Navier Stokes Second Exact Transformation

Transcription

1 Unversal Jornal of Appled Mathematcs (3): , 014 DOI: /jam Naver Stokes Second Eact Transformaton Aleandr Koachok Kev, Ukrane *Correspondng Athor: Coprght 014 Horon Research Pblshng All rghts reserved Abstract In ths artcle second Naver Stokes (NSE) eact transformaton to the smpler eqatons s covered Ths transformaton s eected b classcal methods of Mathematcal Analss It s shown that 3-D NSE can be conversed to a traonal vector form whch looks lke -D Vortct Transport Eqatons Second reslt, as well as frst one, s ver mportant for the solton of Naver Stokes estence and smoothness one of seven Mllennm Pre Problems that were stated b the Cla Mathematcs Insttte The proof of solton estence of sch eqatons s smpler than traonal NSE New eqatons wll smplf the soltons of man other problems of Appled Mathematcs n engneerng, aeronatcs, etc Kewords Incompressble Fld, Naver Stokes Eqatons, Vortct Transport Eqatons, Vector Valed Fncton, Acceleraton Vector, Chan Rle, Psedovector, Antsmmetrc Tensor, Mllennm Pre Problems 1 Introdcton 11 Artcle s Am In prevos artcle [1] frst Naver Stokes (NSE) eact transformatons to the smpler eqvalent eqatons was covered The prpose of preparng ths artcle s to prove that the 3-D NSE eact transformaton to other smpler eqatons s possble b another wa Ths second reslt, as well as frst one, shold facltate the solton of Naver Stokes estence and smoothness one of seven Mllennm Pre Problems that were stated b the Cla Mathematcs Insttte Also, these new eqatons wll smplf the soltons of man other problems of Appled Mathematcs n engneerng, atmospherc scences, aeronatcs, etc Other 3-D NSE eact transformatons to the smpler eqvalent eqatons are not known 1 General Data The eqatons of moton of vscos ncompressble fld also called the Naver Stokes eqatons (NSE) together wth the well known contnt eqaton can be wrtten as follows from [, p 174] ρf grad p µ ρ, (1) dv 0 () Here, F F F - vectors sm of a gven, eternall 1 appled forces (eg gravt F, magnetc F and other), 1 p - pressre, - veloct vector, d / - acceleraton vector, ρ - denst, µ - vscost, - Laplace operator Note that pressre p can be elmnated b takng the operator rot (also known crl) of both sdes of (1) Ths well-known approach belongs to Helmholt As a reslt of sch transformatons the Vortct Transport Eqatons (VTE) were obtaned [3, p 94, 31; 4, p 74] In two dmensons (-D), t s well known that (f F 0 ) ν Ω d Ω, rot, µ (3) Ω ν ρ In three dmensons (3-D) t s known for a long tme that VTE contan the adonal terms [3, p 94] Therefore 3-D VTE are ver dffclt for soltons of mportant mathematcal problems n engneerng Standard methods from PDE appear nadeqate to solve these eqatons After all ths negatvt let s restrct attenton to an postve sggestons for how to solve ths npleasant problem The analss of the adonal terms shows that 3-D VTE can be transformed After transformaton the vector form of 3-D VTE look lke (3) Method of 3-D VTE Transformaton 1 Analss of 3-D VTE Adonal Terms For vsblt let s consder the well known NSE transformaton If we appl the operator rot formla (1) becomes rot F ν rot rot (1*)

2 Unversal Jornal of Appled Mathematcs (3): , To do the analss of the adonal terms referenced above we shold wrte the epressons for components rot ( e rght - hand sde of above eqaton) The well-known form of these components n Cartesan coordnates can be wrtten so (4) Then let's consder the dervaton of epanded epresson for an one component (for eample rot ) The acceleraton vector components can be wrtten followng [, p 39] Let's dfferentate the epressons of acceleraton components to determne rot accordng (4) After transformaton of epresson rot rot, rot, rot d, t d, t d t t t, t t we obtan the 1 rot Ω Ω Ω Ω Ω t 1 Ω Ω (5) dω 1 Ω Ω The epressons for rot, rot can be wrtten b analog Now let s consder well-known traonal transformaton of (5) For -D flow 0 Therefore from (5) we fnd For 3-D flow two terms n brackets of (5) probabl are Ω and the last three terms gve Ω (b dv analog) However, t s onl or spposton Ths spposton wll be proven net Proof of Spposton (6) for 3-D VTE 1 d rot Ω Ω Ω dv Frst note that an vector on Ecldean space (,, ) can be represented as () ς, ςς(,, ) Ths representaton s well known as a vector fncton of scalar argment (also called vector-valed fncton) [5, p 514] Therefore the veloct vector t (,,,) for a fed tme t t can be represented as () ς, ςς(,, ) Then sng formla (1) n [5, p 644] we obtan (6) ς,(,, ) (7) ς Note that formlas (7) also known as chan rle [3, p 77] After transformatons of formlas (7) we obtan sch eqaltes (a complete transformaton can be fond n [1]), j j, (8) j j j j Now we wll need to check a possblt of ths spposton for formla (5) Ω (9) For vsblt of or transformaton t s sefl to wrte together relatons (8) and an eqaltes (mplng from them and reqred for followng transformatons)

