Stream Function/Potential Flow Click here to skip to Potential Flow

Key Concepts: Streamlines are lines in the flow field that are everywhere tangent to the fluid

velocities. So along each streamline the value of ψ(x,y) is a constant. Since flow never crosses streamlines, the difference in the values of ψ between adjacent streamlines equals the volume

rate of flow per unit width, q₁ and q₂ between streamlines.

In a Nut Shell: For steady, incompressible, two-dimensional fluid motion in the xy-plane

conservation of mass (also called the continuity equation) reduces to

∂u/∂x + ∂v/∂y = 0 (1)

It proves useful to define a new function, ψ(x,y) , called the stream function

u = ∂ ψ /∂y and v = - ∂ ψ /∂x

that automatically satisfies the conservation of mass (the continuity equation).

To see this result substitute u and v into eq. (1).

The fluid velocity components in polar coordinates are: (Note here ψ = ψ(r,θ)

v_r = (1/r)∂ψ /∂θ and v_θ = - ∂ ψ /∂r

which likewise automatically satisfies conservation of mass: ∂(rv_r/∂r) + ∂v_θ/∂θ = 0

See the figure below for a two dimensional flow field. Let the streamlines be

ψ = ψ₁ , ψ = ψ₂ , and ψ = ψ₃

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