Motion of an Incompressible Fluid. with Unit Viscosity

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1 Nonl. Analsis and Diffeenial Equaions Vol no HIKARI Ld Moion of an Incompessible Fluid wih Uni Viscosi V. G. Gupa and Kapil Pal Depamen of Mahemaics Univesi of Rajashan Jaipu India gupavguo@ediffmail.com palkapiluo@ahoo.co.in Copigh 013 V. G. Gupa and Kapil Pal. This is an open access aicle disibued unde he Ceaive Commons Aibuion License which pemis unesiced use disibuion and epoducion in an medium povided he oiginal wok is popel cied. Absac In he pesen pape we obain he mos geneal soluion of he ssem of Navie- Sock`s equaion fo he moion of an incompessible fluid wih uni viscosi using he geneal polongaion fomula fo hei infiniesimal smmeies. Kewods: Space and Time invaiance Tanslaion Roaion 1. Inoducion The Navie-Soke equaions ae consideed as he foundaion of fluid mechanics and wee inoduced b C. Navie in 183 and developed b G. Sokes. A finiediffeence mehod fo solving he ime-dependen navie-sokes equaions fo an incompessible fluid is inoduced b Alexande [1]. The Special Class of Analical Soluions o he Thee-Dimensional Incompessible Navie-Sokes Equaions was sudied b Nugoho Ali and Abdul [9]. Musafa [8] was woked on a class of exac soluions o he sead Navie-Sokes equaions fo he incompessible Newonian viscous fluid flow moion due o a disk oaing wih a consan angula speed. New classes of exac soluions of he hee-dimensional unsead Navie Sokes equaions conaining abia funcions and paamees ae descibed b Aisov and Polanin ([] [3]) Polanin [13] Pukhnachev[14]. Gebenev and Obelack [4] wok was devoed o he fis-ode appoximae smme opeao fo he Navie-Sokes equaions. Gunawan Ahmed and Zainal ae sudied he hee-dimensional incompessible Navie-Sokes equaion wih

2 144 V. G. Gupa and Kapil Pal coninui equaions ae solved analicall in [5] and poduced a class of exac soluions in [6]. The classes of nonivial exac soluions ma sill be developed fom he oiginal Navie-Sokes equaions b moe complex pocedues o know moe abou popeies of he exac soluions [1] and he uniqueness and egulai of he soluion can be checked b appling specific bounda and iniial condiions as in [7]. In he ssem of Navie- socks equaion fo he moion of an incompessible fluid wih uni veloci in hee dimensional domains. Hence hee ae fou independen vaiables x = ( x z) being spaial coodinaes and he ime ogehe wih fou dependen vaiables he veloci field u = ( u v w) and he pessue p. The coefficien of he kinemaic viscosi and he densi of he liquid ae aken o be equal o one. In Veco Noaion he ssem has he fom u u. u = p u u = 0 (1.1). Soluion of Navie Sock`s Equaion An infiniesimal ssem of he Navie-Sock`s equaion will be a veco field on X U akes he fom v = ξ x η ζ z τ u v w p whee ξ η ζ τ ae funcions of x u and p using Geneal polongaion fomula (Olve[11]) o deemine he second polongaion p () v of v in (1.1) z u z xx zz x u v xx v zz w x z w z p p () x v = ξ x η ζ z τ u v w p xx u zz z x u zz v z v x x w p x u xx z w u z v x v x x p x p w x u x x w x p u x v x u x z pz w x v x w z p z p z u x v z w z zz w z p v p zz z u u x z v z xx w w z p xx v z p z z u u z v xx zz x v w xx w zz p x p.

