HAM for finding solutions to coupled system of variable coefficient equations arising in fluid dynamics

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1 Available online at Advances in Applied Science Reseach, 214, 5(5):7-81 ISSN: CODEN (USA): AASRFC HAM fo finding solutions to coupled syste of vaiable coefficient equations aising in fluid dynaics T. S. L. Radhika and Ritesh Rai BITS Pilani Hydeabad Capus, Hydeabad, India ABSTRACT The pesent pape applies the hootopy analysis ethod (HAM) fo finding solutions to a coupled syste of vaiable coefficient equations that aise in the pobles of fluid dynaics. Flow of fluids paticulaly nonnewtonian fluids though pipes is a poble that has wide ange of application. The atheatical foulations of these pobles geneally give ise to non-linea (and/o) coupled (and/ o) vaiable coefficient equations. Thus finding exact solutions of these pobles is alost ipossible. Thus eseaches sought to nueical o appoxiate analytical ethod fo solving the. Hee, in the pesent study, the flow of icopola fluid in a igid cicula tube is consideed and an appoxiate analytical solution is found. The effect of the fluid flow paaete, ico otation paaete and the pessue gadient on the velocity and ico otation of the fluid ae studied. The esults ae pesented though gaphs. Keywods: HAM; Micopola fluid;non-linea coupled equations; INTRODUCTION The study of incopessible viscous fluid flow in igid pipes is a enowned classical poble. An exact steady state solution to this poble, unde cetain valid assuptions can be obtained by solving the govening equations called Navie Stokes equations [1]. In pactice, ajoity of the fluids found in natue as well as in industial o edical applications ae non- Newtonian. The flow of these fluids though igid pipes have enoous applications in polye pocessing industies, bio edical engineeing etc., Thus thee is a definite need to undetake the study the flow of these non-newtonian fluids though pipes. Unlike the steady viscous flow, hee, in these pobles, it is difficult to elate the instantaneous velocity pofiles and the voluetic flow ate to the instantaneous pessue gadient. Futhe, in soe cases the stess shea ate elation ay be highly non- linea and ay also be a function of tie. These ake the poble of pipe flow of non- Newtonian fluids even oe coplicated and hence atheatically intactable. Inspite of the difficulties in obtaining exact analytical solutions to the above entioned pobles, due to thei pactical ipotance, these pobles have been attepted by seveal eseaches fo divesified fields and they tied finding the solutions by aking cetain assuptions in ode to get an analytical solution o a nueical solution [2-5]. Vey ecently, soe poising appoxiate analytical ethods such as Hootopy analysis ethod, Hootopy petubation ethod, Optial Hootopy asyptotic ethod (OHAM) have been poposed [6-14]. These ethods can be teated as inteediate ethods fo the exact analytical ethods and the nueical ethods and have seveal advantages ove the. These appoxiate analytical ethods, though cannot povide closed fo of solutions as the exact analytical ethods, can povide good appoxiate expessions fo the solution and hence can solve a wide 7

2 class of pobles whose solutions ae alost ipossible by the classical ethods. Obviously they fo the bette choice than the nueical ethods in any cases, due to the liitations of the nueical ethods. In the pesent pape, the flow of a non- Newtonian fluid naely Micopola fluid is consideed fo study. This fluid odel was poposed by Eigen inode to explain the behavio of eal fluids in cetain contexts. Eingen has poposed this theoy of icopola fluids [15] in 1966 andlukaszewicz descibed this fluid odel as a well-founded and significant genealization of the classical Navie Stokes odel coveing both in theoy and applications, any oe phenoena than the classical one can [16]. Matheatical Foulation: Conside the flow of incopessible icopola fluid in a igid cicula pipe unde a constant pessue gadient. A scheatic diaga of the poble can be found in fig(1). The flow is assued to be laina. Fig(I): Scheatic diaga of fluid flow The equations govening the flow of icopola fluid ae [15] ρ + div( ρ q) = t dq ρ = ρ f gad p + k culν ( µ + k) cul cul q dt + ( λ + 2µ k) gad div q dν ρ j dt 1 + ρ l 2 kν + k cul q γ cul culν + ( α + β + γ ) gad divν = (3) in which q,ν ae velocity and icootation vectos, f, l ae body foce pe unit ass, body couple pe unit ass espectively and p is the fluid pessue at any point. ρ andj ae density of the fluid and gyation paaetes espectively and ae assued to be constants. The ateial constants ( λ 1, µ, k) ae viscosity coefficients and ( α, β, γ ) ae gyoviscosity coefficients. These constants confi to the inequalities The stess tenso t ij and the couple stess tenso t ij ij ae given by ( p+ λ + divq ) δ + ( 2µ + k) e + k ( w v ) = ε 1 ij ij lj (4) ( divν ) δ ij + βvi, j γv j i = α + (5) ij, in which the sybols δ ij, e ij,2 w and voticity vecto and icootation vecto. diffeentiation. v espectively denote Konecke sybol, coponents of ate of stain, ε ij denotes the Levi- Civita sybol and coa denotes covaiant In the absence of body foces and body couple, the equations govening the steady flow is given by (1) (2) 71