3 138 Naver - Stokes Second Eact Transformaton, j j, j j j j (10) The eqaltes after confrm the val of formlas (6) for 3-D flow becase formla (5) becomes 1 rot Ω Ω Ω Ω Ω t Ω (11) As we can see eqaton (11) concdes wth eqaton (6) The epressons for two other components can be wrtten b analog As mentoned above for ncompressble fld dv 0 Therefore 3-D VTE look lke (3) f F 0 In component form we obtan three eqatons (f F 0 ) 1 dω rot F ν Ω, (,, ) (3*) Ths we can see that eqatons (3*) can be derved as the 3-D Naver Stokes eact transformaton f we appl the operator rot (crl) and adonal eqaltes (10) 3 Dscsson of man reslts 31 VTE Contradcton Three eqatons (3*) can be represented as one vector eqaton wth one varable Ω (rot ) 1 dω rot F ν Ω (3**) In mathematcs and phscs the rot (crl) s an operaton whch takes an vector feld A and prodces another vector feld rot A However t s known that rot A (also called psedovector) s eqvalent to an antsmmetrc tensor [, p 183; 8, p 104] In that case nder co-ordnate change the components of the antsmmetrc tensor shold transform dfferentl from the tre vector components Therefore the athor of tetbook [7, p 50] pad attenton that «psedovector from the pont of vew of ts vector prodct on other tre vector s eqvalent to antsmmetrc tensor, bt as vector cannot be eqal to tensor» In other words, the antsmmetrc tensor can be fond as decomposton of an rank- tensor For eample [7, p6, 63] a tensor of partal dervates of veloct vector can be wrtten as a sm of smmetrc and antsmmetrc parts: As we can see the smmetrc part s a veloct deformaton tensor ( ε ), ε ε j j j The antsmmetrc part s antsmmetrc tensor whch contans s components rot (also called psedovector components) Therefore the above epresson can be wrtten so ε ε ε ε ε ε ε ε ε (1) rot rot 1 1 rot 0 rot 1 1 rot rot 0 The above reslts show ver well that new varable Ω (rot ) n the transformed NSE s not a tre vector fncton Therefore the above defnton of (3**) as vector eqaton s ncorrect and mst be reconsdered (for eample, eqaton n vector form ) A complete analss of ths mportant problem and other contradctons n dfferent tetbooks [, 3, 4, 7] we wll consder n another paper 3 Confrmng of Man Reslt b Analog

