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1 15 II (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2 ( 4 ) ( ( by, bx, ) x 2 + y 2 a) u = (u, v, w) = ( ) ba2 y x 2 + y 2, + ba2 x ( x 2 + y 2, ) (5.1) x 2 + y 2 > a a, b (5.1) Rankine [1, p.18] (1) Γ = Γ[C] C Γ[C] = u dr C ( B ) Rankine (5.1) Γ (a) (x, y)- R ( R, Γ ) (b) (,, ) (a,, ) (, a,) (,, ) (c) (,, ) (a,,) (a, a, ) (,a, ) (,,) (2) (i) Rankine (5.1) ω 19 (ii) (x, y) (x, z) (3) Stokes 19 D

2 ( ) 16 4 Rankine (5.1) (5.1) (Rankine ) r ω = 6 (1)(2) (1) (2) (1) du dy u dy = d2 u dy 2 ( u dy = Y ) u(ỹ ) dỹ (6.1) (6.1) Navier-Stokes 7.14 Y u (Y = ) u = ( ) (Y + ) u U (a) Navier-Stokes (6.1)

3 ( ) 17 (b) (i) (6.1) u = U (ii) (c) (6.1) u 2 u (i) (6.1) (ii) (d) u udy U dy = U Y (i) (6.1) (ii) (iii) (e) (6.1) ( u = df/dy, f() = Fortran Mathematica NDSolve ) (2) (3.9) 1 c t = D 2 c x 2. (6.2) (a) (6.2) c = c + θ ( c = const.) < x < l, θ x= = θ x=l = (6.2) 2 (b) (c) θ t= = F(x) =.8sin πx l +.4sin 2πx l.2 sin 3πx l ( < x < l) (GNUPLOT ) (d) Fortran C (6.2) (c) ( ) 7.12( ) 1 GNUPLOT Mathematica ( 8 )

4 ( ) (Euler Navier-Stokes ) ρ (1) Euler div u = (7.1) ( ) u ρ t + u u = grad p + f (7.2) f = (,, ρg) Euler ω = rotu Euler (7.2) { u [7.2 ] = ρ t + grad ( u 2 2 ) } + ω u (7.3) Euler (7.2) p ω t (7.3) + u ω = ω u (7.4) (2) Euler (7.2) Navier-Stokes div u = (7.5) ( ) u ρ t + u u = grad p + f + η u (7.6) ( 7.7 ) Navier-Stokes p ω t + u ω = ω u + ν ω (7.7) η (3) (7.4) (7.7) u = gradφ (7.8) ω = ( ) (7.8) (7.4) ( ) Φ (4) Φ

5 ( ) (ω = ) Φ (7.8) ( ) 21 (1) (x, z) u = (u,, w) (i) Φ = F(x)G(z) 2 (ii) λ (2) z = h, z = Φ 22 (3) λ h ( λ h ( ) [6, 7-3]?? 7.3 λ h u z x t ( z u x u z u = ) w (u ) ρ 2 u 2 < ρgh 23 (1) z (z = h = const.) (z = η) 24 h = h + η y = ±b/2 (i) x 1 < x < x 2 M M = ρb x2 (ii) x 2 x 1 = x x 1 dx (7.9a) M x + O( x 2 ) (7.9b) (iii) M x 25 h t + (hu) =. (7.1) x 21 ( ) ρ u 2 /2 ρgh 24 η 25 (7.1) 1 (3.5)

6 ( ) 2 (2) (i) Euler (7.2) z p p = p + ρg(η z) = p + ρg(h h z) (7.11) (ii) Euler x (7.1), (7.11) h u t (hu) + x = (7.12) (3) (7.1) (7.12) 26 h h = h + ǫ η, u = ǫṽ ; ǫ 1 (7.13) (7.1)(7.12) (7.13) ǫ η = η(ξ), v = v(ξ), ξ = x ct ( η t= = cos kx) ( x, t η = η(x,t) c ( ) h t= = H(x) = h A tanh x λ (A > ) (7.14) (1) 7.3 (7.1)(7.12) A [ ] [ ] [ ] t h Q + A x h Q = (7.15) (2) A pa = ap p a ] p = p ± = [α ± β ± 26 ( 1.4)

