Stream Function

Example 1 The streamlines in a certain incompressible, two-dimensional flow field are all

concentric circles so that v_r = 0. Find the stream function for

1. v_θ = Ar and for b. v_θ = A / r

In general v_r = (1/r) ∂ψ / ∂θ and v_θ = - ∂ ψ / ∂r

For part a: - ∂ ψ / ∂r = Ar Integrate with respect to r gives ψ(r,θ) = - ½ Ar² + f(θ)

And v_r = 0 = (1/r) ∂ψ / ∂θ = df/dθ = 0 So f(θ) = C = constant

Finally, ψ(r,θ) = - ½ Ar² + C (result)

For part b: - ∂ ψ / ∂r = A / r Integrate with respect to r gives ψ(r,θ) = - ½ A ln(r) + f(θ)

And v_r = 0 = (1/r) ∂ψ / ∂θ = df/dθ = 0 So f(θ) = C = constant

Finally, ψ(r,θ) = - ½ A ln(r) + C (result)

Example 2 In a certain two-dimensional flow field, the constant velocity components are

u = - 4 ft/sec and v = - 2 ft/sec

Find the stream function.

In general u = ∂ψ / ∂y and v = - ∂ ψ / ∂x

∂ψ / ∂y = - 4 Integrate with respect to y gives ψ(x,y) = - 4y + f(x)

Now ∂ ψ / ∂x = - v So df/dx = 2 and f(x) = 2x + C

Then ψ(x,y) = - 4y + f(x) and ψ(x,y) = - 4y + 2x + C (result)

Return to Notes on Fluid Mechanics

Copyright © 2019 Richard C. Coddington
All rights reserved.