Lie Subalgebras and Invariant Solutions to the Equation of Fluid Flows in Toroidal Field. Lang Xia

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1 Le Subalgebas and Invaant Solutons to the Equaton of Flud Flows n Toodal Feld Lang a Emal: langxaog@gmalcom Abstact: Patal dffeental equatons (PDEs), patculaly coupled PDE systems, ae dffcult to solve By fa, no systematcal appoach s avalable to guaantee analytcal solutons to an abtay PDE The Le symmety method that s the genealzaton of Galos s appoach fo algebac equatons, howeve, has the potental to analyze and solve a class of dffeental equatons systematcally In the pesent epot, by usng the Stokes- Helmholtz decomposton theoem the -dmensonal Nave-Stokes equaton (NSE) s uncoupled and tansfomed nto a scala equaton fo the velocty potental when the flow feld s toodal The dynamcs of the velocty potental s ndependent of the vecto potental The educton and nvaant solutons to the equaton ae analyzed by the Le symmety method subsequently The Le subalgebas fo the equaton ae dscussed and the coespondng nvaant solutons ae also pesented Keywods: Nave-Stokes equaton, Geometc analyss, PDE, Le goups, Le symmetes Intoducton The Nave-Stokes equaton (NSE) n the classc mechancs s pobably the most mpotant PDE Due to ts complexty, nethe the unqueness no exstence of global solutons has been poved untl now [] Although vey few analytcal solutons ae well-known [], eseaches n the flud mechancs whose phenomena ae govened by the NSE stll advance fast wth the help of numecal computatons [-6] Howeve, analytcal solutons to the NSE ae stll pecous They povde a gauge fo the vefcaton and valdaton of all knd of numecal schemes and algothms Theefoe, many sem-analytcal methods and altenatve analyss ae developed to study the popety of possble analytcal solutons [7, 8] Enlghtened by the success of Galos goups n solvng algebac equatons, Sophus Le poposed contnuous goups fo solvng dffeental equatons [9, 0] Ths method helped Le to dscove the symmetes hdden n the dffeental equatons Meanwhle, t unfed many knds of ad hoc methods of educng dffeental equatons Although Le goups have che meanngs n the felds of dffeental geomety and mathematcal physcs, It s

2 only afte 970 they become a poweful tool to smplfy and solve dffeental equatons [, ] That s because the Le goup method eques an enomous amount of algebac calculatons, and sometmes solvng the lnea detemnng equatons ae even hade than solvng the ognal equatons Thanks to the developments of moden compute systems, those detemnng equatons can be easly solved wth the help of CAS softwae o packages ght now Theefoe, wthn the past 0 yeas, the symmetc goups, Le algebas and the coespondng nvaant solutons to many famous equatons have been gadually documented [, ] In ths pape, we focus on the smplfcaton and Le goup analyss of the NSE Fstly, we deve a elatvely smple scala equaton usng the Stokes-Helmholtz decomposton when the flow feld s toodal Then the Le goup analyss s caed on to the equaton wth the assstance of the machney of jet bundles Fnally, we pesent some solved Le algebas and nvaant solutons to the equaton fo specal cases Smplfcaton of the NSE Let consde the -D non-dmensonal NSE wthout extenal foces n a smply connected doman that s wtten n the fom of + = + t u u u p u u = 0 () whee p s the mechancal pessue, the ynolds numbe and the velocty u C ( ) Thee s moe than one appoach to do the knematc decomposton of flow felds (vecto felds) [8], but the Stokes-Helmholtz mght be the most wdely used one The pupose of ths secton s to deve a dynamcal equaton govenng the flow feld call the Stokes-Helmholtz decomposton theoem, we wte the velocty n the fom of whee the potental component s otatonal, and ψ u = + ψ () the otatonal component s solenodal Hee we also eque beng an analytcal functon so that the decomposton s vald n any smply connected unbounded doman It s well known that the decomposton s not unque [4], because addng any hamonc functon to the otatonal component wll esult n a new decomposton to the velocty feld Ths ssue can be esolved physcally by mposng a bounday condton such that the devatve of the velocty s bounded, moe specfcally, let n be the nomal unt vecto of the bounday, then the Stokes-Helmholtz decomposton s unque f and only f n( ψ ) = 0 (n ths

