Solutions for Euler and Navier-Stokes Equations in Powers of Time

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1 Solutons for Euler and Naver-Stokes Equatons n Powers of Tme Valdr Montero dos Santos Godo valdr.msgodo@gmal.com Abstract We present a soluton for the Euler and Naver-Stokes equatons for ncompressble case gven any smooth (C ) ntal velocty, pressure and external force n N = spatal dmensons, based on expanson n Taylor s seres of tme. Wthout major dffcultes, t can be adapted to any spatal dmenson, N 1. Keywords Lagrange, Mécanque Analtque, exact dfferental, Euler s equatons, Naver-Stokes equatons, Taylor s seres, Cauchy, Mémore sur la Théore des Ondes, Lagrange s theorem, Bernoull s law, non-unqueness solutons. 1 Let p, q, r be the three components of velocty of an element of flud n the - D orthogonal Eucldean system of spatal coordnates (x, y, z) and t the tme n ths system. Lagrange n hs Mécanque Analtque, frstly publshed n 1788, proved that f the quantty (p dx + q dy + r dz) s an exact dfferental when t = t wll also be an exact dfferental when t has any other value. If the quantty (p dx + q dy + r dz) s an exact dfferental at an arbtrary nstant, t should be such for all other nstants. Consequently, f there s one nstant durng the moton for whch t s not an exact dfferental, t cannot be exact for the entre perod of moton. If t were exact at another arbtrary nstant, t should also be exact at the frst nstant. [1] To prove t Lagrange used p = p I + p II t + p III t 2 + p IV t + (1.1) { q = q I + q II t + q III t 2 + q IV t + r = r I + r II t + r III t 2 + r IV t + n whch the quanttes p I, p II, p III, etc., q I, q II, q III, etc., r I, r II, r III, etc., are functons of x, y, z but wthout t. Here we wll fnally solve the equatons of Euler and Naver-Stokes usng ths representaton of the velocty components n nfnte seres, as ponted by Lagrange. We assume satsfed the condton of ncompressblty, for brevty. Wthout t the resultng equatons are more complcated, as we know, but the method of soluton s essentally the same n both cases. We focus our attenton n the general case of the Naver-Stokes equatons, and for the Euler equatons smply set the vscosty coeffcent as ν =. 1

2 To facltate and abbrevate our wrtng, we represent the flud velocty by ts three components n ndcal notaton,.e., u = (u 1, u 2, u ), as well as the external force wll be f = (f 1, f 2, f ) and the spatal coordnates x 1 x, x 2 y, x z. The pressure, a scalar functon, wll be represented as p. As frequently used n mathematcs approach, the densty mass wll be ρ = 1 (otherwse substtute p by p/ρ and by /ρ). We consder all functons belongng to C, beng vald the use of 2 u x k = relaton to space as to tme. 2 u x k and other nversons n order of dervatves, so much n The representaton (1.1) s as the expanson of the velocty n a Taylor s seres n relaton to tme around t =, consderng x, y, z as constant,.e., for 1, (1.2) u = u t= + t t= t + 2 u t 2 t= t2 2 + u t t= t 6 + or (1.) u = u + + k u t k t= tk k! + k u tk k=1 t k t=. k! For the calculaton of, 2 u, u, t t 2 we use the values that are obtaned t drectly from the Naver-Stokes equatons and ts dervatves n relaton to tme,.e., (1.4) and therefore t = p x j=1 u j + ν 2 u + f, (1.5) 2 u t 2 = 2 p t x j=1 ( u j t + u j t ) + ν 2 t + f t, (1.6) (1.7) u = p t t 2 x +ν 2 2 u t 2 j=1 + 2 f t 2, ( 2 u j t 2 4 u = 4 p N t 4 t =1 j N j = t N j 2, N j 2 = 2 u j t 2 N j = u j t + 2 u j t u j t + ν 2 u t + 2 u j t + u j t t t + u j + f t, t + u j 2 u + u t 2 j 2 u t 2, 2 u t 2 ) u t, 2

3 (1.8) 5 u = 5 p 4 N t 5 t 4 =1 j N j 4 = t N j = 4 u j t 4 +4 u j t + 4 u j t u + u t j + ν 2 4 u t f t 4, u j t t 2 4 u t 4, 2 u t 2 + and usng nducton we come to (1.9) k u = k p t k t k 1 x k 1 j=1 N j N k 1 j = N t j k 2 k 1 = ( k 1 l= t u n = u n, t m u n = m u n t m l ) + ν 2 k 1 u t k 1 + k 1 f t k 1, t k 1 l u j t l u,, (k 1 l ) = (k 1)! (k 1 l)! l!. t, 2 u In (1.2) and (1.) t s necessary to know the values of the dervatves,, k u t 2 t k n t = then we must to calculate, from (1.4) to (1.9), (1.1) t t= = p u =1 j + ν 2 u + f, the superor ndex meanng the value of the respectve functon at t =, and (1.11) (1.12) (1.1) 2 u t 2 t= = 2 p 1 t x t= j=1 N j t= + N j 1 t= = u + ν 2 t t= + f t t=, ( u j t t= + u j=1 j t t=), p t t= = t 2 x t= j=1 N 2 j t= + N j 2 t= = 2 u j t 2 4 u + ν 2 2 u t 2 t= + 2 f t= + u j 2 u t 2 t=, 4 p + 2 u j t t= t 2 t=, t t= + t 4 t= = t x t= j=1 N j t= + N j t= = u j t + ν 2 u t t= + f t= + 2 u j t 2 t t=, t= t t= +

4 + u j t t= 2 u t 2 t= + u j u t t=, (1.14) 5 u 5 p 4 t 5 t= = t 4 x t= j=1 N j t= + N j 4 t= = 4 u j t u j t 2 + ν 2 4 u t 4 t= + 4 f t 4 t=, t= t= + u j 4 u t 4 t=, + 4 u j t t= t + 2 u t 2 t= + 4 u j u t t= t t= + and of generc form, (1.15) k u k p N j k 1 t k t= = t k 1 x t= j=1 t= + + ν 2 k 1 u t k 1 t= + k 1 f t k 1 t=, N k 1 k 1 j t= = ( k 1 l= l ) t u n t= = u n, t m u n t= = m u n t m t k 1 l u j t= t l u t=, t=. If the external force s conservatve there s a scalar potental U such as f = U and the pressure can be calculated from ths potental U,.e., (1.16) p = f x = U, x and then (1.17) p = U + θ(t), θ(t) a generc functon of tme of class C, so t s not necessary the use of the pressure p and external force f, and respectve dervatves, n (1.4) to (1.15) f the external force s conservatve. In ths case, the velocty can be ndependent of the both pressure and external force, otherwse t wll be necessary to use both the pressure and external force dervatves to calculate the velocty n powers of tme. The result that we obtan here n ths development n Taylor s seres seems to me a great advance n the search of the solutons of the Euler s and Naver- Stokes equatons. It s possble now to know on the possblty of non-unqueness solutons as well as breakdown soluton respect to unbounded energy of another manner. We now can choose prevously an nfnty of dfferent pressures such that 4

