u = 0 u = ϕ t + Π) = 0 t + Π = C(t) C(t) C(t) = K K C(t) ϕ = ϕ 1 + C(t) dt Kt 2 ϕ = 0

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2 u = (u, v, w) ω ω = u = 0 ϕ u u = ϕ u = 0 ϕ 2 ϕ = 0 u t = u ω 1 ρ Π + ν 2 u Π = p + (1/2)ρ u 2 + ρgz ω = 0 ( ϕ t + Π) = 0 ϕ t + Π = C(t) C(t) C(t) = K K C(t) ϕ = ϕ 1 + C(t) dt Kt

3 C(t) ϕ ϕ 1 ϕ = ϕ 1 p ρ + ϕ t ϕ 2 + gz = K B ( ϕ v) n B = 0 ϕ n B = v n n B v ϕ n = ϕ/ n ϕ F M 0 B F = p n A B M 0 = x p n A B

4 x = (x, y.z) F = B ρ ϕ t n A + 1 ρ ϕ 2 n A + ρ g z n A 2 B B M 0 = B ρ ϕ t x n A + 1 ρ ϕ 2 x n A + ρ g z x n A 2 B B (x, y) (u, v) ψ(x, y) u = ψ y v = ψ x ψ = C C ψ = C y ψ/ x = x ψ/ y = v u

5 ω = v x u 2 y = ψ x 2 2 ψ y 2 ϕ x = ψ y ϕ y = ψ x ϕ, ψ ϕ + iψ i = 1 z = x + iy w(z) w(z) = ϕ(x, y) + iψ(x, y) w z = ϕ x + ψ x = u iv ϕ = C y ϕ/ x = x ϕ/ y = u v U, V x y w(z) = (U iv )z ϕ = Ux + V y ψ = Uy V x Uy V x = (U, V Ux + V y = (U, V

6 ζ = f(z) f(z) z z = x + iy ζ = ξ + iη ζ z z = f 1 (ζ) f (z) f(z) f (z) 0 z ζ (x, y) (ϕ, ψ) ϕ = ψ = w/ z = 0 B (x, y) w(z) ζ = f(z) B B (ξ, η) B W (ζ) B (x, y) w(z) = W (f(z)) B B Im[W (ζ)] B = C Im[A] A (x, y) Im[w(z)] B = Im[W (ζ)] B = C x y ζ = z 2

7 z = x ζ = x 2 ξ z = iy ζ = x 2 ξ ζ W = kζ k z w = kz 2 ϕ = k(x 2 y 2 ) ψ = 2kxy xy = C x 2 y 2 = C z (x, y) w = m 2π z m ln z z = r (iθ) ϕ = m 2π r

8 y x 45 o ψ = m 2π θ u r, u θ u r = ϕ r = m 2πr u θ = 1 r ϕ θ = 0 m > 0 m < 0 m

9 w = i Γ 2π z Γ ϕ = Γ 2π θ ψ = Γ 2π r u r, u θ u r = ϕ r = 0 u θ = 1 r ϕ θ = Γ 2πr x w = µ z µ

10 ϕ = µ r θ ψ = µ r θ ψ = C x 2 + y 2 µ C y = 0 x 2 + (y µ 2C ) 2 = µ 2 4C 2 (0, µ/(2c)) µ/(2c) y x ϕ = C (x µ 2C ) 2 + y 2 = µ 2 4C 2 (µ/(2c), 0) µ/(2c) x y u r = ϕ r = µ r 2 θ u θ = 1 r ϕ θ = µ r 2 θ x y

11 y x x x µ µ

12 w = i µ z x x y µ x α w = i µ α + iµ α z α B w(z) F = (F x, F y ) M 0 F x, F y F x if y = iρ 2 C ( w z ) 2 z M 0 M 0 = ρ 2 [ C z( w z ) 2 z] [A] A

13 n = (n x, n y ) B s F = pn s B y M 0 = B p(xn y yn x ) s x F M 0 B B s F x, F y F x = pn x s F y = pn y s (n x, n y s C B s ( x/ s, y/ s) n = y/ s, x/ s F x = p y F y = p x F x i F y = p y ip x = ip( x i y) = ip z

14 z = x + i y A A B F x if y = i p z C p ρ(u 2 + v 2 ) = k k p = k 1 2 ρ w z 2 F x if y = i 2 ρ C w z 2 z C k z = ( k z) = 0 C B B ψ = ψ = 0 B w = ϕ + i ψ = ϕ = w F x if y = i 2 ρ C w z ( w z ) z = i 2 ρ w C z w = = iρ 2 C w z w = iρ 2 C ( w z ) 2 z B C B F x, F y M 0 = F x y + F y x

15 B M 0 = B z p(x x + y y) = [ z p z ] B M 0 = [ z w B z 2 z ] [ z k z ] = k (x x + y y) = k ( 1 B B B 2 (x 2 + y 2 )) = 0 B x, y U, V w/ z z w z = A 0 + A 1 z + A 2 z = n=0 A n z n A n n = 0, 1, 2,... A 0 A 0 = U α iu α = Ue iα U α x A 1 1/z A 1 = i Γ 2π Γ B

16 A n n = 2, 3,... C 1 z a z = 2iπ n = 1 0 n 1 a C a F x if y = iρ 2 C (A A 0A 1 z + A A 0 A 2 z ) z = = 2πρA 0 A 1 = iρue iα Γ F x = ρuγ α F y = ρuγ α ρuγ A n n = 2, 3,... Γ 0