3 div( q ) = (6) gad p + k cul ν ( µ + k) cul cul q = (7) 2 k ν + k cul q γ cul cul ν = Assuing q = (,, w( ) ) and ν = (, v( ),) p = k d µ + k d dw p d d d z ( v) + = d 1 d dw γ ( v) + k + 2kv = d d d, we get (8) (9) (1) (11) which is a syste of coupled odinay equations with vaiable coefficients. The bounday conditions ae w = on = R (no slip condition) w is finite at = v = on = R (no spin condition) v is finite at = (12) Using the following non-diensionalisation: v w v* = ; w* = ; * = and the non- diensional paaetes given by U U R ico aveage µ + k U aveage pl = (which is the polaity paaete that takes up sall values ), fl = which is efeed to as flow k 2RU γ R dp paaete, vis = the viscosity paaete, Ps 2 2kR = 2kU ico dz the pessue, equations (9) and (1) afte dopping * take the fo ico d d dw d d d ( v) + pl * fl + Ps = d 1 d dw vis ( v) + fl + v = d d d (13) (14) HAM: Conside a nonlinea diffeential equation of the fo: ( ) N u( x ) = (15) whee N is a nonlinea opeato, x is the independent vaiable and u( x ) is the unknown function. Let u ( x) be the initial appoxiation of the exact solution u( x ) and L be an auxiliay linea opeato with the popety that 72

4 L( f ) = when f =. (16) In this ethod, we constuct the Hootopy which is a continuous apping H : u( x) φ ( x; q) H ( φ ( x; q) ; q) ( 1 q) L( φ ( x; q) u ( x) ) hh ( x) qn ( φ ( x; q) ) defined as = (17) Hee H ( x) is an auxiliay function and h is an auxiliay paaete called the convegence contol paaete, q [,1] is an ebedding paaete and φ ( x; q) is the appoxiate solution to the given poble. We notice fo equation (17) that the solution obtained using this ethod, depends on the fou ipotant factos naely the u x, the linea opeato L, the auxiliay function H ( x ) and the auxiliay paaete h. initial appoxiation ( ) When q = and when the Hootopy defined by equation (17) is taken to be zeo, we get the zeoth ode defoation equation given by L φ x; u x = (18) ( ( ) ( )) In view of the lineaity of the opeato L, the zeoth defoation equation is given by ( x;) u ( x) φ = (19) Now, when q = 1, equation (17) takes the fo ( ( )) N φ x;1 = This equation is sae as the given equation povided ( x;1) u ( x) φ = (2) This shows that as the ebedded paaete q vaies fo to 1, ( x; q) seen in equation (19)) to the exact solution u( x ) (as seen equation (2)). Let s now define the th ode defoation deivatives as u x = x q q ( ) [ ] ( ) φ ( ; ) q= φ vaies fo the initial guess u ( ) Then, using the Taylo s theoe, φ ( x; q) can be expanded as a powe seies of q as [ u ] ( x) φ ( x; q) = φ ( x;) + q (22) Witing u ( x) u = 1 [ ] ( x)! = and using equation (2.19), the above takes the fo! = + (23) = 1 ( ; ) ( ) ( ) φ x q u x u x q x (as is With suitable choice of the initial guess, the auxiliay linea opeato, the convegence contol paaete and the auxiliay function, Liao poved that the above powe seies solution conveges fo q = 1[1]. (21) 73