7 ( ) 21 r + t h = α + t + β Q + t, r t h = α t + β Q t (7.16) r ± (3) (7.15) (7.16) [ ] [ ] + (u + c) r + = + (u c) r = ; u = u(r +,r ), c = c(r +,r ). (7.17) t x t x (4) (7.14) r t= = h = h(x, t) r = r = (7.17) h,x, t x h x = x(h,t) ( ) x = x(h, t) = [ t t t ] (7.18) h t x = x(h,t) h(x,) = H(x) t = h/ x ( ) [12, 1] c (1) 7.3 (7.1)(7.12) (2) c (u =, h = h 1 ) c u = u 2, h = h 2 > h 1 (h 1, h 2 ) u 2 = u 2 (h 1,h 2 c = c(h 1,h 2 ) c (Galilei ) h 1, h 2, u 2, c c u 2 (3) (h 2 h 1 )/h 1 + c c 1 (4) h 2 /h 1, c/c 1 u 2 /c 1? 27

8 ( ) (1) 2 t p a 2 2 xp = (7.19) 1 Euler p = p(ρ) (7.2) p = p + ǫp 1, ρ = ρ + ǫρ 1 + ǫ 2 ρ 2 +, u = ǫu 1 + ǫ 2 u 2 + ǫ ǫ O(ǫ 2 ) 1 (2) A r ± = u 1 ± Ap 1 r ± ( t ± a x ) r ± = (7.21) (3) u 1 =, p 1 = exp ( x2 λ 2 ) (λ ) (i) r ± ( x = x/λ, t = at/λ r ± = r ± ( x, t) ) (ii) p 1 (4) a (7.2) 2? T = const. ( ) S = const. ( ) (E.3) p = p(ρ, T) T E (E.2) T p = p(ρ, S) x u, y u, etc.( ) [13, 1-9] [14, p.454] Navier-Stokes 2 ( ) 3 Navier-Stokes (7.6) 28

9 ( ) 23.6 (u,v) (u,v) (ur,vr) (ud,vd) = u = u + u R + u D (1) 1 (2) u ( 2 ) u u R u D 5 r = (x, y ) r = r + (X,Y ) (X,Y ) u Taylor 1 [ ] [ ] [ ] [ ] u u = = u + u R + u D ; u R = 1 ωy ax + by X, u D = = S (7.22) v 2 +ωx bx + cy Y ω, a, b, c u S (2) 2 (Couette ) 29 y =, y = H U = (U,) [ ] [ ] u γy u = =, γ = U = const. (7.23) v H 1 (7.23) u u R u D 5 (3) S [ ] u D = X 1 s 1 p 1 + X X 2 s 2 p 2, = Y X 1 p 1 + X 2 p 2 (7.24) 29

10 ( ) 24 u D X 1 s 1 p 1 X 2 s 2 p 2 (4) 3 Navier-Stokes (7.6) (3.19) 3 τ ij ( τ ij = 2ηS ij 2S ij = u i + u ) j x j x i (7.25) 3 η (3.14) 7.8 (7.7) (3.9) (3.12) 1 (3.12) t T = κ 2 xt. (7.26) 1 (6.2) < x < + t = (T = T = const.) t = T w (= T + T 1 ) T w (x = ) T w x 1 (7.26) (1) (2) (7.26) T = T + T 1 f(s), s = const. κ α x β t γ s α,β,γ / x / t d/ds ( F ) 31 s ( ) f ( G ) T t x (3) t = t (> ) (GNUPLOT ) t = t 1 = 4t t = t 3 ( [15, p.58] ) η µ 31 β = 1

