Application of homotopy perturbation method to the Navier-Stokes equations in cylindrical coordinates

Transcription

1 Computational Ecology and Softwae 5 5(): 9-5 Aticle Application of homotopy petubation method to the Navie-Stokes equations in cylindical coodinates H. A. Wahab Anwa Jamal Saia Bhatti Muhammad Naeem Muhammad Shahzad Sajjad Hussain Depatment of Mathematics Hazaa Univesity Manshea Pakistan Depatment of Mathematics COMSATS Institute of Infomation Technology Abbottabad Pakistan Depatment of Infomation Technology Hazaa Univesity Manshea Pakistan wahabmaths@yahoo.com wahab@hu.edu.pk Received Septembe ; Accepted 8 Febuay 5; Published online June 5 Abstact This pape deals with the appoximate analytical solution of the Navie-Stokes equations in cylindical coodinates. The homotopy petubation method is used to get the analytical appoximation. Depending upon diffeent available choices fo the linea opeato we also have the advantage to choose diffeent initial appoximations to stat ou analysis. The analysis is done without calculating the Adomian s polynomials. Keywods Navie-Stokes equations; homotopy petubation method; iteative appoximation; infinite seies solution. Computational Ecology and Softwae ISSN 7X URL: vesion.asp RSS: E mail: ces@iaees.og Edito in Chief: WenJun Zhang Publishe: Intenational Academy of Ecology and Envionmental Sciences Intoduction The Navie-Stokes equations descibe the motion of fluids that is a substance which can be flow and it aises fom Newton nd law applying to the fluid motion (Squae 95). The Navie-Stokes equations ae widely used in physics they ae used fo modeling of weathe and seas cuents designing of aicafts and cas fo motions of stas they ae used in video games flow of wate in a pipe blood ciculations analysis of powe stations and study of populations (Thope 997). In fluids mechanics the dynamics of a flowing fluid is govened and epesented by the Navie- Stokes equations which ae nonlinea patial diffeential equations. Hee ou case of inteest is to appoximate the govening equations of the flow field in a tube since it is nonlinea in chaacte and it is impossible to solve these equations analytically to get the exact solution. To solve these equations we ae led to adopt some estictive assumptions and some simplifications which involve the suppositions of weak non lineaity to apply taditional petubation methods small paamete assumptions which estict the wide applications of the

2 Computational Ecology and Softwae 5 5(): 9-5 petubation techniques lineaization which is cetainly a handy task discetization to apply numeical techniques etc. In using the taditional numeical methods fo the numeical solution of the Navie- Stokes equations ae vey difficult and it is due to mixing of diffeent length scales involving in the fluid flow which esults in massive out pints. Ou objective hee is to find the continuous analytical solution to the govening equation in cylindical coodinates without massive outspints and estictive suppositions as discussed above which change physical poblem into a mathematical poblem. K. Halda (Halda 995) used Adomian s decomposition method (Adomian 996; Adomian 989) fo the analytical appoximation of the poblem which is most tanspaent method fo the solutions of the Navie-Stokes equation in cylindical coodinates. Howeve the limitations of this method involve a handy task of the calculations of the Adomian polynomials which poved to be too difficult and cause to slow down the application. To ovecome this shotcoming we make use of the homotopy petubation method to get analytical appoximations fo diffeent choices of linea opeatos and the initial guesses available. Recently the homotopy petubation method being a poweful technique was developed by He (He 999 5).The main advantage of this technique is to ovecome the difficulties aising in the pocess of calculations fo the nonlinea tems aising in the poblem. This gives analytical appoximation to the diffeent classes of the nonlinea diffeential equations system of diffeential equations integal and intego-diffeential equation and systems of such equations. Halda applied the Adomian s decomposition method to the Navie- stokes equations in cylindical coodinates fo two dimensional iotational fluid flow in a tube (Hada 997). Ou pesent analysis gives the application of homotopy petubation method without any estictive assumptions and handy calculations of the Adomian polynomials to the Navie Stoke Equations in cylindical coodinates in which the steady two dimensional iotational flow of fluid in a tube of nonunifom cicula coss section can be studied. The Govening Equations Conside the govening equations of motion fo the two dimensional flow field fo a viscous fluid in a tube which ae descibed by the cylindical coodinate tansfomation of the Navie-Stokes equations ead as; u z u v u u u u P z z () u z v v v v v v v P z. () It is suggested that the otational motion of the fluid is negligible. Then the equation of continuity eads is u v () z Whee u is fluid velocity components in the axial x coodinate and v is in the adial coodinate and the fluid pessue is descibed by P the fluid density by and the kinematic viscosity by fo the fluid. Intoducing and labeling the steam function as then we may have u and v z () The equation of continuity is satisfied identically. The dynamical equation of motion in tem of the steam function ae obtained by eliminating P between () and () and making us of the elation () it is ead as;