3 Moion of an incompessible fluid wih uni viscosi 145 Appling he second polongaion o he Navie-Sock equaion (1) we find he following ssem of smme equaions. u x v w z u x u u z = - x ( xx zz ) u x v w z v x v v z = - ( xx zz ) u x v w z w x w w z = - z ( xx zz ) x z = 0 which mus be saisfied wheneve u and p saisf (1). Since () need onl hold on soluions of (1) we can subsiue fo p x p p z and w z wheneve he occu in () using hei expessions fom he equaions in (1). We ma hen equae all he coefficien of he emaining fis ode deivaives of u p in equaion () and solve he esuling ssem of deemining equaions fo ξ η ζ τ. Fom he ssem of deemining equaions we shown ha τ is dependen onl on. The funcion ξ η ζ ae independen of u v w moeove ξ η ζ ae independen of pessue p and also find u =τ ξ x p v = η x w = ζ x u = ξ v =τ η p w = ζ u = ξ z v = η z w =τ ζ z p. These all impl ha nave he geneal soluion fom = (τ ξ x p )u η x v ζ x w ˆ = ξ u (τ η p )v ζ w ˆ = ξ z u η z v (τ ζ z p ) w ˆ whee ˆ ˆ and ˆ depend onl on x and. Theefoe ξ = c 5 x c c 4 z α η = c x c 5 c 3 z β ζ = c 4 x c 3 c 5 z γ = c 5 u c v c 4 w α = c u c 5 v c 3 w β = c 4 u c 3 v c 5 w γ = c 5 p x α β z γ θ τ = c 1 c 5 in which α β γ and θ ae funcion of and c 1 c c 3 c 4 c 5 ae consan. We have hus shown ha he smme goup of he Navie-Socks equaions in hee dimensions is geneaed b he veco fields given as follows v 1 = v = x x vu u v v3 = z z w v v w v = x z u w v5 = x z u v w p p 4 z x w u x z u v w α = α x α u α x p vβ β β v β p v v v θ = γ = γ z γ w γ z p = θ hee v 1 is ime anslaion v v 3 v 4 ae oaions v 5 scaling v α v β v γ p ae moving coodinaes v θ pessue changes. In which α β γ and θ ae abia funcion of. The coesponding one- paamee goups of smmeies of Navie Socks equaion ae hen G1 = ( x ε u p ) G = (( x cosε sin ε cosε xsin ε z) ( u cos ε v sin ε v cos ε u sin ε w) p) G3 = (( x cosε z sin ε zcosε sin ε ) ( u v cos ε w sin ε w cos ε v sin ε ) p ) G4 = (( x cosε z sin ε zcosε xsin ε ) ( u cos ε w sin ε v w cos ε u sin ε ) p ) ε ε ε ε ε ε ε G ε 5 = xe e ze e u e ve w e p e ( ) ε Gα = ( x α ε z ) ( u εα v w ) p ε xα α α

4 146 V. G. Gupa and Kapil Pal ε G β = ( x β ε z ) ( u v εβ w ) p ε β β β ε G γ = ( x z γε ) ( u v w ε γ ) p ε zγ γ γ G = ( x u p εθ ) θ. The smme goup G G 3 G 4 can be wien as he goup SO(3):(R x Ru p) of simulaneous oaion in he boh space x and he veloci field u whee R is an abia 3 3 ohogonal maix. The Smme goup G α G β G γ ae abia funcion of can be wien as he goup ε G α = x αε u εα p ε xα α α whee α = (α β γ) and G α is geneaed b he linea combinaion v α = v α v β v γ of he hee veco fields. If we ake u = f ( x ) and p = g( x ) ae soluion of he (1) Navie-sock equaion (1) hen ae he funcions u = f ( x ε ) 1 ( ) (1) p = g x ε1 () u = ( u cosε vsin ε v cosε u sin ε w) 1 = R f (( x cosε sin ε cosε xsin ε z) ) = R f ( R x ) () p = g (( x cosε sin ε cosε xsin ε z) ) = g ( R 1 x ) (3) u = ( u v cosε wsin ε wcosε vsin ε ) 1 = R3 f (( x cosε z sin ε z cosε sin ε ) ) = R3 f ( R3 x ) (3) p = g (( x cosε z sin ε z cosε sin ε ) ) = g ( R 1 3 x ) (4) u = ( u cosε wsin ε v wcosε u sin ε ) 1 = R4 f (( x cosε z sin ε z cosε xsin ε ) ) = R4 f ( R4 x ) (4) p = (( x cosε z sin ε z cosε xsin ε ) ) = g R 1 4 x g () whee R R 3 R 4 ae an abia 3 3 ohogonal maix. The soluions u u (3) (4) and R f R 1 x and he soluions p () p (3) u ae can be shown as SO(3):u = and p (4) 1 ae wien as p = g ( x ). (5) u = e ε f e ε x e ε p (5) = e ε g e ε x e ε R = g ( x αε z ε ) ε xα α α = g ( x βε z ε ) ε β β β p α p β u β u γ u α (( αε ) ) = f x z εα (( βε ) ) = f x z εβ (( γε ) ) = f x z εγ

