NUMERICAL ANALYSIS OF FORTH-ORDER BOUNDARY VALUE PROBLEMS IN FLUID MECHANICS AND MATHEMATICS

Transcription

1 THERMAL SCIENCE: Year, Vol., No., pp. -9 NUMERICAL ANALYSIS OF FORTH-ORDER BOUNDARY VALUE PROBLEMS IN FLUID MECHANICS AND MATHEMATICS c by Elham HOSSEINZADEH a, Amin BARARI b*, Fama FOULADI c, and Davood Ganji DOMAIRRY c a Division of Thermal Energy, Technical University of Denmark, Lyngby, Denmark b Department of Civil Engineering, Aalborg University, Aalborg, Denmark Department of Mechanical Engineering, Babol University of Technology, Babol, Iran Orig i nal sci en tific pa per UDC: 5.5:5./.:.8 DOI:.98/TSCIH In this pa per He s variational it er a tion method is used to solve some e am ples of lin ear and non-lin ear forth-or der bound ary value prob lems. The first prob lem com - pared with homotopy anal y sis method so lu tion and the other ones with the e act so - lu tion. The re sults show the high ac cu racy and speed of con ver gence of this method. It is found that the variational it er a tion method is a pow er ful method for solv ing of the non-lin ear equa tions. Key words: magnetohydrodynamics, variational iteration method, fourth-order differential equations, boundary value problems Introduction The the o ret i cal study of magnetohydrodynamics (MHD) chan nel flow has been a sub - ject of great in ter est due to its wide spread ap pli ca tions in de sign ing cool ing sys tems with liq uid met als, MHD gen er a tors, ac cel er a tors, pumps, and flow me ters. Be sides the com mon ap pli ca tions of gen er a tion and mo tive power from elec tric ity, which e ist in ev ery thing from an elec tric tooth brush to a por ta ble rec re ational ve hi cle elec tri cal gen er a tor, there are some more un usual ap pli ca tions of the mo tor prin ci ple, or Lenz s law. By e am in ing some of the most com ple uses of elec tro mag ne tism for mo tors and pumps, some very un usual and coun ter-in tu itive ap pli ca tions of elec tro mag netic fields for pro pul sion will be - come ap par ent. The MHD is a field of mag netic pump ing which uses the Lenz s law to pump liq uids us ing only an elec tro mag netic field. This unique con cept al lows MHD to pump con duct ing liq - uids with ab so lutely no mov ing parts. There is sim ply a chang ing elec tro mag netic field de vel - oped through the me dium be ing pumped which will cause it to move. The most com mon ap pli - ca tion of these electromagnetodynamic pumps is in the metal in dus try where mol ten metal can be pumped and stirred with out con tact, right through the walls of the ves sel con tain ing the mol - ten metal. Pumps of this type are uti lized in nu clear re ac tors where liq uid so dium is pumped * Corresponding author; ab@civil.aau.dk, amin8@yahoo.com

2 THERMAL SCIENCE: Year, Vol., No., pp. -9 through the re ac tor core for cool ing. The high tem per a ture and re ac tiv ity of this mol ten salt would de stroy any nor mal im pel ler. By pump ing right through the walls of the pip ing the dan - ger ously ra dio ac tive salt is iso lated from the pumps them selves which is ben e fi cial for safety and main te nance rea sons []. Raptis et al. [] con sid ered the un steady MHD flow of a vis cous and elec tri cally con - duct ing fluid past to a plate by the pres ence of ra di a tion. More over, they de rived an a lyt i cal so lu - tions for the mean tem per a ture, ve loc ity, mag netic field, and the ef fect of the ra di a tion on the tem per a ture. The non-new to nian flu ids are con sid ered as more ap pro pri ate mod els of flu ids in in - dus trial and tech no log i cal ap pli ca tions than New to nian flu ids. Such flu ids e hibit the non-lin ear re la tion ship be tween stress and the rate of strain at ev ery point of flow. Due to non-lin ear de - pend ence of stresses on the rate of strain for non-new to nian flu ids, the flow anal y sis is much more com pli cated in com par i son with New to nian flu ids. The con sti tu tive equa tions are very com ple in volv ing a num ber of pa ram e ters and the so lu tions of the re sult ing equa tions in gen - eral are more dif fi cult to ob tain. This is not only true of e act an a lyt i cal so lu tions but even of nu - mer i cal so lu tions. Sev eral in ves ti ga tors are now en gaged in find ing the an a lyt i cal or nu mer i cal so lu tions for flow prob lems that arise us ing dif fer ent non-new to nian flu ids. One of the im por - tant classes of non-new to nian flu ids is viscoelastic fluid. How ever, even the most com monly used sim plest sub class of viscoelastic flu ids is that of the so-called sec ond or der fluid that can give rise to prob lems which are far from triv ial []. The prob lem which gov erns the MHD bound ary layer flow is []: u u y () u u v u a u u u u v u B v n u y y r v u y y y s y r () u v y at y () u, V H v at y () Here, r is the den sity, n the ki ne matic vis cos ity, s the elec tri cal con duc tiv ity, H the width of the chan nel, a the ma te rial pa ram e ter of sec ond grade fluid, u and v the ve loc - ity com po nents in the and y di rec tions and the fluid in jec tion or e trac tion takes place through the po rous walls with ve loc ity V/. Note that V > cor re sponds to the suc tion case and V < for in jec tion. De fin ing: y, y, H H u V * f ( y ), v Vf ( y ) (5) Equation () is iden ti cally sat is fied and eqs. ()-() re duce to eq. (8) which mod els MHD flow of a sec ond grade fluid in a po rous chan nel and was an a lyzed us ing variational it er a - tion method. In ad di tion to first e am ple, we in ves ti gate two lin ear and non-lin ear fourth or der bound ary value prob lems in the fol low ing. For or di nary and par tial dif fer en tial equa tions, some of the an a lyt i cal/ap pro i mate tech niques that have been de vel oped in clude per tur ba tion [5-], variational it er a tion [8-], de - com po si tion [-], homotopy per tur ba tion [-], etc. At first, al most all per tur ba tion meth -