3 Computational Ecology and Softwae 5 5(): 9-5 Intoducing as a linea opeato which is defined as; z (5) z (6) and the Jacobean defined as L ( ) ( ). z z z z z Now we hee mainly discuss to foms of the linea opeato defined by equation (5). We will split the linea opeato in two pats and discuss the two cases. It is to note that in the homotopy petubation method we ae fee to choose the linea opeato. This mainly depends upon the given fom of the initial o bounday condition and the poblem unde investigation. Theefoe depending upon ou choices and the possibilities fo the appeaance of the auxiliay linea fom of opeato in the poblem we conside two cases hee. Case : The fist fom of the linea opeato extacted fom equation (5) fo the possible fom of the linea opeato is supposed to be;. (8) Then the opeato becomes which implies that then z z (7). z (9) Using (9) in (5) the equation (5) takes the following fom ( ) ( ) ( z) z z z Taking both sides ( ) ( ). z z ( z) z In ode to apply the poposed homotopy petubation method to the given poblem we need to define the nonlinea tem appeaing in the govening equations. Theefoe we define the nonlinea tem as N in the above equation which is given as; ( ) N. (z) z Then we get the following nonlinea fom of equation fo ou analysis

4 Computational Ecology and Softwae 5 5(): 9-5 ( ) ( ) ( z) z z z Opeating on both sides of the above equation ( ) ( z) z z z Using homotopy petubation method (HPM) poposed by J. H. He (He 6) we constuct a homotopy fo equation () as; ( z; ) : [] This satisfies () H v v u A v f and hee is designed to be an embedding paamete ww w. z z z z Suppose the solution of () is of the fom of () z ; () Using () in () we get + z w o. z z z Now we simplify the quantities enclosed in backets z z z () z z ( ) ( ) ( ) z z z z z z o z z z z z z The calculations made in () is accoding to the definition of the Jacobean and ()

5 Computational Ecology and Softwae 5 5(): 9-5 z z z z z z z Combining the tems containing the equal powes of in equation () and (5) C C C z z z z z z z z z z z z (5c) and so on. Now (5) (5a) (5b) (6) z z z z (7) z z z z Now combining the tems containing the equal powes of in equations (6) and (7) D z z (7a) D z z (7b) D z z (7c) and so on. Using equations (5a) (5b) (5c) (7a) (7b) (7c) in equation (). We get fom equation () C C C D D D. Equating the coefficients of equal powes of we have the zeoth ode poblem as: Zeoth Ode poblem: which implies z (8) (9) Hee is defined as the solution of homogenous equation () subject to the pe-pescibed bounday conditions. Now to find the appoximation fo fo which we fist

6 Computational Ecology and Softwae 5 5(): 9-5 find the invese opeato and fo it we conside equation () () We now define & then opeato takes the following fom as Using. equation () we get linea tems Solving fo and that is fo () () Opeating on () and on () we get () (5) and ae the solutions of two homogenous equations and espectively. The invese linea opeatos and ae defined as dd d. Adding (5) and (6) we get sides by to get (6) and dividing both (7) (8) whee then n n. (9)

7 Computational Ecology and Softwae 5 5(): Let the quantity in backets is denoted by n n n () n n n n n () n Now the invese of the linea opeato is defined as n n n () Now we come to equation (9) and define the zeoth ode poblem as. And the st ode poblem as C D. Substituting values fom equation (5a) (7a) we get. z z z z Opeating on both sides of the above equation yields. z z z z Making use of fo the initial guess of HPM methodology. z z z z Opeating with on both sides of the above expession whee z z z z is given in equation (). The nd ode poblem is given as C D. () Using the values ofc and D fom equations (5b) and (7b) we get

8 6 Computational Ecology and Softwae 5 5(): 9-5. z z z z z z Opeating with both sides of the above equation to get z z z z z z Now since in the methodology of HPM we suppose the following expession fo the appoximate solution of the poblem lim (5) whee the components of the seies solution ae defined to be as; z z z z z z z z z z and so on. If once is obtained which can be easily obtained by constucting the homotopy fo the given () poblem and equating the coefficients fo the zeoth ode poblem then we can find in tems of and in the simila fashion can be evaluated in tems of and.the othe highe ode components can be easily obtained having the all othe lowe ode values. Thus all the components of can be calculated. The seies solution n n thus can be given the following fom z z z z z z z z z z (6)

9 Computational Ecology and Softwae 5 5(): Case : We may have the othe available o chosen fom of the linea opeato as; zz. Wheeas and zz then fom equation (5) we have z. z Taking both sides of the above expession z z. z z. (7) Using the methodology of HPM we may constuct a homotopy fo equation (7) as; H q q q z z z ;q : Suppose the solution fo equation (8) is of the following fom Whee as (; zq) q ( z) q ( z) q ( z) (9) q q z q q z z ( q ) q q ( q ) z z q z z z q z z z () (8) q q z z z z q z z z Using () and () in (8) we get ()

10 8 Computational Ecology and Softwae 5 5(): 9-5 q q z z z q z z z q q z z z We fist define the invese linea opeato conside equation (). Then zz zz Multiplying both sides of the above equation by we get In simila way we get. zz zz () zz (). () zz zz zz zz zz zz zz (6) whee and and and then zz (5) ae the solutions of homogenous equations zz zz zz. (7) The invese linea opeatos zz and ae defined by Then we have (8) d dzdz dd zz

