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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 j (ρv i v j )=F (ex) i i p + j j τ ij τ ij = µ ( i v j + j v i ) ρ ( t + v ) v = F (ex) p + µ v (#1) v Dv t Dt 3 L U Reynolds Navier Stokes df dx = lim f(x + x) f(x) x x (#) f(x + x) f(x) = df x ( x ) (#3) dx 3 (#3) x r x r f(r + r) f(r) = (grad f) r (#4) r r grad f A

II 14 B r = OA, r = AB f(b) f(a) = AB grad f (#4 ) grad f f (gradient) (#4) f(r + r) f(r) = f f f x + y + x y z z grad f = xf y f (#5) z f (#5) grad f = f, = e x x + e y y + e z z (#6) f(b) f(a) = AB f (#7) p/ x = x p p x (#6) e x x x e x 4 (pq) p q : x y p v v divergence rotation : divv = v (#8) : rot v = v (#9) x / x x x x balance law V v nds =(divv) V ( V ) (#1) (#1) (#4 ) (#3) (#4 ) AB (#1) 3 rot v3 v =(u, v, w) ω = rot v =( y w z v, z u x w, x v y u) (#11) 3 3 3 (#11) w z ω =(,,ω)=(,, x v y u) (#1) (#3) Γ= v dr (#13) C C xy S v dr =( x v y u) S Γ=ω S C 3 (16.3) f (#7) v grad v Laplace Laplacian f = f = div grad f (#14) (#6) f = ( x + y + z) f (#15)

II 14 3 (#14) v = div(grad v) grad v Laplacian v = rot rot v + grad div v = ( v)+ ( v) div v = 1 v = grad φ =( x φ, y φ) (#16) 6 (6.) 4.4 I (#1) I Bernoulli Lagrange II Reynolds II (6.) II II 4.5 I Reynolds 4.6 Reynolds Reynolds Reynolds p.44 (#11) (#1) 1 1 (#16) µ (#)

II 14 4 1 1 Lagrange t λ νt (46.13 ) λ t λ /ν (#17) A B A B 1 cos θ θ 5 ν t λ ν a t b (46.13 ) a = b =1/ 46.3 x (46.13 ) t x x/u x δ B νx δ B U (#18) L (#18) max δ B νl ν U = L UL Reynolds Re = UL ν max δ B (#19) L Re (#18 ) (#18 ) Reynolds L 6 U =1.5m/s L =1cm Reynolds (#18 ) (#18) Reynolds 1 1 (#18) 48. 1.3 (#18)

II 14 5 Navier Stokes Euler Navier Stokes v I (#16) φ = xφ + yφ = (#) Navier Stokes µ u = µ ( x + y) ( x φ = µ x x + y) φ = µ v = µ ( x + y) ( y φ = µ y x + y) φ = Navier Stokes p p p = p ρ { t φ + 1 ( xφ) + 1 } ( yφ) (#1) (#1) Bernoulli 7 (#16)(#1) Navier Stokes (#16) (#16) Navier Stokes Navier Stokes p Navier Stokes ρ ( tu + u xu + v yu) = xp + µ ( x + y) u (#a) ρ ( tv + u xv + v yv) = yp + µ ( x + y) v (#b) p Navier Stokes p 1 (#b) x (#a) y x (u x v)=( x u) x v + u xu (#1) x u + y v = (#3) t ω + u x ω + v y ω = ν ( x + y) ω (#4) Dω/Dt 3 3 (x, y) z 3 µ = Euler I (6.) ρgz (#1) (#1) Bernoulli (#4)

II 14 6 46 8 Navier Stokes (#1) z? : v =(u, v, w) (#1) u v zw = 9 θ x y g =(g sin θ, g cos θ) Navier Stokes ρ Dv Dt = ρg p + µ v (#5) div v = (#6)? Navier Stokes Fourier ODE U v =(u,, ) y y = const. u : : u = U : u = U : y u = v =(,,w) y = const. x = const. x w = u = u(x) x u 1 y = y = H x = x min p H x = x max p L (<p H ) 11 z = z = H +x U (> ) x

II 14 7 1 (#5) v =(u, ) y =, y = H yu y=h = Reynolds Reynolds p.161 Hagen Poiseuille Hagen Poiseuille a x v =(u,, ) x Navier Stokes Navier Stokes µ ( y + z) u = x p (#7) x u = (#8) ( y p, z p)=(, ) (#9) (#7) x y z (#7) p x 1 dp/dx p L p H, p L dp dx = p H p L L (#7) ( y + z) u = p H p L µl (46.6) u = u y +z =a = (#3) u = u(r), r = y + z (#31) ( y + z) u (y, z) r u y u = r du y dr = y du r dr { } yu = y y u (r) r { = u (r) u } (r) + y y r r = u (r) + y r { d u } (r) r y dr r { = u (r) + y u } (r) u (r) r r r r zu ( y + z) d u du u = + r 1 dr dr (#3) (46.6) d u du + r 1 dr dr = p H p L µl (#33)

