Differential Analysis in Fluid Flow (continued)

Example 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

Use: v_r = (1/r) ∂ψ/∂θ and v_r = ˗ ∂ψ/∂r

For case a: ∂ψ/∂θ = 0 and ˗ ∂ψ/∂r = Ar Integrate:

From ˗ ∂ψ/∂r = Ar , ψ(r,θ) = ˗ (1/2) Ar² + f(θ)

So ∂ψ/∂θ = df/dθ = 0 and f(θ) = C

The result is: ψ(r,θ) = ˗ (1/2) Ar² + C

For case b: ∂ψ/∂θ = 0 and ˗ ∂ψ/∂r = A/r Integrate:

From ˗ ∂ψ/∂r = A/r , ψ(r,θ) = ˗ A ln r + f(θ)

And again ∂ψ/∂θ = df/dθ = 0 and f(θ) = C

The result is: ψ(r,θ) = ˗ A ln r + C

Example 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.

Use: u = ∂ψ/∂y and v = ˗ ∂ψ/∂x and integrate.

From ∂ψ/∂y = ˗ 4 ψ(x,y) = ˗ 4y + f(x) and ˗ ∂ψ/∂x = ˗ 2

df/dx = 2 So f(x) = ˗2x + C

The result is: ψ(x,y) = = ˗ 4y + 2x + C

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