17 M 0 = ρ 2 [ C z(a A 0A 1 z + A A 0 A 2 z ) z] = = ρ 2 [2iπ(A A 0 A 2 )] = 2πρ [i(u iv )A 2 ] Γ = 0 R w z = U i Γ 2πz + A 2 z x un x + vn y = [(u iv)(n x + in y )] = 0 n x, n y n x + in y = (iθ) z = R (iθ) A n n = 2, 3,... A n = A n (iβ n ) U θ + A n R n ((n 1)θ β n ) = 0 n=2 θ nθ β 2 = π UR n n = 2 A n = 0 n > 2 UR 2 Γ

18 w z = U(1 R 2 z 2 ) i Γ 2πz z w = U(z + R 2 z ) i Γ 2π z z = r (iθ) ϕ ψ ϕ = U(r + R 2 r ) θ + Γ 2π θ ψ = U(r R 2 r ) θ Γ 2π r z 2 2iκz R 2 = 0 κ = Γ/(4πU) z 1,2 = iκ ± κ 2 + R 2 Γ < 4πUR κ < R R y Γ > 4πUR κ > R R R y Γ = 4πUR κ = R z = ir F x = 0

19 y b a x α U a b F y = ρuγ M 0 = 0 U a b x y x α x ζ = ξ + iη z = x + iy z = ζ + λ 2 ζ λ ζ ζ = 1 2 (z ± z 2 4λ 2 )

20 z = ±2λ z 2λ = r 1 e iθ1 0 < θ 1 < 2π z + 2λ = r 1 e iθ2 0 < θ 2 < 2π ζ = 1 2 (z + r 1 r 2 (i θ 1 + θ 2 )) 2 2λ < ξ < λ ζ R ζ = Re iθ 0 < θ < 2π z x = (R + λ 2 ) θ 0 < θ < 2π R y = (R λ 2 ) θ 0 < θ < 2π R R+λ 2 /R R λ 2 /R a, b R, λ R = 1 (a + b) 2 λ = 1 2 a 2 b 2 λ a, b a = b λ z = ζ

21 ζ = ξ +iη α ξ α ξ 1, η 1 ξ 1 W = U(ζ 1 + R 2 ζ 1 ) i Γ 2π ζ 1 ξ 1 = ξ α + η α η 1 = ξ α η α ζ 1 = ζe iα ζ = ζ 1 e iα ζ W = U(ζe iα + R 2 ζ eiα ) i Γ 2π ζ W ζ = U(e iα R 2 ζ 2 eiα ) i Γ 2πζ z z ζ w z = W 1 ζ z/ ζ = (U(e iα R 2 ζ 2 eiα ) i Γ 2πζ ) 1 1 λ 2 /ζ 2 F x = ρuγ α

22 F y = ρuγ α A 2 A 2 z C A 2 = 1 z w 2iπ C z z C ζ C ζ A 2 = 1 (ζ + 2iπ Cζ λ 2 ζ ) W ζ ζ = = 1 (ζ + 2iπ Cζ λ 2 ζ )(U(e iα R 2 ζ 2 eiα ) i Γ 2πζ ) ζ A 2 = U(λ 2 e iα R 2 e iα ) R λ A 2 A 2 = 1 4 U((a 2 b 2 )e iα (a + b) 2 e iα ) M 0 = 2πρ [iue iα 1 4 U((a 2 b 2 )e iα (a + b) 2 e iα )] = = 1 2 πρu 2 (a 2 b 2 ) (2α) a b

23 c a = c 2 b = 0 R λ R = λ = c 4 R λ Γ ζ = R (iθ) u iv = (U(e iα e i(α 2θ) ) i Γe iθ 2πR ) 1 1 e 2iθ = = (U(e i(θ α) e i(θ α) Γ ) i 2πR ) 1 e iθ e iθ = = (U (θ α) Γ 4πR ) 1 θ v u θ = 0 u Γ = 4πUR α = πuc α Γ u (θ α) + α u = U θ θ 0 u

24 u θ=0 = U α x u = U (θ/2 α) (θ/2) = U α + U α θ 2 u θ = π F x = πρ c U 2 2 α F y = π ρ c U 2 α α L L = F 2 x + F 2 y = π ρ c U 2 α C L C L = L 1 2 ρu = 2π α 2 c M 0 = 1 8 πρu 2 c 2 (2α) C M C M = M ρc 2 U (2α) = 1 π (2α) 2 4

25 α sinα α α C L C M C L = 2πα C M = π 2 α x = c/4 b = 0 b = 0 b b b 0 p p + p = p p + = 1 2 ρ (u 2 + u 2 ) u + u θ 2π θ p = ρ U 2 2α θ 2 0 < θ < π

26 θ = π θ 2 x, y p ρ U 2 2α 2 x c

27 z ψ P r O y θ P x P r P O θ OP x P P (x, y) ψ OP z x = (x, y, z) r θ ψ r = x 2 + y 2 + z 2 x = r ψ θ y = r ψ θ z = r ψ u u r u θ u ψ u r OP u θ

28 (x, y) OP u ψ OP z OP ϕ u r = ϕ r u θ = 1 ϕ r ψ θ u ψ = 1 r ϕ ψ 1 r 2 2 ϕ (r r r ) ϕ r 2 2 ψ θ r 2 ψ ϕ ( ψ ψ ψ ) = 0 r 1 2 ϕ r 2 (r r r ) = 0 ϕ = K r K u θ u ψ u r u r = K r 2 K > 0 K < 0 m m = 4πK ϕ = m 4πr

29 z z ϕ = µ r 2 ψ u r = 2µ r 3 ψ u θ = 0 u ψ = µ r 3 ψ x y z U R u r ϕ r r=r = 0 ϕ = U(r + 1 R 3 2 r 2 ) ψ z

30 u r = U(1 R 3 r 3 ) ψ u θ = 0 u ψ = U(1 + 1 R 3 2 r 3 ) ψ r = R u ψ (3/2)U ψ 1.5 U 2 U

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