THE NAVIER-STOKES EQUATION: The Queen of Fluid Dynamics. A proof simple, but complete.

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1 THE NAIER-TOKE EQUATION: The Queen of Fluid Dnamics. A poof simple, but complete. Leonado Rubino leonubino@ahoo.it eptembe 010 Rev. 00 Fo Abstact: in this pape ou will find a simple demonstation of the Navie-tokes equation, while, most of times, in books, ou find it boken into its vectoial components whose poofs ae usuall not so clea, so getting confused on the topic. Moeove, in the appendies, ou can also find an oiginal poof of the tokes (oto) theoem, b the autho of this pape. The Navie-tokes Equation in the case o fan incompessibile fluid, that is v s = 0: (this situations is about most of pactical cases) = const and 1 [ + Ω v + v ] = p φ + η v whee Ω = v (voticit), η (osit), φ (gavitational potential), (densit), v (velocit), t (time). Poof: -Let s stat fom the Continuit Equation + ( v ) = 0 t, and we pove it: kg s v = J is the mass cuent densit [ ] (dimensionall obvious) m M = d (held obvious) We have: M = d = d = v d,in fact, in tems of dimensions: l d = l d and so d = d = d v and sign is in case of escaping mass. d d l o: d = the last equalit. ( v) d = ( v) d, afte having used the Divegence Theoem in Theefoe: [ + ( v)] d = 0, fom which we get the Continuit Equation.

2 p -and let s also stat fom the Eule s Equation ( + ( v ) v = ), and we also pove this: (p is the pessue; moeove, this equation is a sketch of the Navie-tokes Equation, wheeas we e not et taking into account the gavitational field and the ous foces) The foce acting on a small fluid volume d is df = p d, with sign -, as we ae dealing with a foce towads the small volume. Moeove: f = p d = p d, afte having used a dual of the Divegence theoem (a Geen s fomula), to go fom the suface integal to the volume one. f We also have: = [ p d = p ], but, in tems of dimensions, it s simultaneousl tue that: f d dv dm dv dv = [ M ] = = = and fom these two equations, we have: d dt d dt dt dv = p. (1.1) dt d d d Now we emind that: d l = ( d, d, d), d d d = (,, ) and v = (,, ), so we can d d d dt dt dt easil wite that: dv d d d dv = = + ( v ) v = and fo (1.1) we finall have: dt dt dt dt dt )v... v... p + ( v ) v = that is the Eule s Equation, indeed. Now, the tems of this Eule s Equation have the dimensiono f an acceleation a ; so, if we want to take into account the gavitational field, too, on the ight side we can algebaicall add the gavitational acceleation g, with a negative sign, as it s downwads. But we know that the gadient of the potential φ is eall g ( φ = g ), so: p + ( v ) v = φ. As the following vectoial identit is in foce: 1 ( v ) v= ( v) v+ ( v v), and if we take the epession fo the voticit, on page 1, ( Ω = v ), we have: v 1 p + Ω v+ v = φ and, so fa, we have also taken into account the gavitational field.

3 In the most geneal case whee we have to do with a ous fluid, we ll also add a ous foce component: v 1 + Ω v + v p = φ + f (1.) wheeas f is divided b the densit because of the dimension compatibilità with othe tems in that equation. (1.) is alead the Navie-tokes Equation, wheeas the ous foce f is still to be evaluated. We will evaluate f in the case of incompessibile fluids, that is fluids with = const, >> = 0 so, fo the Continuit Equations, ( v s s ) = 0, >> v = 0. Calculation of f : ICOITY: v F Fee suface of the fluid d Fig. 1. We know fom geneal phsics that: F bottom v = η, (1.3) d That is, in ode to dag the slab whose base suface is, ove the fluid, at a d distance fom the bottom, and dag it at a v speed, we need a foce F Now, let s wite down (1.3) in a diffeential fom, fo stesses τ and fo components: () F τ = = η, having set v = ( u, v, w), and so: F = η (1.4)

4 We now use (1.4) on a small fluid volume d in Fig. : =d w UP one v d=d ( in _ = d) u =0 ( in _ = 0) 6 DOWN one Fig. : mal volume of fluid d. Fig. 3: Ais, faces and 5. In Fig. 3 we have epoduced what shown in Fig. 1, but in a thee-dimension contet. Faces and 5: so, with efeence to Fig. 3, let s figue out the ous foces (due to vaiations of u) on faces and 5 of the small volume, that is those we meet when moving along the ais, b using (1.4): iscous shea stess on face = + η [ ( in _ = d)] dd This foce acting on face is positive (+) because the fluid ove the point whee it s figued out (UP one) has got a highe speed (longe hoiontal aows) which dags along the positive. On face 5, on the conta, we ll have a (-) negative sign, because the fluid unde such suface has got a lowe speed (down) and want to be dagged, so making a esistance, that is a negative foce: iscous shea stess on face 5 = η [ ( in _ = 0)] dd v _ in _(1.3 ) in _(1.3) d The esultant on is the diffeence between the two equations, o bette, the algebaic sum: [ ( = d) ( = 0)] u F ( ) = η[ ( = d) ( = 0)] dd = η ddd = η d, afte d having multiplied numeato and denominato b d. Theefoe:

