2.25 Fluid Mechanics Professors G.H. McKinley and A.E. Hosoi Stream Functions for planar ﬂow (satisfy \ · iv = 0) Planar ﬂow: Cartesian (x, y,/z ) vx = ∂ψ ∂y vy = − ∂ψ ∂x vz = 0 Planar ﬂow: Cylindrical (r, θ,/z) / vr = 1r ∂ψ ∂θ vθ = − ∂ψ ∂r vz = 0 Axisymmetric ﬂow: Cylindrical (r,/θ, z) vr = − 1r ∂ψ ∂z vθ = 0 vz = 1r ∂ψ ∂r Axisymmetric ﬂow: Spherical (r, θ,// ϕ) ∂ψ 1 vr = r2 sin θ ∂θ 1 ∂ψ vθ = − r sin θ ∂r vϕ = 0 Potential Functions (iv = \φ, requires \ × iv = 0, \2 φ = 0) Cartesian coordinates (x, y, z) vx = ∂φ ∂x vy = ∂φ ∂y vz = ∂φ ∂z Cylindrical coordinates (r, θ, z) vr = ∂φ ∂r vθ = 1 ∂φ r ∂θ vz = ∂φ ∂z Spherical coordinates (r, θ, ϕ) vr = ∂φ ∂r vθ = 1 ∂φ r ∂θ vϕ = ∂φ 1 r sin θ ∂ϕ uniform stream W (z) = (U − iV )z V y U shown for U, V > 0 x φ = Ux + V y vx = U ψ = −V x + U y vy = V source (Q>0) or sink (Q<0) z0 r! θ! r θ shown for Q > 0 W (z) = Q 2π ln(z − z0 ) φ= Q 2π ln r' ψ= Q ' 2π θ free vortex z0 r! θ! W (z) = φ= r θ shown for Γ > 0 −iΓ 2π vr = Q 1 2π r' vθ = 0 ln(z − z0 ) Γ ' 2π θ vr = 0 Γ ψ = − 2π ln r' vθ = Γ 1 2π r' forced vortex z0 r! θ! φ=� r θ W (z) = � shown for K > 0 ψ = − Kr2 vr = 0 '2 vθ = Kr' doublet (x-orientation) r! θ! z0 W (z) = φ= r θ shown for c > 0 doublet (y-orientation) ! 0 c cos θ' r' θ ψ = − c sin r' W (z) = r! θ z c z−z0 θ vr = − c cos r '2 ' θ vθ = − c sin r'2 ' ' ic z−z0 φ= c sin θ' r' θ vr = − c sin r '2 ψ= c cos θ' r' vθ = ' r θ shown for c > 0 c cos θ' r'2 sphere (axisymmetric flow) r! θ! z0 U φ = U cos θ' r' + r θ ϕ shown for U > 0 vr = U cos θ' 1 − W (z) = φ + iψ R3 2r '2 ψ = 12 U sin2 θ' r'2 − R3 r'3 vθ = −U sin θ' 1 + R3 r' R3 2r '3 vϕ = 0 shear flow A vx = 2Ay ' W (z) = @ � z0 y x shown for A > 0 φ=@ � vy = 0 ψ = Ay '2 vz = 0 stagnation point flow W (z) = 12 A(z − z0 )2 z0 shown for A>0 y x vx = Ax' φ = 12 A(x'2 − y '2 ) vy = −Ay ' ψ = Ax' y ' vz = 0 Notes: z = x + iy z0 = x0 + iy0 0 ≤ θ < 2π = W (z) = φ + iψ r' = (x − x0 )2 + (y − y0 )2 θ' = tan−1 y−y0 x−x0 vx = vr cos θ − vθ sin θ vr = vx cos θ + vy sin θ vy = vr sin θ + vθ cos θ vθ = −vx sin θ + vy cos θ 1 2 dW dz = vx − ivy dW dz = (vr − ivθ )e−iθ