3 case, we may call t Helmholtz Hodge decomposton) [5] Futhemoe, by defnng an equvalent class fo the potental o otatonal component mathematcally, that s to / ~ say, (mod ), hee s an abtay hamonc functon we can also avod the ambguty We shall see the dscusson n ou futue pape Howeve, hee we focus on the poblem feed of bounday condtons and teat the decomposton oughly unque To deal wth the vecto potental, let s ecall the polodal-toodal decomposton (also known as Me epesentaton) of the solenodal pat n the sphecal coodnate (,, ) [6] ψ = T+ P= + () In Catesan coodnate, the above equaton has a slghtly complcated expesson wth mean felds beng added [7] We shall thnk the case that the polodal feld vanshes Assume that = ( + j+ k ), and expess the toodal feld n Catesan coodnate locally as ( ( )) T = + j+ k (4) Theefoe, we can assume the vecto potental ψ = (,, ), and then the thee components of the velocty have the followng foms u u u = + x y z = + y z x = + z x y (5) The Eq(5) ndcate the vecto potental s a toodal feld Fom the above analyss, we have the followng esult Theoem: If the NSE has the soluton n the fom of u = + ψ, whee the vecto potental s a toodal feld, then the hamonc functon satsfes the followng equaton + ( + + ) ( + + ) + p = 0 t x y z xy yz zx (6) Poof: The summaton of the velocty components equatons gves se to

4 u + u + u = + + (7) x y z Note that the NSE also has the followng component foms u t u t u t + u u p = + u x + u u p = + u y p + u u = + u z (8) Afte the summaton of the above equatons, substtute Eq(7) nto t, we have ( p p p ) ( ) ( ) ( u + + = ) (9) t x y z x y z x y z x y z Fo a scala functon f, we know that ( fu) = f u + u f Apply the ncompessble condton u = 0, we get u ( + + ) = ( u( + + )) x y z x y z (0) Let φ = + + x y z p p p p = + + x y z () whee φ : = ( x, y, z, t)( + j+ k ) and p : = p( x, y, z, t)( + j+ k ) Thus we obtaned the followng equaton ( φ) + ( u φ) = p + ( φ ) () t The ntegaton of the above equaton s n the fom of φ + u φ = p + ( φ ) + C () t

5 C = 0 hee the ntegaton paamete satsfes By notng the fact that C= u = 0, we may ncopoate the ntegaton paamete nto the velocty tem wthout changng the fom of the equaton Wte the above equaton n the scala foms u u + C + u φ = p+ t x + u φ = p+ t y ( φ) ( φ) + u φ = p+ t z ( φ) (4) Note that fo ncompessble condton, we also have above equatons we fnally obtan = 0 By the summaton of the + ( + + ) ( + + ) + p = 0 t x y z xy yz zx In ths way, we have educed the NSE nto the above smple scala equaton Eq(6) gves us a smple expesson fo the dynamcs of the potental feld of fluds The beneft of usng such potental expesson s that all the velocty components can be late on ecoveed by the devaton of the potental functon Ths equaton ndcates that the vecto potental has no mpact on the dynamcs of the aveaged potental feld due to the vanshng of such component n the equaton In othe wods, the dynamcs of the velocty potental s ndependent of ts vecto potental pat Howeve, the total vecto feld can be ecoveed late on by supemposng the velocty potental and vecto potental pats In the next secton, we ae gong to apply Le goup analyss on the equaton Symmety Analyss Snce lookng at the doman and codoman of a gven dffeental equaton as a poduct manfold can be natually genealzed to jet manfolds, t povdes a moe unfed and concse language to the Le symmety method by studyng jet bundles and coespondng Catan dstbutons of the dffeental equatons, we plan to dscuss the poblem hee usng the machney of jet bundle We shall fst ntoduce some basc defntons elated n the pesent pape, eades who wsh to exploe moe detals can go to [8, 9] Let ( E,, M) be a locally tval smooth bundle ove a smooth manfold M, of whch the = = d M The -jet bundle manfold assocated wth the set of sectons s ( ) { : } bundle can be defned as J ( ) = { j : p M, ( )} p p