5 the calculaton of u and dervatves can be done, for a gven ntal velocty and t external force, although such calculaton can be very hard. It s convenent say that Cauchy [2] n hs memorable and admrable Mémore sur la Théore des Ondes, wnner of the Mathematcal Analyss award, year 1815, frstly does a study on the equatons to be obeyed by three-dmensonal molecules n a homogeneous flud n the ntal nstant t =, comng to the concluson whch the ntal velocty must be rrotatonal,.e., a potental flow. Of ths manner, after, he comes to concluson that the velocty s always rrotatonal, potental flow, f the external force s conservatve, whch s essentally the Lagrange s theorem descrbed n the begn of ths artcle, but t s shown wthout the use of seres expanson (a possble excepton to the theorem occurs f one or two components of velocty are dentcally zero, when the reasonngs on -D molecular volume are not vald). The soluton obtaned by Cauchy for Euler's equatons s the Bernoull's law, as almost always happens. Now a more generc soluton s obtaned, n specal when t s possble a soluton be expanded n polynomal seres of tme. Though not always a functon can be expanded n Taylor s seres, there s certanly an nfnty of possble cases of soluton where ths s possble. If the mentoned seres s dvergent n some pont or regon may be an ndcatve of that the correspondent velocty and ts square dverge, agan gong to the case of breakdown soluton due to unbounded energy. Wth the three functons ntal velocty, pressure and external force belongng to Schwartz Space s expected that the soluton for velocty also belongs to Schwartz Space, obtanng physcally reasonable and well-behaved soluton throughout the space. The method presented here n ths frst secton can also be appled n other equatons, of course, for example n the heat equaton, Schrödnger equaton, wave equaton and many others. Always wll be necessary that the remander n the Taylor's seres goes to zero when the order k of the dervatve tends to nfnty (Courant [], chap. VI). Applyng ths concept n (1.) and (1.9), substtutng t by τ, the remander R,k of order k for velocty component s (1.18) R,k = 1 t (t τ) k k! k+1 u dτ, tk+1 whch can be estmated by Lagrange s remander, (1.19) R,k = tk+1 (k+1)! or by Cauchy s remander, (1.2) R,k = tk+1 k! k+1 u t k+1 (ξ), (1 θ) k k+1 u (ξ), tk+1 5

6 wth ξ t and θ 1. Note that f t s not possble to make a seres around t = (for example, to the functons log t, t, e 1/t2, accordng Courant [], chap. VI) an other nstant t of convergence and remander R,k zero must be found, and then replacng t k by (t t ) k and the calculatons n t = by t = t n prevous equatons. 2 In ths secton we wll buld a seres of powers of tme solvng the Naver- Stokes equatons, dfferently than that used n the prevous secton. From theorem of unqueness of seres of powers (A functon f(x) can be represented by a power seres n x n only one way, f t all,.e., the representaton of a functon by a power seres s unque ; Every power seres whch converges for ponts other than x = s the Taylor seres of the functon whch t represents (Courant [], chap. VIII)), both solutons need be the same, for a same ntal velocty, pressure, external force, compressblty condton and all boundary condtons. Defnng (2.1) u = u + X,1 t + X,2 t X,n t n + = n= X,n t n, X, = u = u (x 1, x 2, x, ), where each X,n s a functon of poston (x 1, x 2, x ), wthout t, and (2.2) p = q x + q,1 t + q,2 t q,n t n + = n= q,n t n, q, = q = p x, p = p(x 1, x 2, x, ), (2.) f = f + f,1 t + f,2 t f,n t n + = n= f,n t n, f, = f = f (x 1, x 2, x, ), we can put these seres n the Naver-Stokes equaton, (2.4) t = p x j=1 u j + ν 2 u + f. The velocty dervatve n relaton to tme s (2.5) = X t,1 + 2X,2 t + X, t nx,n t n 1 + = = n= (n + 1)X,n+1 t n, the nonlnear terms are, of order zero (constant n tme) 6

7 (2.6) j=1 u j of order 1,, (2.7) (u j X,1 of order 2, u + X j,1 j=1 ) t, j=1, (2.8) (u X,2 X j + X,1 u j,1 + X j,2 ) t 2 of order, j=1, (2.9) (u X, X j + X,2 X j,1 + X,1 u j,2 + X j, ) t and of order n, of generc form, equal to n j=1 k= t n, (2.1) X j,k X,n k wth X j, = u j, X, = u. Applyng these sums n (2.4) we have (2.11) n= (n + 1)X,n+1 t n = n= q,n t n and then j=1 n k= X X,n k n= j,k + ν n= 2 X,n t n + + n= f,n t n, (2.12) (n + 1)X,n+1 = q,n X j,k X,n k 7 t n + ν 2 X,n + f,n, n j=1 k= + whch allows us to obtan, by recurrence, X,1, X,2, X,, etc., that s, for 1 and n, (2.1) X,n+1 = 1 n+1 S n, n X S n = q,n X,n k j=1 k= j,k + ν 2 X,n + f,n. You can see how much wll become ncreasngly dffcult calculate the terms X,n wth ncreasng the values of n, for example, wll appear terms n