4 Unversal Jornal of Appled Mathematcs (3): , As we can see 1-D, -D and 3-D NSE n the vector form are dentcal If we appl the operator rot these eqatons are dentcal also Ths conclson follows from (1*) In that case, b analog, 1-D, -D and 3-D VTE mst be dentcal too After compare of eqatons (3), (3*) and (3**) we can see that 3-D VTE n the vector form look lke -D VTE It s an adonal confrmaton that or mathematcal proof s correct, 33 Applcatons of Man Reslt A complete descrpton of ths problem needs carefll dscsson We can see that eqatons (3**) are smpler than sstem of eqatons (1), () Each of three eqatons (3*) nclde onl one varable Ω rot becase the pressre p can be deleted f we appl the operator rot (crl) In two dmensons, ths reslt s well known for a long tme from tetbooks [3, p 31; 4, p 74] However for the proof of -D problem Naver Stokes estence and smoothness ths reslt for some reason has not been dscssed [6] Bt, ths reslt s sed for the soltons of appled mathematcal problems n engneerng [9] Ths second eact transformaton, as well as frst one [1], shold facltate the solton of the Naver Stokes Mllennm Pre Problem However, some athors clam that ths problem even ma be solved b transformng the NSE nto an eqvalent sstem A complete descrpton of ths and other approaches can be fond here [10] The athor of ths cted work clams: Ths, one s left wth onl three possble strateges f one wants to solve the fll problem The formlaton of strateg 1 look lke solve the Naver-Stokes eqaton eactl and eplctl (or at least transform ths eqaton eactl and eplctl to a smpler eqaton) 4 Conclson As follows from the dfferent books [, p 94, 31; 3, p 74] NSE (1) can be conversed to the VTE b takng the operator rot (also known crl) However the well-known vector forms of the 3-D and -D VTE are dfferent In ths paper we have shown that 3-D eqatons n the vector form can be conversed and look lke -D eqatons As we can see the conversaton method s based on an eact transformaton of formlas (7) also known as chan rle In 3 we can see the confrmng of ths reslt b analog It s an adonal confrmaton that or mathematcal proof s correct, Ths eact NSE transformaton, as well as frst one [1], shold facltate the solton of the Naver Stokes Mllennm Pre Problem All proofs of the Naver Stokes estence and smoothness shold se the adonal conons (8) However, some athors (for eample [10]) even clam that the Naver Stokes Mllennm Pre Problem ma be solved onl b transformng the NSE nto an eqvalent sstem Other 3-D NSE eact transformatons to the smpler eqvalent eqatons are not known From new basc eqatons (3*) or (3**) one can go to solve more complcated problems of appled mathematcs n engneerng, atmospherc scences, aeronatcs, etc Eqatons (3**) and (11) n [1] can be sed for the checkng of so called NSE eact soltons All tre eact soltons of the NSE shold satsf these eqatons Otherwse sch NSE eact soltons are false The net npleasant thngs we have also establshed for sch well-known classcal rle of operaton rot (crl) After comparson of nformaton n dfferent tetbooks [, 7] we have establshed that new varable Ω (rot ) n the transformed NSE (3**) s not a tre vector fncton It s known [, p 183; 8, p 104] that components of Ω (sometme called psedovector) s eqvalent to the antsmmetrc tensor components The adonal confrmaton of ths clam we can obtan after analss of formla (1) Hence ths clam probabl s correct The consderaton of ths mportant problem we prolong n another paper Acknowledgements The athor thanks three anonmos referees for man sefl remarks and comments These remarks have helped to mprove the sbmtted manscrpt and to prepare ths new verson REFERENCES [1] Aleandr Koachok Naver Stokes Frst Eact Transformaton, Unversal Jornal of Appled Mathematcs, 1, do: /jam , 013 (Englsh) 16pdf [] LI Sedov Mechancs of Contnos Meda, v 1 Tetbook, Scence, Moscow, 1970http://eqworldpmnetr/r/lbrar/b ooks/sedov_mss_t1_1970rdjv) (Rssan) ISBN-13: (Englsh) [3] JH Henbockel Introdcton to Tensor Calcls and Contnm Mechancs, Trafford Pblshng, 001, ISBN-13: (Englsh) 7 [4] GShlhtng The theor of a bondar laer, Scence, Moscow, 1969(Rssan) chtng1974rdjv [5] MJ Vgodsk Manal on Hgher Mathematcs, (1th eon) Scence, 1977 (Rssan) j1977rdjv ASIN: B001U5VF9O (Englsh) [6] O Ladhenskaa The Mathematcal Theor of Vscos Incompressble Flows (nd eon), Gordon and Breach,

5 140 Naver - Stokes Second Eact Transformaton New York, 1969 (Englsh) [7] LGLojtsjansk Fld and Gas Mechancs, Manal, Scence, Moscow, 1970(Rssan) anskj1950rdjv [8] AI Borsenko, IE Tarapov Vector Analss and begnnngs of Tensors Calcls (3th eon), Manal, Hgher school, Moscow, 1966 (Rssan) rdjv ISBN-13: (Englsh) [9] Fel Kaplansk The vortct eqaton and ts applcaton Talln Unverst of Technolog (Englsh) %0eqaton%0and%0ts%0applcatonsppt [10] Terence Tao Wh global reglart for Naver-Stokes s hard (research and epostor papers, dscsson of open problems, and other maths-related topcs), rt-for-naver-stokes-s-hard/