [16] Lok Y.Y. N. Amin and I. Pop Mied conection flow near a nonorthogonal stagnation point towards a stretching ertical plate International Jornal of Heat and Mass ransfer ol. 50 pp [17] Mahapatra.R. S.K. Nand and A.S. Gpta Magnetohdrodnamic stagnation-point flow of a power-law flid towards a stretching srface International Jornal of Non-Linear Mechanics ol. 44 pp [18] Labropl F. D. Li and I. Pop Non-orthogonal stagnation-point flow towards a stretching srface in a non-newtonian flid with heat transfer International Jornal hermal Sciences ol. 49 pp [19] Li D. F. Labropl and I. Pop Obliqe stagnation-point flow of a iscoelastic flid with heat transfer International Jornal of Non- Linear Mechanics ol. 44 pp [0] ille B.S. and P.D. Weidman Obliqe two-flid stagnation-point flow Eropean Jornal of Mechanics ol. 17 pp [1] Amaoche M. and D. Bokari Inflence of thermal conection on non-orthogonal stagnation point flow. International Jornal of hermal Sciences ol. 4 pp International Scholarl and Scientific Research & Innoation 5() scholar.waset.org/ /101