Download Fluid Mechanics Formula Cheat Sheet and more Cheat Sheet Fluid Mechanics in PDF only on Docsity! 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 = 1 r ∂ψ ∂θ vθ = − ∂ψ ∂r vz = 0 Axisymmetric flow: Cylindrical (r,/θ, z) vr = −1 r ∂ψ ∂z vθ = 0 vz = 1 r ∂ψ ∂r Axisymmetric flow: Spherical (r, θ,//ϕ) vr = 1 r2 sin θ ∂ψ ∂θ vθ = − 1 r sin θ ∂ψ ∂r vϕ = 0 Potential Functions (iv = \φ, requires \ × iv = 0, \2φ = 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 r ∂φ ∂θ vz = ∂φ ∂z Spherical coordinates (r, θ, ϕ) vr = ∂φ ∂r vθ = 1 r ∂φ ∂θ vϕ = 1 r sin θ ∂φ ∂ϕ uniform stream x y U V U, V > 0shown for W (z) = (U − iV )z φ = Ux + V y ψ = −V x + Uy vx = U vy = V source (Q>0) or sink (Q<0) shown for z0 r θ r ′ θ ′ Q > 0 W (z) = Q 2π ln(z − z0) φ = Q 2π ln r ' ψ = Q 2π θ ' vr = Q 2π 1 r' vθ = 0 free vortex shown for Γ > 0 r θ r ′ θ ′ z0 W (z) = −iΓ 2π ln(z − z0) φ = Γ 2π θ ' ψ = − Γ 2π ln r ' vr = 0 vθ = Γ 2π 1 r' K > 0 forced vortex shown for r θ r ′ θ ′ z0 W (z) = � φ = � ψ = −Kr'2 2 vr = 0 vθ = Kr' 1 � � doublet (x-orientation) shown for r θ r ′ θ ′ c > 0 z0 W (z) = c z−z0 φ = c cos θ ' r ' ψ = − c sin θ ' r ' vr = − c cos θ ' r '2 vθ = − c sin θ ' r '2 doublet (y-orientation) shown for r θ r ′ θ ′ c > 0 z0 W (z) = ic z−z0 φ = c sin θ ' r ' ψ = c cos θ ' r ' vr = − c sin θ ' r '2 vθ = c cos θ ' r '2 sphere (axisymmetric flow) shown for r θ r ′ θ ′ U U > 0ϕ z0 W (z) = φ + iψ φ = U cos θ ' r ' + R 3 2r '2 ψ = 1 2 U sin 2 θ ' r '2 − R3 r ' vr = U cos θ ' 1 − R3 r '3 vθ = −U sin θ ' 1 + R 3 2r '3 vϕ = 0 shear flow x y shown for A A > 0 z0 W (z) = φ = ψ = Ay '2 vx = 2Ay ' vy = 0 vz = 0 stagnation point flow x y shown for A > 0 z0 W (z) = 1 2 A(z − z0) 2 φ = 1 2 A(x '2 − y '2) ψ = Ax ' y ' vx = Ax ' vy = −Ay ' vz = 0 Notes: z = x + iy z0 = x0 + iy0 0 ≤ θ < 2π = r ' = (x − x0)2 + (y − y0)2 1 2 θ ' = tan−1 y−y0 x−x0 W (z) = φ + iψ dW dz = vx − ivy dW dz = (vr − ivθ)e −iθ vx = vr cos θ − vθ sin θ vy = vr sin θ + vθ cos θ vr = vx cos θ + vy sin θ vθ = −vx sin θ + vy cos θ 2 @ @
