Reduction of over-determined systems of differential equations

Transcription

1 Redction of oer-determined systems of differential eqations Maim Zaytse 1) 1, ) and Vyachesla Akkerman 1) Nclear Safety Institte, Rssian Academy of Sciences, Moscow, Rssia ) Department of Mechanical and Aerospace Engineering, West Virginia Uniersity, USA It is shown how the dimension of any arbitrary oer-determined system of differential eqations can be redced, which makes the system sitable for nmerical soltion modeling. Specifically, oer-determined eqations of hydrodynamics are presented. 1. We start with a set of two linear partial differential eqations, which are oer determined by an additional eqation. For eample: H G H, (1) G H + G, () H G. (3) This system has a joint triial soltion. Indeed, let s sbstitte the epression for deriatie G from Eq. (1) into Eq. (3), H 1 + H. (4) Differentiating Eq. (4) with respect to : H H H + ( 1 ) + H + 1

2 and sbstitting on the deriatie H from formla (), we find G G H G + H. (5) Let s fi the point. Then we obtain the system of ordinary differential eqations (4), (5) eoling in the point precisely. Differentiating Eq. (5) with respect to : 3 G G G G G H H ( ) G (6) and sbstitting the deriatie G and H from formlae (1) and () into (6) we find H H G G H G H 3 3 t (7) Let s epress the deriaties H t, H t, 3 H t 3, G t from formlae (4) and (5). We hae, H ( 1) H (8) H ( 1) H (9) H ( 1) 3 H ( 1) 1 ( 1) ( 1) G G G + H (1) (11) We sbstitte Eqs. (8) - (11) into (7). Conseqently, G ( 1) G + H. (1)

3 Let s fi the point again. Then we hae an oer determined system of ordinary differential eqations (4), (5) and (1) eoling in the point precisely. Let s find its soltion. After differentiating (1) with respect to t and sbstitting (8) - (11) we hae ( + 1) ( 1) G ( 1) G + H (13) Let s repeat the same procedre with Eq. (13): + 4 G G + H (14) ariables G As a reslt we fond the linear system of three eqations (1) - (14) with three t, G and H which has triial soltion only becase its determinant is not zero. Conseqently, an oer-determined system of eqations (1) (3) has also zero soltion only.. Consider the set of p partial differential eqations which are oer determined by single independent eqation now H k S S S,,..., 1... p, k 1... p, (15) r S S G S,,.... (16) r Let s switch to the coordinates (,, ) τ τ n in a point M on a fied srface (see Fig. 1). 1 Then eqations (15) and (16) can be written as H k S S S S S,,,,..., 1... p, k 1... p, (17) τ1 τ n S S S S G S,,,,.... (18) τ1 τ n We epress the normal deriaties S n from the eqations (17) in an eplicit form k S k, S k, S F S, S..., 1... p, k 1... p, (19) n τ1 τ 3

4 We sbstitte the epression (19) into (18). Then G ( 1) S S S S,,,.... () τ1 τ We sbseqently differentiate Eq. () in the direction n and sbstitte Eq. (19). Then G S S S S,,,.... (1) τ1 τ Let s make the same procedre p times. As a reslt, we get p eqations on the srface as G ( 1) S S S S,,,..., τ1 τ. () G p p p p S p 1 p 1 p τ1 τ S S S,,,.... We fond a closed system of p differential eqations () along the bondary of the srface (see Fig. 1) and the same nmber of ariables S k, k 1... p, that eole oer time. Formally, a similar procedre can be tilized to obtain more than p eqations on the srface (), i.e. find the oer-determined system of eqations which is in it. Therefore, to redce the dimension of the srface, etc. p to analytic soltions. Howeer, this does not mean that sch a procedre is possible to find an analytic soltion to any oerdetermined system of eqations. For eample, the following oerdetermined system of eqations has the general soltion α e +, t + α. Howeer, frther redction of dimension one can not moe forward. Introdce the notation 4