3 THERMAL SCIENCE: Year, Vol., No., pp. -9 ods are based on an as sump tion that a small pa ram e ter must e ist in the equa tion. This so-called small pa ram e ter as sump tion greatly re stricts ap pli ca tions of per tur ba tion tech niques. As is well known, an over whelm ing ma jor ity of non-lin ear prob lems have no small pa ram e ters at all. Sec - ondly, the de ter mi na tion of small pa ram e ters seems to be a spe cial art re quir ing spe cial tech - niques. Fur ther more, the ap pro i mate so lu tions solved by the per tur ba tion meth ods are valid, in most cases, only for the small val ues of the pa ram e ters. It is ob vi ous that all these lim i ta tions come from the small pa ram e ter as sump tion. Variational it er a tion method (VIM) [8-] was in tro duced by He [8-] based on the use of re stricted vari a tions and cor rec tion functionals which has found a wide ap pli ca tion for the so lu tion of non-lin ear or di nary and par tial dif fer en tial equa tions. This method does not re quire the pres ence of small pa ram e ters in the dif fer en tial equa tion, and pro vides the so lu tion (or an ap - pro i ma tion to it) as a se quence of it er ates. The method does not re quire that the non-linearities be dif fer en tia ble with re spect to the de pend ent vari able and its de riv a tives. Ba sic idea of He s variational iteration method To clar ify the ba sic ideas of VIM, we con sider the fol low ing dif fer en tial equa tion: Lu + Nu = g(t) () where L is a lin ear op er a tor, N a non-lin ear op er a tor, and g(t) an inhomogeneous term. Ac cord - ing to VIM, we can write down a cor rec tion func tional as fol lows: t un ( t) un ( t) l [ Lun ( t ) Nu ~ n ( t) g ( t)] d t () where l is a gen eral Lagrangian mul ti plier which can be iden ti fied op ti mally via the variational the ory. The sub script n in di cates the n th ap pro i ma tion and u ~ n is con sid ered as a re stricted vari a - tion d u ~ n. Numerical eamples E am ple Con sider the fol low ing non-lin ear fourth-or der dif fer en tial equa tion: y M y( ) Re[ y( ) y( ) y ( )] a[ y( ) y ( ) y ( ) y( ) yiv ( )] (8) with the bound ary con di tions y( ), yn ( ) ; y( 5. ) 5. ; y( 5. ) (9) where M s B H /m, Re = HV/v, and a = a /H m []. B is a con stant mag netic field, ap plied per pen dic u lar to the chan nel walls and the elec tric field is con sid ered to be zero. The in duced mag netic field is ne glected for small mag netic Reynolds num ber. Re > in di cates the suc tion case and Re < stands for in jec tion. It should be noted that for a =, eq. (8) is re lated to New to - nian fluid. Its correction variational func tional can be e pressed as fol lows: yn ( ) yn ( ) y M y y l[ ( t) ( t) Re ( t) y( t) y ( t)] a[ y( t) y ( t) y ( t) y( t) yiv ( t)] d t ()

