II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier Stokes 1 balance law t(ρu i) + X j j(ρu iu j) = F (ex) i ip + X j jτ ij τ ij = µ ( iu j + ju i) ρ ( t + u ) u = F (ex) p + µ u ( 1) 2 u u u t Du Dt 3 Navier Stokes ρ Du Dt = ρg p + µ 2 u div u = ( 2) ( 3) 3 g = (g sin θ,, g cos θ) ( 2)( 3) 4 L U Reynolds df dx = lim f(x + x) f(x) x x f(x + x) f(x) = df x ( x ) ( 4) dx
II 29 7 2 ( 4) x r x r f(r + r) f(r) = (grad f) r ( 5) r grad f f (gradient) ( 5) f(r + r) f(r) = f f f x + y + x y z z f ( 8) u grad u 2 Laplace Laplacian 2 f = 2 f = div grad f ( 11) ( 7) f = ( x 2 + y 2 + z) 2 f ( 12) x f grad f = y f z f x / x ( 6) = e x x + e y y + e z z grad f = f ( 6) ( 7) ( 8) ( 11) u = div(grad u) ( 13) 2 grad u Laplacian 5 (pq) p q u u divergence rotation : div u = u ( 9) : rot u = u ( 1) lim u = rot rot u + grad div u = ( u) + ( u) ( 14) div u = 1 r = x y = r cos θ r sin θ ( 15) z z 2 A B r A x z
II 29 7 3 OA = r ( 15) B (r + r) cos(θ + θ) OB = (r + r) sin(θ + θ) = r + r z + z r r = r r r r + θ + r θ z z = cos θ r sin θ sin θ r + r cos θ θ + r ( 16) 1 ( 16) e r = cos θ sin θ, e θ = sin θ cos θ, e z = ( 17) 1 r = e r r + re θ θ + e z z ( 16 ) f f(b) f(a) = (grad f) r ( 5 ) grad f ( 5 ) ( 16 ) f(b) f(a) = f(r + r, θ + θ, z + z) f(r, θ, z) = f f f r + θ + r θ z z e r grad f = r f re θ grad f = θ f e z grad f = z f ( 19a) ( 19b) ( 19c) ( 19) grad f grad f = e r r f + e θ r θf + e z z f. ( 2) 6 r ( 17) 3 {e r, e θ, e z} r 9 = e r r + e θ r θ + e z z ( 21) 7 {e r, e θ, e z} e r e θ = e z, e θ e r = e z, e θ e z = e r, e z e θ = e r, e z e r = e θ, e r e z = e θ {e x, e y, e z } r e r e θ r θ e r = e θ, θ e θ = e r. ( 18) ( 2) ( 19) ( 2) f = ( r) f f f + ( θ) + ( z) r θ z ( 15) r = p x 2 + y 2, θ = tan 1 y x r 2 3 2 x r = 4 y 5 p x/ p 3 2 3 x 2 + y 2 x 2 + y 2 = 4y/ p cos θ x 2 + y 2 5 = 4sin θ5 z ( 2) 8 ( 18) u u = u r e r + u θ e θ + u z e z ( 22)
II 29 7 4 u r = u r (r, θ, z) u θ = u θ (r, θ, z) u z = u z (r, θ, z) ( 21) div u ( div u = e r r + e ) θ r θ + e z z u = e r r (u r e r + u θ e θ + u z e z ) + e θ r θ (u r e r + u θ e θ + u z e z ) + e z z (u r e r + u θ e θ + u z e z ) ( 23) rot u ( rot u = e r r + e ) θ r θ + e z z (ue z ) = e r r (ue z ) + e θ r θ(ue z ) + e z z (ue z ) = e r ( r u) e z + + = u r e θ 1 u = v e θ, v = v(r) ( 27) θ e r e θ θ ( 18) θ (u r e r + u θ e θ + u z e z ) = ( θ u r ) e r + u r θ e r + ( θ u θ ) e θ + u θ θ e θ + ( θ u z ) e z = ( θ u r u θ ) e r + ( θ u θ + u r ) e θ + ( θ u z ) e z {e r, e θ, e z } e θ r θ (u r e r + u θ e θ + u z e z ) = θu θ + u r r ( 24) ( 23) div u = r u r + u r r + 1 r θu θ + z u z ( 25) e r e θ θ ( 25) u r /r ( 23) rot u = u u u = ue z, u = u(r) ( 26) θ u θ = rot u = ( rv + v/r) e z Laplacian ( 11) ( 14) ( 3) div u = ( 25) r u r + u r r + 1 r θu θ + z u z = ( 28) balance law balance law ( 28) balance law ρ = const. 6 = r e r = r + r +e r = θ e θ = θ + θ +e θ = z e z = z + z +e z
II 29 7 5 ( ) ( ) S[( )] S[( )] S[( )] = r θ z S[( )] = (r + r) θ z 12 r θ z u = ue r, u = u(r, t); ρ = ρ(r, t) t(rρ) + r(rρu) = +(u r t) S[( )] = +r u r r θ z t (u r t) S[( )] = (r + r) u r r+ r θ z t [ ] ( ) ( ) = ρ ( ) r u r r (r + r) u r r+ r θ z t = ρ (r + r) u r r+ r r u r r r ρ (ru r) r V t r V = (r + 12 ) r r θ z V t r + 1 2 r ( 29) 2 (ρ V ) t+ t = (ρ V ) t [ 1 + ρ r r (ru r ) + 1 ] r θu θ + z u z V t (ρ V ) t+ t = (ρ V ) t ( 28) 11 ( 28) Navier Stokes Euler Navier Stokes u 2 46 13 Navier Stokes ( 1) z? : u = (u, v, w) ( 1) u v zw = Euler Navier Stokes
II 29 7 6 x w = 14 Navier Stokes ( 2) x? 15 θ ( 2) u = (u,, ) z =, z = H Navier Stokes 2 16 x = ±L/2 +y u = (, v, ) y = y min y = y max p atm v x ODE U 17 y = y = H x = x min p H x = x max p L (< p H ) u = (u,, ) y y = const. u : : u = ( 3) U : u = U ( 31) : y u = ( 32) u = (,, w) y = const. x = const. 18 z = z = H x U 2 (x min, H) (x max, H) p atm x y Reynolds Reynolds p.161
II 29 7 7 Hagen Poiseuille Hagen Poiseuille z ( 15) = e r r + e θ r θ + e z z ( 21 ) z u = ue z ( 33) ( 21) ( 33) div u = u = z u z u = u z ( 34) ( 35) t θ u r : u = u(r). Navier Stokes ( 1) [ ( 1) ] = ρ (u )u = ρ u z ue z = = ( rot u = e r r + e ) θ r θ + e z z (ue z ) = e r r (ue z ) = ( r u) e θ ( 36) u = rot rot u + grad div u ( = e r r + e ) θ r θ + e z z ( ) ( r u)e θ + = e r r (( r u) e θ ) + e θ r θ (( r u) e θ ) = e r ( 2 r u) e θ + e θ r ( ( ru)e r ) = ( 2 r u + r 1 r u ) e z ( 37) p [ ( 1) ] = p + µ u = (e r r p + e z z p) + µ ( 2 r u + r 1 r u ) e z r : = r p ( 38) z : = z p + µ ( 2 r u + r 1 r u ) ( 39) u = u(r) ( 39) r d/dr 19 Navier Stokes ( 33) ( 38)( 39) ( 37) u 2 Laplacian ( 33) e z ( 37) ( 33) ( 38) p r ( 39) µ ( 2 r u + r 1 r u ) = z p ( 39 ) z r ( 39 ) p z 1 L p H, p L z p = p H p L L ( 39) d 2 u du 2 + r 1 dr dr = p H p L µl ( 4) 2 ODE u r a