5 A S t, 1... p. (3) Using (3) let s represent of (19), () as Sk n F k ( A...), 1... p, k 1... p, (4) ( 1 ) G A.... (5) We differentiate Eq. (5) in the direction n and denote terms containing the highest time deriaties in the reslting epansion. ( 1) S p G A n (6) 1 or, sing (4) p ( 1) G F S A A 1 ( +... G ) (...). (7) 1, 1 1 Do the same procedre p times. Then the system of eqations (), which highlighted the terms containing the highest time deriaties, can be written as p ( 1) l G F F S A A A 1 l 1 l ( l G ) (...), l 1... l 1,... l 1 1 l p, (8) We differentiate each l - th eqation of the system (8) with respect to time t ( p l) times. Then we get the following system ( 1) F F S, l 1... p. (9) p p G 1 l 1 l p 1,... l 1 A A 1 A l System of srface (9) is linear with respect to the highest time deriaties p p S t, 1... p. The condition that these deriaties can be eplicitly epressed from (9), is as follows a, (3) j i where 5

6 a ji p G ( 1) F F 1 j 1..., 1 1,... j 1 1 A A A 1 i j >, a 1i ( 1) G, j 1. A i In this case, (9) can be written as p p 1 p 1 S k S S Qk,... p p 1 p, 1... p, k 1... p. (31) τ1 To the system of srface eqations (31) it is already possible to pt the corresponding Cachy problem. According to the general Cachy-Koaleskaya theorem, this problem has a niqe soltion. [1] We see that if the condition (3) holds, the bondary conditions do not need to pt. For or eample (1) - (3) the corresponding determinant has the form a j i ( ) ( ) 1. 1 To accont for a srface, moing with elocity Vn, where V Ft F and n F F - the nit normal to the srface F (,..., 1 ) m t (see Fig. ), it is enogh to se the following obios relation to the srface: j 1 j j d S S S V j 1. j 1... p, 1... p (3) j j 1 dt n Instead of the system of eqations () is a system of p + p eqations (19), () и (3) and p + p nknown: j j S t, j... p, 1... p. 6

7 Figre 1 Stationary srface F,..., 1 m t Figre Moing srface 7

8 3. Naier-Stokes eqations I 1 ω + P + ν ω, (33) ω, (34) di. (35) ρ + ( + α) ρ + ρ di ( + α), (36) [ α ω + ( α) + α ( α) ] + ν ω, (37) r r ω + α ω ( r ), z (38) r r ω y + α y ω ( r ), z (39) 1 (, y, z) 1. (4) ρ (, y, z) + + α, (41) Preiosly, we hae an oer-determined system of 15 eqations (33) - (41) "Naier-Stokes eqations", and 14 ariables α,, ω, P, ρ and r, where ω - the initial distribtion of the orticity ω.[]. The Naier-Stokes eqations (33) - (41) in a olme in this way can be redced to a system of eqations in the plane { z c }, and een get an oer-determined system of the srface eqations. Conseqently, it can be redced to an oer-determined system of eqations on the line{ z c, y b }, then at the point{ z c, y b, a }and finally to an oer-determined system of ordinary differential eqations (abot seeral hndred thosands of eqations) whose soltion gies an analytic soltion at{ a b c t } 4. Oerriding a complete system of hydrodynamic eqations I ρ + di ( ρ), (4) 8

9 P σ F ω + +, (43) ρ ρ ρ ω, (44) ψ di, (45) s ρt + ( ) s Q di + Φ q, (46) ( ρ s) T T,, (47) ( ρ s) P P,, (48) где 1 4 σ µ + ( ) µ + ψ 3 ω 3. (49) µ i k l Φ + µ. (5) k i 3 l ρ + ( + α) ρ + ρ di ( + α), (51) [ α ω + ( α) + α ( α) ] σ F T s. (5) ρ ρ r r ω + α ω ( r ), z (53) r r ω y + α y ω ( r ), z (54) 9

