International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS)

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1 Inernaional Associaion of Scienific Innovaion and Research (IASIR) (An Associaion Unifying he Sciences, Engineering, and Applied Research) Inernaional Journal of Emerging Technologies in Compuaional and Applied Sciences (IJETCAS) ISSN (Prin): ISSN (Online): Variaional Homoopy Perurbaion Mehod for Longiudinal Dispersion arising in Fluid Flow hrough Porous Media Amrua Daga 1, Khyai Desai 1 and Vikas Pradhan 1 1 Deparmen of Applied Mahemaics & Humaniies S.V.Naional Insiue of Technology, Sura India. Absrac: In his paper, Variaional Homoopy Perurbaion Mehod is employed o solve dispersion equaion arising o predic he conaminan concenraion disribuion of unseady unidirecional seepage flows hrough semi-infinie, homogeneous, isoropic porous media. The resuls obained are presened numerically and graphically. I shows remarkable high precision. Thus Variaional Homoopy Perurbaion Mehod can solve a large number of non-linear problems effecively and accuraely having wide applicaions in engineering and physics. Keywords: Variaional Ieraion Mehod (VIM), Homoopy Perurbaion Mehod (HPM) and Variaional Homoopy Perurbaion Mehod (VHPM), Dispersion, Burger s equaion. I. Inroducion Dispersion is essenially macroscopic phenomenon caused by a combinaion of molecular diffusion and hydrodynamic mixing occurring wih laminar flow hrough porous media. The paern of poin races as is move downsream from is source ends o a normal disribuion boh longiudinal and ransverse. Furhermore, he longiudinal componen is larger han ha of he ransverse so ha he major axis of mixing occurs in he direcion of flow. Hydrodynamic dispersion is normally referred o as mixing of miscible fluid. I is a hydraulic mixing process by which wase concenraion is aenuaed while wase conaminans are being ranspored downsream. A he macro scale level, conaminan ranspor is defined by he average groundwaer velociy. However, a he micro scale level he acual velociy of waer may vary from poin o poin and can be eiher lower or higher han he average velociy. This difference in micro scale level waer velociies arises due o pore size, pah lengh and fricion in pores. Because of hese differences in velociies mixing occurs along he flow pah. This mixing is known as mechanical dispersion or hydrodynamic dispersion [4]. The dispersion sources in uniform groundwaer flow were repored by Hun [8]. In general, he heory of dispersion in porous media was presened by Scheidegger [14]. Dispersion phenomena occur in many problems of ground waer flow, in chemical engineering processes, in oil reservoir engineering, ec In his paper a heoreical model is developed for he dispersion problem in porous media in which he flow is one dimensional and he average flow is unseady. Analysis of flow agains dispersion in porous media was presened by Al-Niami and Rushon [1]. The sudy of flow agains dispersion in non-adsorbing porous media was presened by Marino. Hun [8] applied he perurbaion mehod o longiudinal and laeral dispersion in no uniform seepage flow hrough heerogeneous aquifers. Analyical soluion of one dimensional ime-dependen ranspor equaion was also presened by Basha and Habel. An analyical soluion for solue ranspor wih deph dependen ransformaion or sorpion coefficien was developed by Flury e al. [4]. Analyical soluions o ransien, unsauraed ranspor of waer and conaminans hrough horizonal porous media were discussed by Sander Braddock [13]. Analyical soluion for conservaive solue ranspor in one-dimensional homogeneous porous formaions wih ime dependen velociy was also derived by Singh e al. [15]. R. Meher e al[9] solved dispersion problem in porous media in which he flow is one dimensional and he average flow is unseady using Adomain decomposiion mehod Misry Piyush [1] solved by Variaion Ieraion Mehod. The presen paper discusses he analyical soluion of he nonlinear differenial equaion for longiudinal dispersion phenomena which akes places when miscible fluids mix in he direcion of flow by means of Variaional Homoopy Perurbaion Mehod [3]. Variaional Homoopy Perurbaion Mehod (VHPM) which is formulaed by he coupling of variaional ieraion mehod and He s polynomials. The proposed VHPM provides he soluion in a rapid convergen series which may lead he soluion o a closed form. The proposed ieraive scheme akes full advanage of variaional ieraion and he homoopy perurbaion mehods and absorbs all he posiive feaures of he coupled echniques. IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 13