11 ( ) 25 (4) T > T w T 1 /2 x = x H x H t (GNUPLOT ) 7.9 Poiseuille Navier-Stokes (7.6) (7.7) (Poiseuille ) y = y = b (> ) 2 x z (1) (2) x u = ω x ω = (7.7) (3) (7.27) 2 yω = (7.27) u = (7.28) ( 32 ) u (X, Y ) = (x/b, y/b) u = U U = max U (4) u = U ω = rotu (5) t = x = t = b (6) ( (i) u = U + (ũ, ṽ,), ω = rotu + (,, ω) (7.27) (ii) (ũ, ṽ) = (+ y ψ, x ψ) ψ = e st E(X) F(Y ) (Orr-Sommerfeld ) Reynolds 32 [16, p.114]

12 ( ) Poiseuille 33 (1) (2) (i) (x,y,z) Descartes (r,θ, ζ) (x,y,z) = (r cos θ, r sinθ,ζ) (7.29) r, θ, ζ x, y, z [ e x2 x φ x e x2 φ!] (ii) 34 x, y, z r, θ, ζ (3) 35 u = ue z (e z ) f Navier-Stokes (7.5) (7.6) ( ) (4) [ r ] = [z ] 2 u (r = a) (5) (i) (z, r) (ii) (x, y) (6) Q a? ( Hagen-Poiseuille 36 ) (7) a =.4 mm 15 cm (i) 3 1ml (ii)? ( ) 7.11 ( ) [7, p.289] (1) h [ ] u = u, u = (g sinβ)h2 2ν { 2z ( z ) } 2 h h (7.3) Nusselt ( ) Reynolds [6, 8-1] 36 Hagen Poiseuille [17, pp.31-36] Hagen-Poiseuille (7.3)

13 ( ) 27 β x y [ ] [ ] u + sin β u =, f = ρg cosβ Navier-Stokes (7.6) x u = Navier-Stokes p y=h = p ( p ) p (z = ) (y = h) τ xy = (7.31) τ ij = pδ ij + τ ij τ ij (7.25) u (2) h h/ x Nusselt (7.3) (i) (7.1) h t + Q x = (7.32) Q h (ii) (7.32) Q (3) h t + C h =, C = C(h) (7.33) x h t= = h Atanh x λ (7.34) ( A > ) h = h(x, t) x h x = x(h,t) (7.34) h x t= (7.33) (7.18) x = x(h, t) t = h/ x Stokes 3.6 (7.7) 3.5 2

14 ( ) 28 (1) (7.7) u ω = ν 2 ω (7.35) (2) 37 (i) 3.6 (7.35) (ii) ( ) (3) (7.35) u 2 u Stokes Stokes (7.35) ω ψ (3.26) ω (4) 38 (5) ω u (GNUPLOT Mathematica ) ( 3.6) 7.13 Oseen Stokes (Oseen ) u U x (1) (i) Oseen ( 2 x + 2 y α 2) Ω = (7.36) (ii) α Reynolds ω = Ωe +αx α (iii) αd 1 Reynolds (2) (7.36) (3) (i) ψ = y F(r) ω θ (ii) Ω ω Ω (4) Bessel K 1 (αr) e αr αr 39 F ru/ν 1 39 [7, 81]

15 ( ) 29 (5) Stokes ( [13] ) 7.14 x y u = (U,), p = p = const. (7.37) (y = ) (7.37) (1) 2 Navier-Stokes (2) (7.8) s (u, v) = (+ y ψ, x ψ), ψ = Uy Ψ(s), s = const. ν α y β( x ) γ (7.38) U s α,β,γ 4 (3) x u = Uf (s) (i) f Ψ (ii) v f (4) xu/ν 1 Navier-Stokes x (5) Navier-Stokes y y p (x ) y p = (6) (6.1) ( 3 ) 8.1 Feynmann Feynmann ( [9] 2 ) 41 ( [1] 9 1 [8, 18] ) Reynolds? 2 Couette (Taylor-Couette )? Reynolds? ( ) 4 β = 1 41 [15] 15 [1] p.169