5 u = η d (ous foce on due to vaiations of u along ) (1.5) F ( ) Faces 3 and 6: =0 =d ( in _ = 0) Fig. 4: Ais, faces 3 and 6. ( in _ = d) imilal to the pevious case, we have, as a esult: u =η d (ous foce on due to vaiationd of u along ) (1.6) F ( ) Faces 1 and 4: Fo what case F ( ) is concened, that is the ous foce on due to vaiations of u (which is a component on ) along itself, we will not talk about shea stesses, as, in such a case, the elevant foce is still about, but acts on =dd, which is othogonal to ; so, it s about a NORMAL foce, a tensile/compession one, and we efe to Fig. 5 below: epansion ( in _ = 0) ( in _ = d) compession =d =0 Fig. 5: Ais, faces 1 and 4. Anwa, nothing changes with numbes, with espect to pevious cases, and we have:

6 u =η d (ous foce on due to vaiations of u along itself) (1.7) F ( ) Now that we have thee components of the ous foces acting along (that is those due to vaiations of the u component (comp. ) of speed v, with espect to, and itself), let s sum them up and get F : u u u u u u = η d + η d + η d = η d ( + + ) = η d u, and we ewite F it below: F = η d u (1.8) Now we ca out the same easonings foa n evaluation of obviousl get ( v = ( u, v, w) ): F = η d F = η d v w F and of F, and (1.9) (1.10) fom which, finall, b adding (1.8), (1.9), and (1.10), we have: F F ˆ + F ˆ + F = ˆ = η d[ ˆ u + ˆ v + ˆ w] = η d v che isciviamo: F =η d v (1.11) Now, such a F must be used in (1.), afte having divided it b and b d (that is, fo M = d ), as both sides of (1.) have got the dimensiono f a foce pe a mass, indie, so: 1 + Ω v + v p = φ + η v And theefoe, finall, the Navie-tokes Equation, and we wite it bette again: + Ω v + 1 v p = φ + η v (1.1) F = +... M t Geneal acceleation Pessue foces Gavit foces iscous foces

7 Appendies: Appendi 1) Compessible fluids ve ae cases: fo those cases, const, >> 0, >> ( v s ) 0, and to (1.1) we have to add the ( η+ η') following tem: + ( v ), but (1.1) alead enclose a big seies of pactical cases Appendi ) Divegence Theoem (pactical poof): B d d C D E F G H dτ E d = dive d 0 A d E Fig. 6: Fo the Divegence Theoem. Nameφ the flu of the vecto E ; we have: dφ = E d = E (,, dd ( means mean ) ABCD ) EFGH = E( + d,, ) dd dφ, but we obviousl know that also: (as a development): E(,, ) E( + d,, ) = E(,, ) + d so: E(,, ) dφ EFGH = E(,, ) dd + ddd and so: E dφ ABCD + dφefgh = d. We similal act on aes and : E dφ AEHD + dφbcgf = d E dφ ABFE + dφcghd = d And then we sum up the flues so found, having totall: E E E dφ = ( + + ) d = ( div E) d = ( E) d theefoe: φ ( E ) = dφ = E d = dive d = ( E) d that is the statement. φ

8 Appendi 3) Roto o tokes Theoem (pactical poof-b Rubino!): d B l B = otb d = B d -d d -d d 0 -d Walk Fig. 7: Fo the Roto Theoem (poof b Rubino). Let s figue out B : On d B is B ; on d B is B ; on d B is B ; B B on -d B is B + d d, fo 3-D Talo s development and also because to go fom the cente of d to that of d we go up along, then we go down along and nothing along itself. B B B B imilal, on -d B is B d + d and on -d B is B d + d. B summing up all contibutions: B B B B B = Bd ( B + d d) d + Bd ( B d + d) d + Bd B B B B B B B B + ( B d + d) d = ( ) dd + ( ) dd + ( ) dd = otb d B d wheeas hee d v = = has got components [ ˆ( dd), ˆ( dd), ˆ( dd) ] that is, the statement: l B = v otb d = B d, afte having eminded of: otb = B = ˆ B ˆ B ˆ B.

9 1 Appendi 4) The Benoulli s Equations: v + p + g = 0 If we ae in a stationa situation, wheeas v f (t) >> v = 0, and then = const, and whee thee s no ous foces, the Navie-tokes Equation fo sue educe sto the Eule s one (but added with the gavitational component): p + ( v ) v = g, and, bette, as we said that = 0, we have: p ( v ) v = g. (1.13) If now we conside the divegence and the gadient in tems of diectional deivative, on diection dv v dp, specificall, then we have in (1.13): instead of, and instead of p and then, still in (1.13), the gavitational acceleation g (which eets along, downwads) must be pojected along d ( is the elevant diection cosine), and so (1.13) becomes: dv 1 dp d 1 v = g, fom which: vdv + dp + gd = 0 and b integating it: 1 p v + + g = 0, and b multipling b the densit, we get: 1 v + p + g = 0 that is, eall the statement! Bibliogaph: 1) (C. Mencuccini and. ilvestini) FIICA I - Meccanica Temodinamica, Liguoi. ) ( Y. Nakaama) INTRODUCTION TO FLUID MECHANIC - Buttewoth Heinemann. 3) (L. D. Landau & E. M. Lifshit) FLUID MECHANIC - Pegamon Pess. 4) ME INTERMEDIATE FLUID DYNAMIC (Lectues). 5) (R. Fenman) THE FEYNMAN PHYIC II Zanichelli. 6) (L. Rubino) Publications on phsics in the Italian phsics website fisicamente.net

dq 1 (5) q 1 where the previously mentioned limit has been taken.

dq 1 (5) q 1 where the previously mentioned limit has been taken. 1 Vecto Calculus And Continuum Consevation Equations In Cuvilinea Othogonal Coodinates Robet Maska: Novembe 25, 2008 In ode to ewite the consevation equations(continuit, momentum, eneg) to some cuvilinea