5 Now, to find the solution using equation (2.23), we need to find the functions u ( ) x fo =1,2,3 Liao [1] has deived that these functions ae given by the th ode defoation equation defined as follows: L u x u x = hh x R u x (24) ( ( ) χ 1 ( )) ( ) 1 ( ) ( ) whee 1 1 R ( u 1 ( x) ) = 1 ( N ( φ ( x; q ))) ( 1)! q q= (25) and, 1 χ = 1, othewise (26) Afte deteining u ( ) u x u x u x ( ) ( ) ( ) = 1 x fo =1,2, an appoxiate solution to the poble given in (14) is + (27) Solution: Let L w w (28) 2 Lv v + x w (29) Hee N ( v) + pl * fl + Ps = w d d dw d d d d 1 d dw Nv vis ( v) + fl + v = d d d (3) Constuct the hootopy fo the above nonlinea coupled equations as: H w ; q ; q = 1 q L w ; q w hn w ; q ( ( ) ) ( ) ( ( ) ( )) w ( ( )) ( ( ; ); ) = ( 1 ) ( ( ; ) ( )) v ( ( ; )) H v q q q L v q v hn v q Using zeo initial appoxiations and assuing the solutions as = + = 1 = + = 1 ( ; ) ( η ) ( η ) w q w w q ( ; ) ( η ) ( η ) v q v v q w ( ) and v ( ) ae obtained using equations (24)- (26). Using MATHEMATICA, the fist fou appoxiations fo the fluid velocitycoponent w( ) and otation coponent v( ) ae calculated and the plots ae pesented. (31) (32) the ico 74

6 RESULTS AND DISCUSSION is To find the values of the convegence contol paaete h in equations (31), the h-gaphs fo w i.e w ( h) plotted at η = an in Fig 1. It can be seen that fo fo Ps=.25;pl=.1;fp=.1;Vp=.1,the contol paaete h satisfies -.5<h<.5 Siilaly, to find the convegence contol paaete h fo finding v, the h-gaph fo v is plotted as in Fig Fig(1) Plot of w ( h) to find the convegence paaete (-.5<h<.5) =.;=.;=.;=.; Fig(2) Plot of w( ) =.;=.;=.;=.; 75

7 =.;=.;=.;=.; Fig(3) : Plot of v ( h) (-.5<h<.5) Fig(4) : Plot of v( ) =.;=.;=.;=.; fo =.;=.;=.;=.; Fig(5) Plot of w ( h) 76

8 Fig(6): Plot of w( ) =.;=.;=.;=.; =.;=.;=.;=.; Fig(7) : Plot of v ( h) Fig(8): Plot of v( ) =.;=.;=.;=.; 77

9 Fig(9) Plot of w ( h) fo =.;=.;=.;=. to find the convegence paaete (-1<h<) Fig(1): Plot of ( ) w =.;=.;=.;= =.;=.;=.;=. Fig(11) : Plot of v ( h) 78

10 Fig(12): Plot of v( ) =.;=.;=.;= fo =.;=.;=.;=. Fig(13) Plot of w ( h) Fig(14): Plot of ( ) w fo =.;=.;=.;=. 79

11 =.;=.;=.;=. Fig(15) : Plot of v ( h) Fig(16): Plot of v( ) =.;=.;=.;=. CONCLUSION Fo the above gaphs, we daw the following conclusions: 1. As the pessue gadient (Ps) inceases, the velocity of fluid at the cente of the tube inceases as is seen in fig(2) and fig(6). 2. As the ecipocal of the ico polaity (pl) deceases, the velocity of the fluid inceases as is seen in fig(2) and (1). 3. Also as the ecipocal of the ico polaity (pl) deceases, the ico otation of the fluid inceases as is seen in fig(4) and (12). 4. As the fluid flow paaete inceases, the velocity of the fluid deceases(fig (2) and (14) ) and the ico otation inceases (fig (4) and (16). REFERENCES [1] Heann Schlichting, Klaus Gesten, Bounday Laye Theoy, 8th ed. Spinge-Velag24. [2]Ra P. Bhati, Dalton J.E. Havie, Malcol R. Davidson:Poceedings of FEDSM28,28 [3] H. Zhu, D.De.Kee :Jounal of Non-Newtonian fluid Dynaics, 27143, [4] H. Zhu Y.D Ki,D.De.Kee: Jounal of Non-Newtonian fluid Dynaics, 25, 129, [5] N.Zeaibi, M. Aoua, A. Benzaoui and M. Gaeche: Intenational counication in Heat and Mass tansfe,27,34, [6] S.J Liao: Applied atheatics and coputation, 24, 147, [7] S.J Liao: Beyond Petubation: Intoduction to Hootopy Analysis ethod, Chapan and Hall/ CRC pess, Boca Raton, 23. [8] J. H. He :Cop. Meth. App. MechEng, 1999, 178, [9] J. H. He :I. J. Non-Lin. Mech. 2, 35,

12 [1] J. H. He : Int. J. Non. Sc. NuSi, 21, 2, [1] J. H. He, Intenational Jounal of Moden Physics, B, 29, 2, [11] Eingen, A. C: J. Math. Mech.,1966, 16,1-18. [12] Lukaszewicz,G.: Micopola Fluids, Theoy and Applications, Bikhause, Boston, Basel, Belin,