A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD
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1 IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: pissn: A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish V Desale, V H Pradha R.C.Patel Istitute of Techology, Shirpur, Idia satishdesale@gmail.com S.V.Natioal Istitute of Techology, Surat, Idia pradha65@yahoo.com Abstract The problem of the MHD boudary layer flow of a icompressible viscous fluid over a o-liear stretchig sheet is discussed. Usig similarity trasformatio goverig equatios are trasformed i to oliear ordiary differetial equatio. The Implicit fiite differece Keller box method is applied to fid the umerical solutio of oliear differetial equatio. Graphical results of fluid velocity have bee preseted ad discussed for the related parameters. Keywords: MHD, boudary layer, oliear differetial equatios, ad Keller box method *** INTRODUCTION I 973, McCormack & Crae [8] itroduced the stretchig sheet problem. The stretchig problems for steady flow have bee used i various egieerig ad idustrial processes, like o-newtoia fluids, MHD flows, porous plate, porous medium ad heat trasfer aalysis. Magetohydrodyamics (MHD) is the study of the iteractio of coductig fluids with electromagetic pheomea. The flow of a electrically coductig fluid i the presece of a magetic field is importat i various areas of techology ad egieerig such as MHD power geeratio, MHD flow meters, MHD pumps, etc. Chiam [7] ivestigated the MHD flow of a viscous fluid bouded by a stretchig surface with power law velocity. He obtaied the umerical solutio of the boudary value problem by usig the Ruge Kutta shootig algorithm with Newto iteratio. Hayat et al. [6] applied the modified Adomia decompositio method with the Padé approximatio ad obtaied the series solutio of the goverig oliear problem. Rashidi i [4] used the differetial trasform method with the Padé approximat ad developed aalytical solutios for this problem. I [, ] applied the HAM i order to obtai a aalytical solutio of the goverig oliear differetial equatios.. GOVERNING EQUATIONS Cosider the Magetohydrodyamic flow of a icompressible viscous fluid over a stretchig sheet at y = 0. The fluid is electrically coductig uder the ifluece of a applied magetic field B(x) ormal to the stretchig sheet. The iduced magetic field is eglected. The resultig boudary layer equatios are as follows [6] u v + = 0 x y u v u B ( x) u + v = ν σ u x y y ρ where u ad v are the velocity compoets i the x - ad y directios respectively, ν is the kiematic viscosity, ρ is the fluid desity ad σ is the electrical coductivity of the fluid. I equatio (), the exteral electric field ad the polarizatio effects are egligible ad followig Chiam [7] we assume that the magetic field B takes the form B ( )/ ( x) = B x The boudary coditios correspodig to the o-liear stretchig of the sheet are [6] 0 () () I this paper, MHD boudary layer equatio is solved with the u( x,0) = cx, v( x,0) = 0 help of implicit fiite differece Keller box method ad various results are discussed graphically. Volume: 0 Issue: Dec-03, 87
2 IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: pissn: u( x, y) 0, y where c ad are costats. usig the followig substitutios c( + ) t = x y v Now, write the fiite differece approximatios of the ordiary differetial equatios (5)-(6) for the midpoit x, t of the segmet usig cetered differece x, t derivatives, this is called ceterig about. u = cx f ( t) c( + ) v = x f ( t) t f ( t) v + + Equatio ()-() are trasformed to f f u = h u u v + v = h (9) (0) ( ) ( ) + ( ) ( ) β ( ) ( ) = 0 f t f t f t f t Mf t with the followig boudary coditios f f f (0) = 0, (0) =, ( ) = 0 σ B0 β =, M = Where + ρc( + ). 