4 THERMAL SCIENCE: Year, Vol., No., pp. -9 Af ter some com pu ta tions, we ob tain the fol low ing sta tion ary con di tions: l( iv ) l () l, l () t t l, l () t t The Lagrangian mul ti plier can therefore be iden ti fied as: ( t ) l( t) and the variational it er a tion for mula is ob tained in the form: yn ( ) yn ( ) ( t ) { yn ( t) M yn ( t) Re[ yn ( t) yn ( t) yn ( t)] dt a[ yn ( t) yn ( t) yn ( t) yn ( t) yiv n ( t)]} We start with the ini tial ap pro i ma tion of y (), but since no ini tial ap pro i ma tion of y () is avail able, we make one in the form of a poly no mial as: () (5) y ( ) a b c d () which de pends on the or der of dif fer en ti a tion, and a, b, c, and d are un known con stants to be later de ter mined. Us ing the above it er a tion for mula (5), we can di rectly ob tain other com po nents as: y ( ) 5. a. a Re c a d. bd Re a b. a 5. ab Re 5. b Re 5. a. am. a b bm. ab 8. c Re 8. cm. a c. a b bcre. ad Re d () For a spe cial case: M =, Re =, and a =., y () will be: y ( ) 5. a c. ad a b. a 5. a. a. ab. b. b. c. ac. ab d (8) In cor po rat ing the bound ary con di tions, eq. (9), into y (), we have: y () = d = (9) y ( ) a. ad b. b. c () y ( 5. ). a 58. c 5. ad. b 5. a 58. ab. b 8. ac d 5. ()

5 THERMAL SCIENCE: Year, Vol., No., pp y ( 5. ). a. c. ad. b 5. a 8. ab. b 5. ac () Solv ing the sys tem of equa tions si mul ta neously, we ob tain: a =.8, b =., c =.895, d = () There fore, we ob tain the fol low ing first-or der ap pro i mate so lu tion for spe cial case, M =, Re =, a =.: y () = () In the same man ner, the rest of the com po nents of the it er a tion for mula can be ob tained. Figure. Variation of y() for different values of M for a =., Re = : (HAM and VIM) Figure. Variation of y() for different values of Re for a =., M = : (HAM and VIM) Fig ures - show com par i son of homotopy anal y sis method HAM [] and VIM re sults which re veals e cel lent agree ment be tween the meth ods. E am ple Figure. Variation of dy()/d for different values of Re for a =., M = : (HAM and VIM) Now con sider an other non-lin ear fourth-or - der BVP: yiv ( ) y( ) y( ) (5) Sub ject to the bound ary con di tions: y( ) y ( 5. ) y ( 5. ) y( ) ()

6 THERMAL SCIENCE: Year, Vol., No., pp. -9 Its correction variational func tional in can be e pressed as: yn ( ) yn ( ) l[ yiv n ( t) yn ( t) y ( t) n ] dt () Af ter some com pu ta tions, we ob tain the fol low ing sta tion ary con di tions: l iv = (8) l t l t (9) l l () t t The Lagrangian mul ti plier can, therefore, be iden ti fied as: ( t) l( t) and the variational it er a tion for mula is ob tained in the form: ( t) yn ( ) yn ( ) [ yiv n yn yn ( t) ( t) ( t) ] dt Now we as sume that the ini tial ap pro i ma tion has the form: () () y ( ) a b c d () where a, b, c, and d are un known con stants to be fur ther de ter mined. Us ing the it er a tion for mula (), we can di rectly ob tain the other com po nents as fol lows: y ( ) ab 8. ac 8. b 8. ad 8. bc. bd 5 8. c 5. cd a 9. a b c d In cor po rat ing the bound ary con di tions, eq. (), into y (), we have: y ( ) d (5) y ( ) 8. ac 9. ab 8. bc 9. a (). 8b 8. c 5. a b c () y ( 5. ) 5. ac. ab 55. bc 5. 5 a. b 8. c a b. () y ( 5. ) 9. ac 9. ab 55. bc. a 95. b 5. ca. (8) Solv ing the sys tem of equa tions si mul ta neously, we ob tain: a., b 55., c 5., d (9) There fore, we ob tain the fol low ing first-or der ap pro i mate so lu tion, in the form: y ( ) ()