2 II. Mahemaical formulaion of he problem The problem is o find he concenraion as a funcion of ime and posiion x as he wo miscible fluid flow hrough porous media on eiher sides of he mixed region. The single fluid equaion describes he moion of fluid. Here he mixing akes place longiudinally as well as ransversely a = and a do of fluid having C concenraion is injeced over he phase. The do moves in he direcion of flow as well as perpendicular o he flow. Finally i akes he shape of he ellipse wih C n. According o Darcy s law he equaion of coninuiy for he mixure in case of incompressible fluids is given by.( v ) (1) Where ρ is he densiy for mixure and v is he pore seepage velociy. The equaion of diffusion for a fluid flow hrough a homogeneous porous medium wihou increasing or decreasing he dispersing maerial is given by C C.( Cv ). D () where C is he concenraion of a fluid in a porous media. D is he Coefficien of dispersion wih nine componens D ij. In a laminar flow for an Incompressible fluid hrough homogeneous porous medium, densiy ρ is consan. Then equaion () becomes, C v. C. DC (3) Le us assume ha he seepage velociy v v ux, is along he x- axis, hen and he nonzero componens will be D 11 D L = γ (Coefficien of longiudinal dispersion) and oher Componens will be zero. Equaion (3) becomes, C C C u x x (4) where u is he componen velociy along x axis which is ime dependen as well as concenraion along x axis in x direcion x axis and D L > and i is he cross secional flow velociy in porous media., where x and for C 1 by Meha (6). Equaion (4) becomes C C C C x x (5) This is he non linear Burger s equaion for longiudinal dispersion of miscible fluid flow hrough porous media. The heory ha follows is confined o dispersion in unidirecional seepage flow hrough semi-infinie homogeneous porous media. The seepage flow velociy is assumed unseady. The dispersion sysems o be considered are subjec o an inpu concenraion of conaminans C. The governing parial differenial equaion (5) for longiudinal hydrodynamic dispersion wih in a semi-infinie non adsorbing porous medium in a unidirecional flow field in which γ is he longiudinal dispersion coefficien, C is he average cross-secional concenraion, u is he unseady seepage velociy, x is a coordinae parallel o flow and is ime. The iniial and boundary condiions are, x C x, f x e, x C, (where 1),.1 (6-b) Since Concenraion is decreasing wih disance x. Therefore for he sake of convenience f(x) is considered as negaive exponenial funcion [9]. (6-a) IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 14

3 III. Variaional ieraion and homoopy perurbaion mehod To illusrae he basic conceps of he VIM and HPM, a firs we consider he following nonlinear differenial equaion Lu Nu g( x) () Where L = A linear operaor, N = A nonlinear operaor, g(x) = An inhomogeneous erm According o he VIM [6,16] we can consruc a correcion funcional as follows: n un 1 ( x) un( x) ( ) Lun Nu g d (3) where λ(τ) is a general Lagrange muliplier, which can be idenified opimally via variaional heory. The second erm on he righ hand side is called he correcion and is considered as a resriced variaion, i.e., δ u n =. By his mehod, i is required firs o deermine he Lagrangian muliplier λ(τ) ha will be idenified opimally. The successive approximaions u n+1 (x,), n of he soluion u(x,) will be readily obained upon using he deermined Lagrangian muliplier and any selecive funcion u (x,). Consequenly, he soluion is given by u( x, ) lim un( x, ) n (4) To illusrae he basic concep of homoopy perurbaion mehod, consider he following nonlinear sysem of differenial equaions A(U) f(r), r (5) wih boundary condiions u B(U, ), r n (6) Where A = A differenial operaor, B = A boundary operaor, f(r) = A known analyic funcion, Γ = The boundary of he domain Ω. The operaor A can be divided ino wo pars L (linear) and N (nonlinear operaor). Therefore (5) can be rewrien L U N U f r We consruc a homoopy v( r, p) : [,1] R which saisfies (7) H( v, p) (1 p)[ L( v) L( u )] p[ A( v) f ( r)] Or p[,1], r H( v, p) L( v) L( u ) pl( u ) p[ N( v) f ( r)] (9) where u is iniial approximaion of equaion (6). In his mehod, using he homoopy parameer p, we have he following power series presenaion for v v pv1 p v... Seing p=1 yields in he approximae soluion of equaion (1) o (8) (1) u lim v v v v... p1 1 The convergence of series (11) is discussed by J. Biazar, H. Ghazvini [17]. (11) IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 15

4 IV. Variaional Homoopy Perurbaion Mehod All In he homoopy perurbaion mehod [7], he basic assumpion is ha he soluions can be wrien as a power series in p i u p u u pu p u... i i 1 To illusrae he concep of he variaional homoopy perurbaion mehod [18] we consider he general differenial equaion (5). We consruc he correcion funcional (6) and apply he homoopy perurbaion mehod o obain [7,17]. i () i p ui u( x, ) p ( ) Np uix, g x, d (13) i n As we see, he procedure is formulaed by he coupling of variaional ieraion mehod and homoopy perurbaion mehod. A comparison of like powers of p gives soluions of various orders. (1) V. Implemenaion of VHPM o Solve Dispersion Equaion According o Variaional Homoopy Perurbaion mehod, we consruc he correcion funcional for equaion (5) Cn Cn C n Cn 1 x, Cn x, Cn d x x Making he above funcional saionary, he Lagrange muliplier can be deermined as λ = 1, which yields he following ieraion formula Cn Cn C n Cn 1 x, Cn x, Cn d x x Applying he variaional homoopy perurbaion mehod, we have C pc1 p C... C pc1 p C... C pc1 p C... x C pc1 p C... Cn x, d C pc1 p C... x Comparing he coefficien of like powers of p, we have p C x e x : (, ) x p : u ( x, ) e (1 e ) 1 x 1 1 p : C ( x, ) e (3 e (6 e )) 3x x x 1 p : C ( x, ) e (16 51e 8 e e ) 6 3 4x x x 3x 3 3 Similarly furher approximaions can be obained up o desired accuracy. The soluion becomes C( x, ) C C C C C IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 16