16 ( ) Reynolds 3 Reynolds ( ) Reynolds L = [ ] = 6 cm U = [ ] = 1 m/s (ν =.15 cm 2 /s) Reynolds Re = UL ν = (6 cm) (1 m/s).15 cm 2 /s = cm m/s cm 2 /s = 4 1 = (8.1) [1].. 4., [2]., [3].., 198. [4]. One Point 3., [5] Glansdorff and Prigogine.., [6].. 2., [7]. ( ). 14., [8]., [9] Feynman, Leighton, and Sands.. IV., [1],.., [11],.. 6., [12] Landau and Lifshitz. 2., [13]. One Point 11., [14]., (http//

17 ( ) 31 [15].. 21., [16]. 2., [17]. One Point 16., [18]., [19] Feynman, Leighton, and Sands.. III., [2] Donald A. McQuarrie. Statistical Mechanics, chapter 17, Continuum Mechanics. HarperCollins, [21] E. Kreyszig.. 2., [22]., Feynman [19] ( ) [1] [6] [15] [7] 1997 [5, 2] A A α alpha B β beta Γ γ gamma δ delta E ǫ e-psilon Z ζ zeta H η eta Θ θ theta I ι iota K κ kappa Λ λ lambda M µ mu N ν nu Ξ ξ xi O o o-micron Π π pi R ρ rho Σ σ sigma T τ tau Υ υ u-psilon Φ φ phi X χ chi (khi) Ψ ψ psi Ω ω o-mega B Γ = u dr C C C r = r(s) (s a b ) s

18 ( ) 32 dr ( s) u s 44 dr nds S 2 nds C a nds ( n ) r = (x, y, z) 2 (θ, φ) r = (a sin θ cos φ, asin θ sin φ, a cosθ) (C.1) (C.1) r e r r = ae r, e r = (sin θ cos φ, sin θ sin φ, cosθ) r θ θ r φ φ (C.1) nds = r = (acos θ cos φ, a cosθ sinφ, asin θ) θ r = ( asin θ sin φ, a sinθ cosφ, ) φ ( ) ( ) r r θ dθ φ dφ = = e r a 2 sin θ dθ dφ (C.2) 45 (C.1) [21, p.182] (u,v) r = (a cos v cos u, a cosv sin u, asin v) nds ( ) ( ) r r nds = u du u dv = = e r a 2 cosv du dv (C.4) 44 nds ds 45 2 ( ) (C.3)

19 ( ) 33 D df dx = lim f(x + h) f(x) h h 46 f(x + h) f(x) = h df dx ( h ). (D.1) (D.1) x r = (x,y, z) f(r + h) f(r) = h f, h = ( x, y, z) (D.2a) f f (D.2a) ( ) f f = x, f y, f (D.2b) z (D.2) f f f 2 47 p q = [p x p y p z ] x y z [q x q y q z ] p ( q) (D.3) (D.3) p q = (p )q = (p x x + p y y + p z z )q (D.4) p p (D.2a) h 2 n S C u dr = (n S) rotu (D.5a) C rotu u 48 (D.5a) ( uz rotu = u = y u y z, u x z u z x, u y x u ) x (D.5b) y 46 lim 47 (3 ) rot curl

20 ( ) 34 (D.5) [21, 3-11] (D.5b) V S q nds = ( V ) divq S (D.6a) divq q (D.6a) divq = q = q x x + q y y + q z z (D.6b) Stokes Stokes (D.2a) C grad f dr = f(r B ) f(r A ) (r A = [C ], r B = [C ]) (D.7) C (D.5a) (rotu) nds = u dr (C = [S ]) (D.8) S C (D.6a) (divq) dv = q nds (S = [V ]) (D.9) V S (D.8) ( )Stokes (D.9) Gauß n 49 Stokes Stokes E U S U = c V dt = c V T cv R S = T dt + V dv = S + c V log T + R log V T V (E.1) (E.2) 49 [2, 38]