II 29 7 8 2 ( 4) u r=a = ( 41) u = A log r + B p H p L 4µL r2 ( 42) ( 41) u A, B u 46.1 Q = u ds = a u 2πrdr ( 4) Q Q = p H p L L πa4 8µ (46.8) Hagen Poiseuille (46.8) 21 (46.8) Navier Stokes ( div u = u = e r r + e ) θ r θ + e z z (ve θ ) = e r r (ve θ ) + e θ r θ(ve θ ) = e r ( r v)e θ + e θ r ( ve r) = ( 44) ( 43) ( 14) u u = ( r 2 v + r 1 r v v ) r 2 e θ ( 45) [Navier Stokes ] = p + µ u = e r r p + µ ( r 2 v + r 1 r v v ) r 2 e θ u = (v/r) θ [Navier Stokes ] = ρ ( t + u )u = ρ t (ve θ ) + ρv r θ(ve θ ) = ρ( t v)e θ ρv2 r e r r : ρv2 = r p ( 46) r θ : ρ t v = µ ( r 2 v + r 1 r v v ) r 2 ( 47) r d/dr ( 15) e θ u = ve θ, v = v(r, t); p = p(r, t) ( 43) 22 ( 45) 23 Navier Stokes ( 46)( 47) ( 37) ( 45) Laplacian v/r 2 ( 43) e θ θ
II 29 7 9 u = ve θ, v = Γ 2πr (19.9) 9 27 (19.9) z r = (19.9) Euler Navier Stokes δ δ νt (46.13) δ 24 t δ 2 /ν ν t δ ν a t b (46.13) a = b = 1/2 ( 48) (19.9) Navier Stokes v = Γ 2πr f(s), s = r νt ( 49) ( 47) s r, t, ν v = κ/r κ = Γ/(2π) Γ 7/17 v t = Γ s df 2πr t ds = Γ 4π df νt 3 ds ( ) d Γ f + Γ s df dr 2πr 2πr r ds = Γ 2πr 2 f + Γ 2πr df νt ds ( ) v r v r = 2 v r 2 = r = Γ πr 3 f Γ πr 2 νt df ds + Γ d 2 f 2πrνt ds 2 ( 47) d 2 ( f 1 ds 2 + 2 s 1 ) df s ds = ( 5) 2 ODE ( 5) df/ds = g f s= = f t= =, f s= = f t= = 1 ( 5) v = f = 1 e s2 /4 Γ [ )] 1 exp ( r2 2πr 4νt ( 51) ( 51) ( 49) s PDE ODE ( 51) v r 2 νt r/ νt = 2.2418 (19.9) r < 2 νt 2 νt (46.13)
II 29 7 1 25 (x, z) y = < y < + t = +x U u = U (1 erfη), η = y 2 νt (46.11) : Fourier (46.11) ω z = u y = U ) exp ( y2 πνt 4νt ( 52) ( 52) y (46.11) ( 52) (46.13) Burgers ( 51) r z ( 43) u = ve θ + αze z βre r ( 53) Navier Stokes α β v = v(r), p = p(r, z) div u = 2β + α β = 1 2 α 2 4 (46.13) 4 δ νt Navier Stokes z u ( 45) r d/dr u u ( α 2 = 4 r v2 r ) e r α ( r dv ) 2 dr + v e θ + α 2 ze z ( ) α 2 r : ρ 4 r v2 = r p ( 54) r z : ρα 2 z = z p ( 55) θ : αρ ( r dv ) 2 dr + v ( d 2 v dv = µ 2 + r 1 dr dr v ) r 2 ( 56) ( 56) (19.9) v = Γ 2πr f(r) ( 57) ( 49) f s r v = Γ [ ( 1 exp α 2πr 4ν r2)] ( 58) ( 51) ( 51) ( 53)( 58) Burgers 26 Navier Stokes ( 53) ( 56) Burgers ( 58) 46.3
II 29 7 11 (46.13) t x x/u x δ νx U ( 59) 2 4 L ( 59) δ max νl ν U = L UL Reynolds Re = UL ν ( 6) δ max L Re ( 59 ) ( 59 ) Reynolds L 27 U = 1.5 m/s L = 1 cm Reynolds ( 59 ) ( 59) Reynolds 1 1 ( 59) 48.2 ( 59)