7 THERMAL SCIENCE: Year, Vol., No., pp. -9 The e act so lu tion is y() =. Plot ting the e act and VIM so lu tions, it is clear that the re - sults are in e cel lent agree ment (fig ). E am ple Con sider the lin ear bound ary value prob lem as fol lows: yiv ( ) y( ) y ( ) e ( ) () Sub ject to the fol low ing con di tions: y( ), y( ) y( ), y( ) e () Its correction variational func tional can be e pressed as: y y y y e n ( ) n ( ) l[ iv n ( t) y n ( t) n ( t) ( )] dt () Af ter some com pu ta tions, we ob tain the fol low ing sta tion ary con di tions: l () l t l t (5) The Lagrangian mul ti plier can, there fore, be iden ti fied as: ( t ) l( t) and the variational it er a tion for mula is ob tained in the form: ( t ) yn yn yn yn y n e ( ) ( ) [ iv ( t) ( t) ( t) ( ) ] dt Now we as sume that the ini tial ap pro i ma tion has the form: () () y () = a + b + c + d (8) Where a, b, c, and d are un known con stants to be fur ther de ter mined. Us ing the above it er a tion for mula (), we can di rectly ob tain the other com po nents as: y ( ) 9. a. b ( 5. a 8. c) 5 (. 8b. d) ( a. ) ( b 5. ) ( c ) e ( ) d (9) In cor po rat ing the bound ary con di tions, eq. (), into y (), and solv ing the equa tions, we ob tain: y ( ) d (5) y ( ) 59. a 8. b 8. c e 8. (5) Figure. The comparison between the eact and VIM solutions

8 8 THERMAL SCIENCE: Year, Vol., No., pp. -9 as: y ( ) c e (5) y ( ) 58. a 5. b. e (5) Solv ing the sys tem of equa tions si mul ta neously, we ob tain: a =.58, b =.89, c =, d = (5) In the same man ner, the rest of the com po nents of the it er a tion for mula can be ob tained y ( ) e ( ) 8 (55) The e act so lu tion for this prob lem is: ( )e Fig ure 5 shows the com par i son be tween VIM and e act so lu tion (fig. 5). Con clu sions Figure 5. The comparison between the eact and VIM solutions In this work, we stud ied the ap pli ca tion of the VIM fourth or der bound ary value prob lems. The fig ures clearly show that the re sults by VIM are in e cel lent agree ment with the HAM and e act so lu tions. VIM pro vides highly ac cu - rate nu mer i cal so lu tions in com par i son with other meth ods, and it is e pected here that VIM as a pow er ful math e mat i cal tool can solve a large class of lin ear and non-lin ear dif fer en tial sys tem and equa tions used in en gi neer ing and phys ics. References [] Karban, P., et al., Pumps of Mol ten Metal Based on Magnetohydrodynamic Prin ci ple for Cool ing High-Tem per a ture Nu clear Re ac tor, Przeglad Elektrotechniczny, 85 (9),, pp. -5 [] Raptis, A., Massalas, C. V., Magnetohydrodynamic Flow Past a Plate by the Pres ence of Ra di a tion, Heat and Mass Trans fer, (998), -, pp. -9 [] Hayat, T., et al., On the MHD Flow of a Sec ond Grade Fluid in a Po rous Chan nel, Computers and Mathe - mat ics with Ap pli ca tions, 5 (),, pp. - [] Cortell, R., Flow and Heat Trans fer of an Elec tri cally Con duct ing Fluid of a Sec ond Grade over a Stretch - ing Sheet Sub ject to Suc tion and to a Trans verse Mag netic Field, Int.J.Heat Mass Trans fer, 9 (), -, pp [5] Kevorkian, J., Cole, J. D., Mul ti ple Scale and Sin gu lar Per tur ba tion Meth ods, Springer-Verlag, New York, USA, 99 [] Nayfeh, A. H., Per tur ba tion Meth ods, John Wiley & Sons, New York, USA, 9 [] Nayfeh, A. H, Mook, D. T., Non lin ear Os cil la tions, John Wiley & Sons, New York, USA, 99 [8] He, J.-H., Variational It er a tion Method for Autonomous Ordinary Differential Systems, Applied Mathe - mat ics and Com pu ta tion, (), -, pp. 5- [9] He, J.-H., Some As ymp totic Meth ods for Strongly Non lin ear Equa tions, Int. J. Mod. Phys. B, (),, pp. -99