6 whee the -th equvalence class the fbe manfolds jp s called -jet of at J ( ) = j p ( ) pm p Thus, the dsjont unon of foms a smooth vecto bundle ( J ( ),, M) that s called jet bundle The constucton k also nduces affne bundles ( J ( ),,, J ( )), hee Suppose J ( ) tangent to the gaph planes k k and J 0 ( ) = E, Let J ( ) be the gaph of -jets, then the span of all planes s called the Catan plane The dsjont unon of the Catan = J ( ) s an ntegable dstbuton that s called the Catan dstbuton, whch s the basc geometc stuctue on the manfold J ( ) Gven an patal dffeental equaton as ( x y z xy yz ) F t, x, y, z,,,,,, = 0 Then the set ( ) = {( x,, ) : F x,, = 0} j j j j defnes a submanfold n J = The Catan dstbuton on the -jet ( ) (,, ) manfold n the local coodnates then can be chaactezed by the followng contact foms = d + dx of whch the estcton on the equaton [8, 0] nduces the Catan dstbuton on the equaton ( ) = T whee T s the tangent space on the equaton at A maxmal ntegal manfold of the Catan dstbuton ( ) s called the geneal soluton to the equaton

7 As fo the equaton n ths pape, we can defne a nowhee vanshng vecto feld on spanned by J 0 ( ) = x y z t whch s also geneated by a one-paamete goup Le pont symmety f ts lftng on J ( ) s n the fom of 0 C ( J ( )) Then s called the () j = + + j such that ( ) () F x j = j F = 0,, 0 whee j = D D j = D D j jk j j k and the opeato of total dffeentaton s n the fom of D = + + j x j Futhemoe, f C ( J ( )), we call the Le contact symmety One can easly check that the lftng of Le symmetes peseve the Catan dstbuton [8, 0], eg =, ( ( )) ( ) C J M call the one-paamete Le tansfomaton goup of the fom [] x = x + ( x, y, z, t, ) + O( ) * y = y + ( x, y, z, t, ) + O( ) * z = z + ( x, y, z, t, ) + O( ) * t = t + ( x, y, z, t, ) + O( ) * 4 = + ( x, y, z, t, ) + O( ) * (5) and the coespondng nfntesmal geneato

8 = x y z t (6) whee * dx = ( x, ) = d * d = ( x, ) = d = 0 = 0 (7) and denote ( x ) = ( x, y, z, t), =,,,4 Also, the second polongaton (lftng) of the vecto feld can be wtten as () j = + + j (8) By applyng ( ) () F x, j, j = F = 0 0 (9) the detemnng equatons fo and can be obtaned Usually, solvng the detemnng equatons s not ease than solvng the ognal dffeental equatons It was the eason why the Le goup method was not of much pactcal use befoe the developng of moden computes Rght now by usng CAS softwae o elated packages, we can eadly solve the systems of lnea patal dffeental equatons Fo the scala equaton Eq(6), the detemnng equatons ae obtaned and solved by usng MAPLE sults The coespondng Le subalgebas and nvaant solutons ae pesented case by case as follows Case : p = 0 In Eq(6), we can teat the pessue as the souce tem wth settng the pessue to be zeo fo the smplest case = x + t + ( y z) + = y + t + ( z x) + = z + t + ( x y) + = t + 4 = x

9 whee the Geek alphabets ae abtay constants By settng each constant coeffcent to be zeo sequentally, the Le subalgebas of the equaton, n ths case, ae spanned by and the possble nvaant solutons =, =, =, 4 = x y z t x 5 = t + t + t + x y z 6 = ( y z) + ( z x) + ( x y) x y z 7 x y z = t t x y z x y z = A, = + A, = + A, = + A 4t 4t 4t z z y = + Bef + A 4t t z z x = + Bef + A 4t t da = B + z/ yaln( a) a Aa + R e Case : p = p0 = const Usng the same pocedue (heeafte we omt the pesentaton of nfntesmal geneatos) we obtan the followng Le subalgebas

10 and the possble nvaant solutons =, =, =, 4 = x y z t x 5 = t + t + t + x y z 6 = ( z x) 7 = ( y x) 8 = ( z y) + ( x z) + ( y z) x y z 9 ( y) x y z p0 x = t t x y z = A pt, = A x p 0 = A y p, = A z p z = p0t + A 4t Case : p = p() t In ths case, we assume that the pessue depends on tme only The Le subalgebas of the equaton fo ths case ae =, =, =, 4 = p( t) x y z t x 5 = t + t + t + x y z 6 = ( z x) 7 = ( y x), 8 = 9 = ( z y) + ( x z) + ( y z) x y z x y z 0 = t tp() t t x y z and the possble nvaant solutons