21 ( ) 35 [15, p.46] V = 1/ρ c V R 5 γ R = (γ 1)c V γ 1.4 (E.1)(E.2) (S,U,V ) 51 (E.1) (E.2) T U ( ) V 1 γ ( ) S S U = U exp V du = T ds p dv ( ) U p = = = ρrt (E.3) V S (E.3) c V F 1 f = f(x), x = x(t) t x f df dt = dx dt f = f(x, y), x = x(s, t), y = y(s,t) df dx. (s, t) (x, y) f? (D.1) f(x + x, y + y) = f(x, y) + f = f(x,y) + f f x + x y y x(s + s, t + t) = x(s,t) + x = x(s,t) + x x s + s t t y(s + s, t + t) = y(s, t) + y = y(s, t) + y y s + s t t (F.3) x (F.4) y (F.2) s t f(s + s, t + t) = f(s, t) + f = f(s, t) + f f s + s t t f s = x f s x + y f s y f t = x f t x + y f t y (F.1) (F.2) (F.3) (F.4) (F.5) (F.6a) (F.6b) 5 R n pv = n RT M ρ = Mn/V p = ρ RT/M (E.3) R = R/M 51

22 ( ) 36 (F.6) f = f(x,y) s = x s x + y s y t = x t x + y t y (F.7a) (F.7b) ( ) G Gauß erf x = 2 e x2 dx = 2 x e s2 ds, erfcx = 2 e s2 ds π π π x (G.1) 52 GNUPLOT Mathematica GNUPLOT plot erf(x), erfc(x) plot [2.5] [1.2] erf(x) title "Gauss error function" Ei( x) = e s x s ds (G.2) ( x > ) x Ei( x) = γ + log x x + x2 4 x (G.3) γ Euler γ

23 ( ) 37 H Helmholtz Bessel φ ± α 2 φ = (H.1) = 2 (H.1) Helmholtz 53 Bessel 2 Helmholtz (H.1) Helmholtz 1 + Helmholtz (H.1) 2 x φ + α2 φ = (H.2) (H.2) φ = A cos αx + B sinαx = C 1 exp(+iαx) + C 2 exp( iαx) exp(±iαx) = cos αx ± isin αx (H.3a) (H.3b) (H.3b) 1 (H.1) 2 xφ α 2 φ =. (H.4) (H.4) φ = exp(+αx) φ = exp( αx) φ = A exp(+αx) + B exp( αx) (H.5) Bessel 2 (H.1) λ θ d 2 Θ dθ 2 + λθ = λ = m 2 ( m ) r ( 1 r R ) ± α 2 R m2 r r r r 2 R = (H.6) (Bessel ) (H.6) + 1 (H.2) R = AJ m (αr) + BY m (αr) (H.7) 54 1 J m (αr) Bessel 2 Y m (αr) Neumann ( 2 Bessel ) Mathematica 53 (H.1) + Helmholtz ( ) [22, p.9] 54 Y m N m

24 ( ) 38 Plot[ BesselJ[,s], {s,,5} ] Do[ Plot[ {BesselJ[m,s], BesselY[m,s]}, {s,,15}, PlotRange {-3.5, 1.2} ], {m,, 5} ] Neumann 1 (H.3b) (H.6) (Hankel ) H (1) n (αr) = J n (αr) + i Y n (αr) H (2) n (αr) = J n (αr) i Y n (αr) (H.8a) (H.8b) Bessel (H.6) 55 1 (H.4) R = AI m (αr) + BK m (αr) αr + 1 I m (αr) 2 K m ( [13] A.4 ) Bessel (H.9) Bessel Bessel Bessel Bessel K m ξ > K m (ξ) = 1 2 [ exp ξ 2 ( p + 1 )] dp p p m+1 (H.1) p = e s K 1 K 1 (ξ) = ξ exp( ξ coshs) sinh 2 sds (H.11) coshs = 1 + σ/ξ (H.11) ( K 1 (ξ) = e ξ e σ σ 1 + 2ξ ) 1/2 dσ (H.12) ξ σ ξ (H.12) ξ = K 1 (ξ) = e ξ ξ K 1 (ξ) e ξ ξ e σ σdσ = e ξ ξ. [ ( ξ γ log ξ ) ] ξ γ (G.3) Euler (γ.577) 55 (H.6) Bessel (H.13) (H.14)

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Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j