9 THERMAL SCIENCE: Year, Vol., No., pp [] He, J. -H., Variational It er a tion Method a Kind of Non-Lin ear An a lyt i cal Tech nique: Some E am ples, Int. J. Non lin ear Mech. (999),, pp [] He, J. -H., Wu, X. H., Con struc tion of Sol i tary So lu tion and Compacton-Like Solution by Variational It er - a tion Method, Chaos Soliton Fract, 9 (),, pp. 8- [] He, J. -H., Wu, X. H.,Variational It er ation Method: New Development and Applications, Computers & Math e mat ics with Ap pli ca tions, 5 (),-8, pp [] He, J. -H., Variational It er a tion Method Some Re cent Re sults and New In ter pre ta tions, Jour nal of Com - pu ta tional and Ap plied Math e mat ics, (),, pp. - [] He, J. -H., An El e men tary In tro duc tion to Recently Developed Asymptotic Methods and Nanomechanics in Te tile En gi neer ing, In ter na tional Journal of Modern Physics B, (8),, pp [5] Barari, A., et al., An Ap pro i mate So lution for Boundary Value Problems in Structural Engineering and Fluid Me chan ics, Jour nal of Math e matical Problems in Engineering (8), Ar ti cle ID 9, pp. - [] Sweilam, N. H., Khader, M. M., Variational Iteration Method for One Dimensional Nonlinear Thermoelasticity, Chaos Soliton Fract, (),, pp. 5-9 [] Momani, S., Abuasad, S., Ap pli ca tion of He s Variational It er a tion Method to Helmholtz Equa tion, Chaos Soliton Fract, (), 5, pp. 9- [8] Odibat, Z. M., Momani, S., Ap pli ca tion of Variational It er a tion Method to Non lin ear Dif fer en tial Equa - tions of Frac tional Or der, Int. J. Non lin ear Sci. Numer. Simul., (),, pp. - [9] Bildik, N., Konuralp, A., The Use of Variational Iteration Method, Differential Transform Method and Adomian De com po si tion Method for Solving Different Types of Nonlinear Partial Differential Equations, Int. J. Non lin ear Sci. Numer. Simul., (),, pp. 5- [] Barari, A., et. al., Ap pli ca tion of Homotopy Perturbation Method and Variational Iteration Method to Non lin ear Os cil la tor Dif fer en tial Equa tions, Acta Applicanda Mathematicae, (8),, pp. - [] Barari, A., et. al., Variational It er a tion Method and Homotopy-Perturbation Method for Solving Different Types of Wave Equa tions, Jour nal of Ap plied Sci ences, 8 (8),, pp. - [] Momani, S., Odibat, Z., Nu mer i cal Comparison of Methods for Solving Linear Differential Equations of Frac tional Or der, Chaos Soliton Fract, (), 5, pp [] Konuralp, A., The Steady Tem per a ture Dis tri bu tions with Dif fer ent Types of Nonlinearities, Computers & Math e mat ics with Aplications, 58 (9), -, pp [] Adomian, G., Sto chas tic Sys tems, Ac a demic Press Inc., New York, USA, 98 [5] Adomian, G., Non lin ear Sto chas tic Operator Equations, Ac a demic Press Inc., New York, USA, 98 [] Adomian, G., Solv ing Fron tier Prob lems of Phys ics: The De com po si tion Method, Kluwer Ac a demic Publischer, Boston, Mass., USA, 99 [] He, J.-H., Homotopy Per tur ba tion Method: A New Nonlinear Analytical Technique, Applied Mathematics and Com pu ta tion, 5 (),, pp. -9 [8] He, J.-H., Ad den dum: New In ter pre tation of Homotopy Perturbation Method, International Journal of Mod ern Phys ics B, (), 8, pp [9] He, J.-H., Re cent De vel op ment of the Homotopy Per tur ba tion Method, Topological Methods in Nonlinear Anal y sis, (8),, pp. 5-9 [] He, J. H., Homotopy Per tur ba tion Tech nique, Computer Methods in Applied Mechanics and Engineering. 8, (999), -, pp. 5- [] Abdoul, R., et al., Solv ing Ra tio-de pendent Predator-Prey System with Constant Effort Harvesting Using Homotopy Per tur ba tion Method, Journal of Mathematical Problems in Engineering, (8), Ar ti cle ID 95, pp. -8 [] Hosseinnia, S., et. al, Homotropu Perturabation Based Linearization of Non lin ear Heat Trans fer Dy namic, Jour nal of Ap plied Math e mat ics and Computing, 9 (9), -, pp. - [] Khatami, I., Pashai, M. H., Tolou, N., Com par a tive Vi bra tion Anal y sis of a Parametrically Non lin ear E - cited Os cil la tor Us ing HPM and Nu merical Method, Journal of Mathematical Problems in Engineering, (8), Ar ti cle ID 95 [] Mehmood, A., Ali, A., An Ap pli ca tion of He's Homotopy Per tur ba tion Method in Fluid Me chan ics, Int. J. Nonlin. Sci. Numer. Simul, (9),, pp. 9- Paper submitted: June, 9 Paper revised: September, 9 Paper accepted: September, 9