II 14 8 ODE u r 13 u r=a = (#3 ) (#3) 14 (#33) u = A log r + B p H p L 4µL r (#34) (#3) u A, B u 46.1 Q = u ds = a u πrdr (#33) Q Q = p H p L L πa4 8µ Hagen Poiseuille 15 (46.8) (46.8) (46.8) y = <y<+ t = +x U x Navier Stokes Navier Stokes t u = ν yu (46.1) u t= = (#35) u y= = U (#36) (46.1) 16 Navier Stokes (46.1) Fourier (46.1) PDE Fourier u(y, t) F (k, t) t F (k, t)+νk F (k, t) = (#37) ODE (#36) (#36) u = u(y, t) =U {1 f(y, t)} (#38)

II 14 9 u f (#36) f y= = (#36 ) Fourier <y<+ + (#36 ) F = F (k, t) = π f = f(y, t) = π f(y, t)sinky dy (#39) F (k, t)sinky dk (#4) (#38) (46.1) (#35) t f = ν yf (46.1 ) f t= =1 (#35 ) (#4) (46.1 ) π { t F (k, t)+νk F (k, t) } sin ky dk = (#37) (#37) F (k, t) =F e νkt, F = F (k, ) (#41) ODE F F k t F (#35 ) (#39) f t= F t= F (k, ) = π sin ky dy = (#41) π 1 k (#4) (#4) f 7/18 f = π u = U e νkt sin ky k { 1 erf ( y νt 46. 17 ( ) y dk =erf νt )} (46.1) 18 (46.11) 1 (46.11) Fourier [ ] [ ] x X = 1 [ ][ ] 1 x (#43) y Y 5 1 y (#39) (#4) [ ] [ ] X x = 1 [ ][ ] 1 X (#44) Y y 5 1 Y Fourier (#43) 1/ 5 ±1 (#44) (#39)(#4) /π 1 /π /π F (k, t) = /π k e νk t Stanley Farlow 1.1 (#4) (46.1)

II 14 1 u = u(s), s = y λ(t) (#45) (t, y) s u t u = s du t ds = λ (t) {λ(t)} yu (s) = λ (t) λ(t) su (s) y u = s du y ds = u (s) λ(t) yu = y { u (s) λ(t) } = = u (s) {λ(t)} (46.1) λ(t)λ (t) ν = u (s) su (s) (#46) (#45) s, t, y 3 (46.1) 3 (#46) t s y t y s α λ(t)λ (t) =αν (#47) u (s) = αsu (s) (#48) (#47) (#35) λ t= = λ = ανt (#49) (#48) exp ODE ϕ(s) =u (s) 1 A, B s ( u = ϕ ds = A exp α s) d s + B (#5) (#35)(#36) u(s) u() = U, u(+ ) = A, B α A = U, B = U (#51) π (#49) α/ s = η, α/ s = η u α U =1 π =1 η π s ( exp α s) d s e η d η =1 erf η α α α (#47)(#48) α = λ = νt (#49 ) (46.11) 19 Navier Stokes (46.11) Fourier (46.13 )

II 14 11 Blasius 46.3 Navier Stokes (#) y Navier Stokes : y x µ yu µ xu Navier Stokes (#) t u + u x u + v y u = 1 ρ xp + ν yu (48.9) (48.9) u x u + v y u = ν yu (48.11) (#3) y + u U (48.11) Fourier y/δ B δ B λ u = u(s), s = y λ(x) (#5) (x, y) s u x u = s du x ds = = (x) λ λ(x) su (s) (#53) y u = s du y ds = u (s) λ(x) (#54) yu = y ( y u)= = u (s) {λ(x)} (#55) v (#53) (#3) v = λ (x)v (s) (#56) (#56) y y v = λ (x) s dv y ds = λ (x) λ(x) V (s) (#3) λ(x) s V (s) =su (s) s V (s) = su (s)ds = su(s) u(s)ds (#57) (#56) v u(s) λ(x) (48.11) λ (x) λ(x) u (s) u(s)ds = ν {λ(x)} u (s) u(s) f(s) = U ds, u(s) =Uf (s) (#58) x λ(x) s f(s) U ν λ(x)λ (x) = f (s) f(s)f (s) (#59) U U/ 1 f U (48.15)(48.16) U/

II 14 1 y s x s x y (#46) 1 λ 1/ (48.11) λ dλ dx = ν U (#6) d3 f ds 3 + f d f = ds (#61) ODE (#6) 7 6 5 4 s 3 1.7 λ = νx U (#18 ) 1 (#18) (#61) 1 Blasius 1 u / U 1 (#61) Navier Stokes (#6)(#61) µ yu µ xu xu U x, yu U νx = Ux ν U x Ux/ν 1 yu x Reynolds UL/ν x = (#18) u y = y <y<y y u dy Uy y u dy =(y δ D ) U (y + ) δ D y = δ D (#61) δ D δ D νx =1.7 δ D =1.7 νx/u U 1 (#18)

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