5 Moion of an incompessible fluid wih uni viscosi 147 ε p γ = g (( x z γε ) ) ε zγ γ γ The soluions of wien as follows u α = f ( x αε ) εα and p α p β p α ε = g ( x αε ) ε xα α α ( θ ) ( θ ) u = f ( x ) p u α u β u γ can be p γ can shown as = g x εθ. and Conclusion The Goup analsis is he onl igoous mahemaical mehod o find all he smmeies of a given diffeenial equaion. In ou invesigaion he smme goup G 1 shown he ime anslaions. The smme goup G G 3 and G 4 ae epesens he oaion in x z and zx diecion especivel. Goup G 5 epesen he scale ansfomaion and G α G β G γ epesen he abia anslaion in x z diecion especivel. The goup G θ shows ha he pessue p is onl defined up o he addiion of an abia funcion of. This complees he lis of smmeies (1) (1) of Navie-Socks in hee-dimension. u = f ( x ε1) p = g ( x ε1) R f R 1 x 1 (5) p = g ( u = e ε f e ε x e ε (5) p = e ε g e ε x e ε u = u α = f x εα ( θ ) u = f x ( αε ) ( θ ) ( ) and p g ( x ) R x ) p α ε = ( x αε ) ε xα α α = θ ε g A he end he mos geneal soluion is in he fom given below ε5 ε5 1 ε 5 = e R f e R x α e ε α ( 1) 1 ( 1 ) u p e ε e ε = g R ( x α ) e ε ε xα α α θ. whee ε 1 and ε 5 ae eal consan and R is abia 3 3ohogoanl maix α = (α β γ) and θ ae abia soluion of Navie-Sock equaion. Refeences 1. Alexande Joel ChoinNumeical Soluion of he Navie-Sokes Equaions Mahemaics of Compuaion Oc. Vol. No. 104 (1968) Aisov S. N. Polanin A. D. New classes of exac soluions of heedimensional Navie Sokes equaions axiv.og/abs/ v1(sep. 009) 3. Aisov S. N. Polanin A. D. Exac soluions of unsead hee-dimensional Navie Sokes equaions. Doklad Phsics Vol. 54 No. 7 (009) Gebenev V. N. and Obelack M. Appoximae Lie smmeies of he Navie-Sokes equaions JNMP 14 (007)

6 148 V. G. Gupa and Kapil Pal 5. Gunawan N. Ahmed M. S. Ali and Zainal A. Abdul Kaim Towad a New Simple Analical Fomulaion of Navie-Sokes Equaions Wold Academ of Science Engineeing and Technolog 51 (009). 6. Gunawan N. Ahmed M. S. Ali Zainal A. Abdul Kaima A Class of exac soluion o he hee-dimensional incompessible Navie-Sokes equaions Applied Mahemaics Lees Elsevie (010). 7. Kozono H. Yasu N.Exension Cieion via Two-Componens of Voici on Song Soluions o he 3D NS Equaions Mah. Z. 46 (004) Musafa T ukilmazoˇglu Exac Soluions fo he Incompessible Viscous Fluid of a Roaing Disk Flow PAM 1 1 (011) Nugoho G. Ali A.M.S. and Abdul Kaim Z.A. On a Special Class of Analical Soluions o he Thee-Dimensional Incompessible Navie-Sokes Equaions AML Elsevie (009) doi /j.aml Olve PJ. Smme Goups and Goup Invaian Soluions of Paial Diffeenial Equaion. J. Diff. Geom. 14 (1979) Olve PJ. Applicaion of Lie Goups o Diffeenial Equaions Second Ediion Spinge Velag: New Yok Inc. (1993). 1. Popovich R. On Lie Reducion of he Navie-Sokes Equaions Nonlinea Mahemaical Phsics Vol. No.3-4 (1995) Polanin A. D. Exac soluions o he Navie Sokes equaions wih genealized sepaaion of vaiables. Doklad Phsics46(10)(001) Pukhnachev V. V. Smmeies in he Navie Sokes equaions [in Russian]. Uspekhi Mekhaniki [Advances in Mechanics] Vol. 4 No. 1 (006) Received: June 1 013