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Stream Function/Potential Flow (continued)

Special Case of Fluid motion in the xy-plane (continued)

There is no flow across streamlines since the fluid velocity is tangent to each streamline. The

flow between streamlines equals the flow rate per unit width into the paper. See the figure below.

Assume ψ₁> ψ₂ . Then q₁ as shown and q₁ = ψ₁ - ψ₂

Assume ψ₂> ψ₃ . Then q₂ as shown and q₁ = ψ₂ - ψ₃

For the case where ψ₁< ψ₂ , then q₁ = ψ₂ - ψ₁ in the flow is in the opposite direction.

Two types of applications involving stream functions occur as given in the table below.

Given

Find

Strategy

u(x,y) and v(x,y)

ψ (x,y)

Integrate

∂ ψ /∂y = u(x,y) and

- ∂ ψ /∂x = v(x,y)

Subject to boundary conditions

ψ (x,y)

u(x,y) and v(x,y)

Take derivative

u(x,y) = ∂ ψ /∂y and

v(x,y) = ˗ ∂ ψ /∂x

Use a similar strategy for fluid velocity components in polar coordinates: (Note here ψ = ψ(r,θ)

v_r = (1/r)∂ψ /∂θ and v_θ = - ∂ ψ /∂r