11 Computational Ecology and Softwae 5 5(): zz zz zz zz zz zz n zz zz zz n. Let us denote the quantity within the backets by then the following expession is obtained n n n (9) (5) n zz zz zz n. n n n n Thus the invese linea opeato can be easily identified as; n zz n zz zz zz n n (5) Now the zeoth ode poblem is. (5) The st ode poblem is: Opeating with both sides of the above equations;. z z z z In ode to find the initial guess of HPM we make use of the zeoth ode poblem as:. z z Opeating on both sides to get z z (5)

12 5 Computational Ecology and Softwae 5 5(): 9-5 The nd ode poblem is: z z z z and so on. The seies solution fom of the poblem eads as z Whee the following quantities ae defined z z lim ;q z z z z (5) and so on. The seies fom of the solution can be witten as; z z z z z z z z z z n n Conclusion We have consideed two cases fo the available linea opeatos and obtained the appoximation fo ou poblem. Of couse the selection of the linea opeatos mainly depends upon the given initial o bounday conditions. We can see that fo the fist case the available linea opeato was split in two pats and fo the second case we consideed the full linea fom of the opeato without splitting it into pats. Thus on the basis on methodology of the Adomian decomposition and the homotopy petubation method (Halda 995) the pesent analysis can be applied to a wide ange of the physical and engineeing poblems (Shakil et al. ; Wahab et al. ; Wahab et al. ; Siddiqui et al. ). As compaed to the Adomian decomposition method fo the analysis of the poblem (Halda 995) we have the geat advantage of the selection of the initial guess which can be chosen on the basis of the pevious knowledge and most impotantly the initial appoximation should satisfy the given initial o bounday conditions which leads us to the unifomly valid appoximately seies solution. While the Adomian decomposition method does not have such advantage because we have to select the initial guess based on the ecusive elation poduced by the method. But this initial appoximations sometimes may lead to nonunifomly valid seies solution which also may contain the secula tems in the seies. In homotopy petubation method the initial guess satisfying the given conditions may give a unifomly valid seies solution. On the othe hand the calculation of the Adomian polynomials is not an easy task fo the nonlinea tems

13 Computational Ecology and Softwae 5 5(): appeaing in the poblems. Howeve thee ae some compute pogams which can calculate the Adomian polynomials but they ae fo some specific cases. In ou analysis we avoid such handy calculation because homotopy petubation method tansfoms a non-linea poblem into a small numbe o sub-linea poblems with pescibed conditions. No matte of concen with the existence of the paamete small o lage. This is again a dominating advantage of the method ove Adomian decomposition method (Shakil et al. ; Wahab et al. ; Wahab et al. ; Siddiqui et al. ). Refeences Adomian G Application of the decomposition method to the Navie-Stokes equations. Jounal of Mathematical Analysis and Applications 9: -6 Adomian G Nonlinea Stochastic Systems: Theoy and Applications to Physics. Kluwe Academic Publishes USA Adomian G. 99. Nonlinea tanspot in moving fluids. Applied Mathematics Lettes 6(5): 5-8 Afzal M Wahab HA Bhatti S. and Naeem Queshi MT.. A Mathematical Model fo the Rods with Heat Geneation and Convective Cooling. Jounal of Basic and Applied Scientific (6): Halda K Application of adomian appoximations to the Navie-Stokes equation in cylindical coodinates. Applied Mathematics Lettes 9(): 9- He JH Homotopy petubation technique. Compute Methods in Applied Mechanics and Engineeing 78(/): 57-6 He JH.. A coupling method of homotopy techniques and petubation technique fo nonlinea poblems. Intenational Jounal of Non-Linea Mechanics 5(): 7- He JH. 6. Some asymptotic methods fo stongly nonlinea equations. Intenational Jounal of Moden Physics B -99 Shakil M Khan T Wahab HA Bhatti S.. A Compaison of Adomian Decomposition Method (ADM) and Homotopy Petubation Method (HPM) fo Nonlinea Poblems. Intenational Jounal of Reseach in Applied Natual and Social Sciences (): 7-8 Siddiqui AM Wahab HA Bhatti S Naeem M.. Compaison of HPM and PEM fo the flow of nonnewtonian fluid between heate paallel plates. Reseach Jounal of Applied Sciences Engineeing and Technology 7(): 6- Squie HB. 95. Some viscous fluid flow poblems. Philosophical Magazine : 9-95 Thope JF Development in theoetical and applied mechanics. Shaw WA. Pegamon Pess Oxfod UK. Wahab HA Shakil M Khan T Bhatti S Naeem M.. A compaative study of a system of Lotka-Voltea type of PDEs though petubation methods. Computational Ecology and Softwae. (): -5 Wahab HA Khan T Shakil M Bhatti S Naeem M.. Analytical appoximate solutions of the systems of non linea patial diffeential equations by Homotopy Petubation Method (HPM) and Homotopy Analysis Method (HAM). Jounal of Applied Sciences and Agicultue 9():