8 ν n, u, etc. If ν > 1 (say, 1/ρ > 1 or ν/ρ > 1) certanly there s a specfc problem to be studed wth relaton to convergence of the seres, whch of course also occurs n the representaton gven n secton 1. The same can be sad for t. In fact, I do not understand why a partcle flud ntally n moton, wthout any collson wth another partcle and submtted to a permanent mpulsve force need always be wth fnte velocty as t. For example, a constant resultng force f, not equal to zero, appled all tme on a body wll produce an nfnte velocty u to ths body when t, supposng possble such force and a way no obstacles, etc. The prevous solutons show us that we need to have, for all ntegers 1 and n, (.1) 1 n! n u t n t= = X,n, and both members of ths relaton are very dffcult to be calculated, ether equaton (1.15) as well as (2.1). Add to ths dffculty the fact that besdes the man Naver-Stokes equatons (1.4)-(2.4) must be ncluded the condton of ncompressblty, u x (.2) u = =1 =. Usng (2.1) n (.2) we have (.) u = =1 n= X,n t n = n= ( =1 X,n ) t n =. x As ths equaton need be vald for all t we have (.4) =1 X,n = X n =, x defnng X n = (X 1,n, X 2,n, X,n ),.e., all coeffcents X n must obey the condton of ncompressblty n the vector representaton of velocty, (.5) u = n= X n t n. Followng Lagrange [1], gettng two dfferentable and contnuous functons α and β of class C 2 and defnng x (.6.1) u 1 = α z, u 2 = β z, u = ( α x + β y ), 8

9 (.6.2) u 1 = α z, u 2 = β z, u = ( α x + β y ), wth α = α(t = ) and β = β(t = ), we have satsfed the condton (.2), whch t s easy to see. Other manner s when u s derved from a vector potental A,.e., (.7.1) u = A, (.7.2) u = A, wth A = A(t = ). The relatons (.6) are very useful and easy to be mplemented and we wll use them to solve the Euler and Naver-Stokes equatons when the ncompressblty condton s requred. Gven any contnuous, dfferentable and ntegrable vector components u 1 and u 2 then (.8.1) α = u 1 dz, (.8.2) β = u 2 dz, and thus u and u need to be accordng (.9.1) u = ( u 1 (.9.2) u = ( u 1 + u 2 x y + u 2 x y ) dz = (α ) dz = (α + β ), x y + β ), x y whch remnds us that the components of the velocty vector mantans condtons to be compled to each other,.e., t s not any ntal velocty whch can be used for soluton of Euler and Naver-Stokes equatons n ncompressble flows case. In the equatons of the sectons 1 and 2, nstead u 1 we wll use α z, nstead u 2 wll be β, and (α + β z x y ) nstead u, as well as the correspondents ntal values, replacng u 1 by α z, u 2 by β z, and u by ( α x + β ). Of ths y manner, we wll be developng seres for α, β and (α + β ), so that u =. z z x y Then ths s a prelmnary problem to be solved, the calculaton of α and β gvng u 1, u 2 and u when u = and t s necessary that u =,.e., (.1.1) α = u 1 dz, (.1.2) β = u 2 dz, wth the valdty of (.9.2). Done ths, the exact soluton for the prncpal problem can be calculated from reasonng exposed here, f there s not an equvalent 9

10 soluton descrbed n a most smplfed formulaton, for example, accordng Bernoull s law and Laplace s equaton. September-11, December-16,21, What good would lvng on a planet wthout destructon, greed and envy, where the natons were dedcated to buldng a beautful world and to the salvaton of those n need. That there were no enemes and everyone could be happy where they lve, n ther own way. References [1] Lagrange, Joseph L., Analytcal Mechancs, Boston Studes n the Phlosophy of Scence, 191. Boston: Sprnger-Scence+Busness Meda Dordrecht (1997). [2] Cauchy, Augustn-Lous, Mémore sur la Théore des Ondes, n Oeuvres Completes D Augustn Cauchy, sére I, tome I. Pars: Gauthers-Vllars (1882). [] Courant, Rchard, Dfferental and Integral Calculus, vol. I. London and Glasgow: Blacke & Son Lmted (197). 1