10 1 (, y, z) 1 ρ (, y, z) ρ ( r ). (55) + + α, (56) q κ T. (57) Conseqently, we find an oer-determined system of first order partial differential * eqations (4) - (48), (51) - (57) of 1-th nknown ρ, ρ,, ω, α, P, T, s, ψ, r, q [1]. With this system (4) - (48), (51) - (57) one can redce the fll system of hydrodynamic eqations describing the nsteady flow arond a solid body in threedimensional flow to a system of eqations on the srface. These eqations can greatly simplify the nmerical simlations and inestigate the deeper featres of the process. First, they redce the dimensionality of the problem by one as there is no need to sole the hydrodynamic eqations in the bondary layer. Second, in addition to the gas elocity at the srface, they jst allow determining how all the other parameters, characterizing the flow at the border sch as, P, ω and so on, change. It shold be noted that not all the reslting eqations of oer-determined systems of eqations in the blk are time-dependent time. Therefore, to determine the reslting stress distribtion, in addition to the initial data is also reqired to consider bondary conditions. Throgh them the information abot the eternal flow, of corse, affects the eoltion of the impact of flow on the body. 5. Viscos incompressible flid in two-dimensional flow + A + B, (58) + C + D, (59) where B D + CB AD, (6) C A 1

11 + A + + ω, B, C + ω, D α ω ν ω ω y and ω. ω We hae an oer-determined system of differential eqations of first order (58) - (59) of the 1st nknown ω [3]. 6. Naier-Stokes eqations II Consider the Naier-Stokes eqation in the form: ω + ( ) ω ( ω ) ν ω, (61) ω, (6) di. (63) These eqations can be transformed to ω + (( + α) ) ω, (64) where the ector α is determined by the system of linear eqations with respect to it ( α ) ω ( ω ) ν ω. (65) Consider the change of ariables dr (, t) (, t) dt r + α r, r r ( r, t) и r r ( r, ), r t r. (66) t Eqations (64) take the form in these ariables dω dt (67) or 11

12 ω ω r, (68) where ω ( r ) - the initial distribtion of the orticity ω. The change of ariables (66) means that r + + α r. (69) We see that (58) is a conseqence of (61) - (63) and (69). Therefore, one can make the following one independent eqation oer-determined system of 1 partial differential eqations (68), (69), (6), (63), and 9 nknown, ω, r. This system of eqations with the integral (68) is particlarly sefl for simplifying the flow problem (redction per nit of dimension). 7. Oerriding a complete system of hydrodynamic eqations II Consider the hydrodynamic eqations (4) - (5) as ρ + di ( ρ) (7) 1 P σ F [ ω ] + +, (71) ρ ρ ρ ω P σ F + ( ) ω ( ω ) + +, (7) ρ ρ ρ ω, (73) ψ di, (74) s ρt + ( ) s Q di + Φ q, (75) T( ρ s), P P( ρ, s) T,, q κ T, (76) where 1

13 1 4 σ µ + ( ) µ + ψ 3 ω 3 (77) µ i k l Φ + µ. (78) k i 3 l Eqs. (7) can be transformed to ω + (( + α) ) ω, (79) where the ector α is determined by the system of linear eqations with respect to it P σ F α ω ω + +. (8) ρ ρ ρ Consider the change of ariables dr (, t) (, t) dt r + α r, r r ( r, t) и r r ( r, ), r t r. (81) t Eqations (79) take the form in these ariables dω dt (8) or ω ω r, (83) where ω ( r ) - the initial distribtion of the orticity ω. The change of ariables (81) means that r + + α r. (84) Conseqently, we hae an oer-determined system of 18 differential eqations of (7) - (76), (83) and (84) of 17 nknown ρ,, ω, P, T, s, ψ, r, q []. 13