3. KELLER BOX METHOD Equatio (3) subect to the boudary coditios (4) is solved umerically usig implicit fiite differece method that is kow as Keller-box i combiatio with the Newto s liearizatio techiques as described by Cebeci ad Bradshaw [3]. This method is ucoditioally stable ad has secodorder accuracy. I this method the trasformed differetial equatio (3) are writes i terms of first order system, for that itroduce ew depedet variable u, v such that f = u u = v Where prime deotes differetiatio with t. + β = 0 v fv u Mu with the ew boudary coditios (4) (3) (5) (6) (7) Now, ordiary differetial equatio (7) is approximated by the x ceterig about the mid-poit, t of the rectagle. x, t I first step we ceter about without specifyig t. ( v ) + ( fv ) β ( u ) Mu ( ) ( ) β ( ) = v + fv u Mu Now ceterig about (for simplicity, remove ) v v f + f v + v + h u u u β M = v v ( fv) β ( u ) Mu / + / h / () Now liearize the oliear system of equatios (9)-() usig the Newto s quasi-liearizatio method [5] For that use, f (0) = 0, u(0) =, u( ) = 0 (8) Volume: 0 Issue: Dec-03, 88
3 IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: pissn: f = f + δ f + u = u + δu + v = v + δ v + Equatio (8) to(0) ca be rewritte as h f f ( u ) = ( r ) () δ δ δ δ h u u ( v + v ) = ( r ) (3) δ δ δ δ This block tridiagoal structure ca be solved usig LU method explaied by Na [5]. 4. RESULT AND DISCUSSION Graphically, effects of magetic parameter ad odimesioal parameter β o velocity profile are show i followig figures. Fig. ad shows, as the magetic parameter M ad o-dimesioal parameter β icreases, the fluid velocity decreases. Also it ca be observed that as magetic parameter M icreases velocity profile decreases very rapidly as compared to icrease of o-dimesioal parameter β. Fig 3 ad 4 shows the effect of magetic parameter o f. ( a ) δv + ( a ) δv + ( a ) δ f + ( a ) δ f ( ) ( a ) δu + ( a ) δu = r (4) The liearized differece system of equatios ()-(4) has a block tridiagoal structure. I a vector matrix form, it ca be writte as [ ] [ ] A C [ ] [ ] [ ] B A C [ ] [ ] [ ] [ ] [ ] [ δ ] [ r ] [ δ ] [ r ] [ δ ] [ r ] 3 3 B A C : : : = : [ B ] [ A ] [ C ] : : B A [ δ ] [ r ] Fig -:.Effect of Magetic Parameter o fluid velocity obtaied by Keller Box Method for β = Volume: 0 Issue: Dec-03, 89
4 IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: pissn: Fig -: Effect of o fluid velocity obtaied by Keller Box Method whe M = Fig -3: Effect of Magetic Parameter o f (t) obtaied by Keller Box Method for β = Fig -4: Effect of Magetic Parameter o f (t) obtaied by Keller Box Method for β = Volume: 0 Issue: Dec-03, 90
5 IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: pissn: CONCLUSIONS I this study, MHD viscous flow over a stretchig sheet is cosidered. Keller Box Method is applied to solve the goverig oliear differetial equatio. The results idicate that the velocity will be reduced by icreasig the two parameters of M ad β; however, effect of M i reductio of the velocity compoets is more tha β. Therefore, it ca be cocluded that Keller Box is oe of the best method to study o MHD viscous flow umerically ad get the appropriate results. REFERENCES [] A.Mehmood, S. Muawar, A. Ali, Commets to: Homotopy aalysis method for solvig the MHD flow over a o-liear stretchig sheet (Commu. Noliear Sci. Numer. Simulat. 4 (009) ), Commuicatios i Noliear Sciece ad Numerical Simulatio, 5 (00) [] A.R. Ghotbi, Homotopy aalysis method for solvig the MHD flow over a o-liear stretchig sheet, Commuicatios i Noliear Sciece ad Numerical Simulatio,4 (009) [3] Cebeci T., Bradshaw P. Physical ad computatioal Aspects of Covective Heat Trasfer, New York: Spriger (988), [4] M.M. Rashidi, The modified differetial trasform method for solvig MHD boudary-layer equatios, Computer Physics Commuicatio, 80 (009), 0-7. [5] Na T.Y., Computatioal Methods i Egieerig Boudary Value Problem. New York: Academic Press (979), -8. [6] T. Hayat, Q. Hussai, T. aved, The modified decompositio method ad Padé approximats for the MHD flow over a o-liear stretchig sheet, Noliear Aalysis: Real World Applicatios, 0 (009), [7] T.C. Chiam, Hydromagetic flow over a surface stretchig with a power-law velocity, It.. Eg. Sci. 33 (995), [8] McCormack P.D., Crae L., "Physics of Fluid Dyamics", New York, Academic Press, (973). Volume: 0 Issue: Dec-03, 9
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