TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES

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1 TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES M Sghiar To cite this versio: M Sghiar. TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES. Iteratioal Joural of Egieerig ad Advaced Techology, Blue Eyes Itelligece Egieerig Scieces Publicatio Pvt. Ltd., 06, 6 (, pp < <hal > HAL Id: hal Submitted o 4 Oct 06 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES M. Sghiar Phoe umber : , allée capitaie J. B. Bossu, 40, Talat, Frace October 4, 06 Abstract : I give a resolutio of the Navier-Stokes [] equatio by usig the series of Fourier. Résumé: Je doe ue résolutio de l'équatio de Navier-Stokes [] par les séries de Fourier. Keywords : Navier-Stokes, Fourier, Séries de Fourier. I. Itroductio The Navier Stokes equatios is cosidered to be the first step to uderstadig the elusive pheomeo of turbulece, the Clay Mathematics Istitute i May 000 made this problem [] oe of its seve Milleium Prize problems i mathematics. I this article I will prove that the Navier-Stokes equatio have a solutios ad I will give techiques to resolve this beautiful equatio. The Navier-Stokes equatio, established i the ieteeth cetury by the Frech Navier ad the British Stokes. It is a equatio that describes the velocity field of a fluid. More specifically, it is a differetial equatio whose velocity field is ukow. The Navier-Stokes equatio is also used to predict the weather, the oceas simulate, optimize aircraft wigs... Kowig that a lik betwee the Boltzma equatio ad the Navier-Stokes equatio was established, by studyig the latter problem, I foud that for to solve it we ca reduce the problem of the heat-equatio which is kow ca be solved by several methods : oe of the first methods of solvig the heat-equatio was proposed by Joseph Fourier i his treatise aalytical Theory of heat [] i 8. After givig a specific solutios to the Navier-Stokes equatio, I will demostrate how to fid all solutios of this equatio if they exist, ad I give the ecessary ad sufficiet coditios for their existece.

3 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. It will be see i a remark that if the turbulece fuctio is egligible, the the fluid will ted to behave like a ideal gas. II. Recall, otatios ad defiitios Here are the Navier-Stokes equatio: ρ ( u t +(u. u = p+μ u div u=0 Where u is the velocity field, p is the pressure,the desity of the fluid, ad viscosity. μ its Ad : = (,, x x (u. u = u= div u= u i u u x i u I the followig, by dividig by ρ, the Navier-Stokes equatios is of the form : u t +(u. u =α p +β u div u=0 III- Existece of the solutios for the Navier-Stokes equatio : O each axis i, try to fid the solutios of the form :

4 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. This is equivalet to: t ( + u u i i =α p i + β u i ( t + u i α p i =β u i If u i is a solutio of the equatio : t =β u i Such solutios u i exist because the equatio is aalogous to the heat-equatio which is resolvable by the Fourier series []. If p i is such u i =f i (t, the ( u i =0, ad the equatio is solved. We do the same for all axes i util i=. For the axe i= : Let be u = u i. We have : u t = β u, ad if p is such u α p =f (t, the

5 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. ( u α p =0 x, ad the equatio is solved for the axe. It is clear that if e i is the vector for the axes i, the u= u i e i Naviers-Stokes equatio if div u=0. is oe solutio of the Else, to have div u=0, we take u i of the form : u i =e β t + x ( i, i {,, } So we have solutios of the Navier-Stokes equatio. IV- Necessary coditios : Ay solutio (u, p of the Navier-Stokes equatio verifies that : u i =f i (t, i {,, } Ideed : If (u, p is a solutio of the Navier-Stokes equatio, we must have : Therefore : ( t + u i α p i =β u i

6 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. ( u i t = u i α p i β x i Ad : u x i i t ( = u i β u i Whe the fluid flows i oe directio, the the space-time flows i the opposite directio with the same speed value: We deduce that : u i u i = ( u i β u i Therefore : 0= ( u i α p i β u i Ad : u i β =f i (t So : u i β u i =f (t

7 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. Ad cosequetly u i =g (t because div u=0. We deduce therefore that : u i =h i (t, i {,, } because : i {,,}, u i =l i (x i,t, ad we must have (u i x j =0 j {,, }. V- Coclusio : Theorem: The Navier-stokes equatio have a solutio, Moreover, ay solutio (u, p must check: u i =f i (t ad t =β u i, i {,,}. where u= u i e i ad p= p i e i. Coversely ay pair (, p satisfyig these coditios with Navier-Stokes equatio. div u=0 is solutio of the Remarks : - We ote i the above equatios the depedece betwee pressure, desity, speed vector fields ad viscosity - By dividig by α i the equatio u i =f i (t we deduce that : ρ u i p i =ρ f i (t where α =ρ is the desity of the fluid. Let's ρ= m V, we will have : mu ² i Vp i =mf i (t. Ad the equatio mu ² i Vp i =mf i (t is likig eergy, mass, pressure, temperature, volume ad time... This may ot be surprisig sice a lik betwee the Boltzma equatio ad the

8 (IJEAT, ISSN: (Olie, Volume-6 Issue-, Page No.: 79-80, October 06. Navier-Stokes has bee established. Whe f i (t teds to 0, we will have mu ² i Vp i, there is therefore a tedecy towards the law of a ideal gas, ad the fuctio f i (t ca be regarded as a turbulet fuctio. Refereces [] Joseph Fourier, Théorie aalytique de la chaleur, Firmi Didot Père et Fils (Paris-8. Rééditio Jacques Gabay, 988 (ISBN []