5 VI. Graphical and Numerical Represenaion Table I Numerical Values of Concenraion for longiudinal dispersion of miscible fluid flow hrough porous media by VHPM T x=.1 x=. x=.3 x=.4 x=.5 x=.6 x=.7 x=.8 x=.9 x= Figure 1 Curves refers Concenraion values < x < 1 for fixed =,.1, Concenraion C Disance x Figure Curves refers Concenraion values where < <.5, for fixed x=,.1,---1 Concenraion C Time IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 17

6 Figure 3 The 3D behaviour of Concenraion versus x for differen values of ime by VHPM VII. Conclusion I can be easily concluded ha he mehod is elegan and reduces compuaional work wih high accuracy; he numerical values have been abulaed. The graphical represenaions have been developed for predicing he possible concenraion of a given dissolved subsance in unseady unidirecional seepage flows hrough semiinfinie, homogeneous, isoropic porous media subjec o he source concenraions ha vary negaive exponenially wih disance which shows Concenraion decreases wih disance and slighly increases wih ime. In previous paper auhors have already used Adomian polynomials o decompose he nonlinear erms in equaions. The soluion procedure is simple, bu he calculaion of Adomian polynomials is complex. To overcome his shorcoming, Variaional Homoopy Perurbaion Mehod is used. References [1] A. N. S.Al-Niami and K. R Rushon, Analysis of Flow agains dispersion in porous media, J. Hydrol.,vol.33, pp.87-97,1977 [] J.Bear., Dynamics of fluids in porous media, American Elsevier Publishing Company [3] A.R Daga and V.H. Pradhan Variaional homoopy perurbaion mehod for solving nonlinear reacion-diffusion convecion problems, Inernaional Journal of Advanced Engineering Research and Sudies,vol.,no.,pp.11-14,13. [4] M.Flury, Q. J. Wu, L Wu and L Xu., Analyical soluion for solue ranspor wih deph dependen ransformaion or sorpion coefficien, Waer Resources Research, vol.34,no.11,pp ,1998. [5] J. J.Fried, Groundwaer Polluion, Elsevier Scienific Pub. Comp., Amserdam [6] J.H He, Variaional ieraion mehod -- a kind of non-linear analyical echnique: some examples, Inerna. J. Nonlinear Mech. vol.34,pp [7] J.H He, Homoopy Perurbaion mehod -- a new non-linearr analyical echnique, some examples, Inerna. Journal of applied Mahemaics and Compuaion,vol.135,no.1,pp [8] B Hun, Dispersion sources in uniform ground waer flow, J Hydraulic Div., vol.14,no.1,pp , [9] R Meher, M.N Meha, S.K.Meher Adomain Decomposiion Mehod for Dispersion Phenomena Arising in Longiudinal Dispersion of Miscible Fluid Flow hrough Porous Media. Journal of Advances in Theoreical and Applied Mechanics,vol. 3,no.5,pp [1] P.R Misry, Variaional ieraion mehod for dispersion phenomena arising in longiudinal dispersion of miscible fluid flow hrough porous media, Inernaional Journal of Advanced Compuer and Mahemaical Sciences, vol.4,no.1,pp , 13. [11] Z M. Odiba, A sudy on he convergence of variaional ieraion mehod,mahemaical and Compuer Modelling : An Inernaional Journal,vol.51,no.1,pp ,1. [1] M.Sahimi,D. H Barry., L.E Scriven. and H. T Davis., Dispersion in flow hrough porous media- One phase flow, Chemical Engineering Sciences, vol.41,no.8, pp [13] G. C. Sander and R. D.Braddock, Analyical soluions o he ransien, unsauraed ranspor of waer and conaminans hrough horizonal porous media, Adv. Waer Resour. 8, [14] A. E.Scheidegger, General heory of dispersion in porous media. J. Geophysics, Res., vol.66,no.1, pp [15] M. K Singh, V.P.Singh, P.Singh and D.Shukla, Analyical soluion for conservaive solue ranspor in one-dimensional homogeneous porous formaions wih ime dependen velociy, J Engg. Mech., ASCE,vol.135,no.9,pp , 9. [16] Wazwaz A. M., The variaional ieraion mehod: A Powerful scheme for handling linear and nonlinear diffusion equaions. Compuer and Mahemaics wih Applicaions 54,pp ,7. [17] J.Biazar and H.Ghazvini. Convergence of he homoopy perurbaion mehod for parial differenial equaions, Nonlinear Analysis: Real World Applicaions, vol.1,pp ,9. [18] M. Mainfar, M. Saeidy, and Z. Raeisi, "Modifed variaional ieraion mehod for hea equaion using He's polynomials",bull. Mah. Anal. Appl.vol. 3, no., pp. 38-4, 11. IJETCAS 13-56; 13, IJETCAS All Righs Reserved Page 18