14 The following problem is of interest. How to transform the ariables and nknowns in the system (7) - (76), (83), (84) that the condition (3) is alid. Then one does not need to specify the bondary conditions to the corresponding system of eqations with a lower per nit dimension. 8. Compressible non-niformly heated liqid in the two-dimensional stream Consider the eqations of hydrodynamics in two dimensions as follows: 1 + P + y +, (85) ρ y y y 1 P + + y +, ρ (86) ρ y + + ρ y + +, (87) s + s s + y, s s ( ρ, P). (88) Transform (85), (86) to the form 1 + P + y +, (89) ρ ω y 1 P P ω ω ρ ρ + + y + ω + +, (9) ρ y. (91) y ω From (87) and (9) one has ω ρ 1 P P ω ρ ω ρ ρ ρ + + y +. (9) ω ρ ω ρ ω ρ ωρ From (9) it follows y α, (93) where 14

15 ω ρ 1 ρ P ρ P + ω ρ ωρ α, ω ρ ω ρ ω ρ ω ρ. (94) ω ρ ω ρ After sbstitting (93) in (91) and (87) we obtain ω, (95) 1 ρ 1 ρ 1 ρ 1 ρ α. (96) ρ ρ ρ ρ Epressing, y from eqations (95) и (96), we find + A + B, (97) where + C + D, (98) 1 ρ 1 ρ + + ρ ρ A 1 ρ 1 ρ + + ω + α ρ ρ,, B 1 ρ 1 ρ + ρ ρ C 1 ρ 1 ρ + ω + α ρ ρ. (99), D Differentiating (97) and (98) with respect to and y one finds A B + AC + AD, (1) C D + AC + CB. (11) From eqations (1) and (11) one follows 15

16 B D + CB AD. (1) C A Epressions (97) and (98) if there sbstitte (1), obtained from the (85) - (87) rigorosly. Bt this does not mean that (97) and (98) are conseqences of each other. Ths, we hae an oer-determined system of eqations (88), (89), (93), (97), (98) for ω, ρ, P, where there is possible to redce the dimension by one That is we can redce the system of eqations (88), (89), (93), (97), (98) in the plane to the system on any cre. In particlar, this allows s to write the system of eqations describing the eoltion of shock waes in two dimensions. 9. Naier-Stokes eqations III Consider the eqations of hydrodynamics in three dimensions as follows: P + + y + z +, (13) z y y y y P + + y + z + y, (14) z z z z z P + + y + z + z, (15) z z y z + +. (16) z Transform (13) and (14) to the form ω ω ω ω z z z z z z y z z y z + + y + ωz + + z + ωz, (17) ω y z y. (18) From (16) and (17) it follows ωz ωz ωz z ω z z y z + + y ωz + z + ωz. (19) z z z z 16

17 From (19) one finds, y α, (11) where α ωz z ωz z y z ωz ωz + z + z z z z ωz, ωz ωz (111) After sbstitting (11) in (16) and (18) we obtain ω, (11) α z + +. z (113) Epressing, y from (11) and (113), we find + A + B, (114) + C + D, (115) where + A + + ω z z, B,, C + ω + z z D. (116) Differentiating (114) and (115) with respect to and y accordingly, we hae A B + AC + AD, (117) 17

18 C D + AC + CB. (118) From eqations (117) and (118) one follows: B D + CB AD. (119) C A Similar to the preios section, we hae an oer-determined system of 6 eqations (13), (15), (16) (11), (114), (115) for the 5-nknown ω z, z,, y, P, where there is possible to redce the dimension by one References 1. R. Corant, D. Hilbert. Methods of Mathematical Physics, John Wiley & Sons, 8.. V.B. Akkerman and M.L. Zaytse, Dimension Redction in Flid Dynamics Eqations, Comptational Mathematics and Mathematical Physics, Vol. 8 No. 51, pp (11) 3. M.L. Zaytse, V.B. Akkerman, Free Srface and Flow Problem for a Viscos Liqid, JETP, 113, No 4, pp (11) 18