11 = = A p( t) dt z A p( t) dt + 4t Case 4: p = p( x) In the last, let s consde the stuaton that the pessue depends x thee tanslaton goups ae coespondng to the followng Le subalgebas only, = = = t y z and the possble nvaant solutons ae n the fom of Dscusson = A p( x) dx = A + p( x) dx Fom the above Le goup analyss, we have obtaned the Le subalgebas fo some specal cases, as well as the nvaant soluton Although the physcal meanng of each the soluton has not yet been fully exploted, the appoach n ths pape has shed lght on how to smplfy coupled dffeental equatons By the summaton of each component n the equatons, we keep only the scala nfomaton, whch s moe mpotant n most of the cases, eg n wave equaton the ampltude of the wave and n heat equaton the ntensty of the heat If we specfy the pessue p() t = ( x + y + z) t we can constuct the nvaant solutons fo ths specal case as 5 t t = A ( x + y + z) 0 5 z t t = ( y + x) + A 4t 5 as well as p( t) = sn ( xyz t), and the coespondng nvaant solutons

12 = = A cos( t xyz) z A + cos( t xyz) 4t Wth the ad of the Stokes-Helmholtz decomposton, we have uncoupled the - dmensonal Nave-Stokes equaton nto a scala equaton of the velocty potental Ths equaton gves a neat way to descbe the dynamcal behavos of flows n the toodal feld The appoach of smplfcaton ndcates an aveagng magntude of the velocty vectos The vecto potental does not affect the dynamcs of the aveage potental feld snce t s vanshed n the equaton afte aveagng The Le goup analyss has helped us to constuct some nvaant solutons to the equaton wthn some nstances We appled an aveaged velocty potental to smplfy the NSE n the pape, of whch the technque may also help smplfy othe smla patal dffeental equatons feences Feffeman, CL, Exstence and smoothness of the Nave-Stokes equaton The mllennum pze poblems, 006: p Wang, C, Exact solutons of the steady-state Nave-Stokes equatons Annual vew of Flud Mechancs, 99 (): p Taylo, C and P Hood, A numecal soluton of the Nave-Stokes equatons usng the fnte element technque Computes & Fluds, 97 (): p Gha, U, KN Gha, and C Shn, Hgh- solutons fo ncompessble flow usng the Nave- Stokes equatons and a multgd method Jounal of computatonal physcs, 98 48(): p Qan, Y, D d'humèes, and P Lallemand, Lattce BGK models fo Nave-Stokes equaton EPL (Euophyscs Lettes), 99 7(6): p Temam, R, Nave-Stokes equatons: theoy and numecal analyss Vol 4 00: Amecan Mathematcal Soc 7 u, H, et al, Homotopy based solutons of the Nave Stokes equatons fo a poous channel wth othogonally movng walls Physcs of Fluds (994-pesent), 00 (5): p Wu, J-Z, H-Y Ma, and M-D Zhou, Votcty and votex dynamcs 007: Spnge Scence & Busness Meda 9 Schwaz, F, Solvng second-ode dffeental equatons wth Le symmetes Acta Applcandae Mathematca, (): p 9-0 Olve, F, Le symmetes of dffeental equatons: classcal esults and ecent contbutons Symmety, 00 (): p Ibagmov, NK and NK Ibagmov, Elementay Le goup analyss and odnay dffeental equatons Vol : Wley Chcheste Ovsannkov, LVe, Goup analyss of dffeental equatons 04: Academc Pess Bluman, G and S Kume, Symmetes and dffeental equatons Vol 54 0: Spnge Scence & Busness Meda 4 Batchelo, GK, An ntoducton to flud dynamcs 000: Cambdge unvesty pess

13 5 Chon, AJ, JE Masden, and JE Masden, A mathematcal ntoducton to flud mechancs Vol 990: Spnge 6 Backus, G, Polodal and toodal felds n geomagnetc feld modelng vews of Geophyscs, 986 4(): p Schmtt, BJ and W von Wahl, Decomposton of solenodal felds nto polodal felds, toodal felds and the mean flow Applcatons to the Boussnesq-equatons, n The Nave- Stokes Equatons II Theoy and Numecal Methods 99, Spnge p Kasl shchk, I and A Vnogadov, Symmetes and consevaton laws fo dffeental equatons of mathematcal physcs Saundes, DJ, The geomety of jet bundles Vol 4 989: Cambdge Unvesty Pess 0 a, L GEOMETRIC ANALYSIS OF DIFFERENTIAL EQUATIONS - A BRIEF 05 a, L, Symmety ducton and Soluton to the Thee-Dmensonal Incompessble Nave- Stokes Equatons Chnese Quately of Mechancs, 0 4: p 007