Engr Help

Combining Forms of Plane Potential Flows

In a Nutshell: Each potential function satisfies the Laplace equation (a linear equation). Thus the

sum of different potential functions also satisfies the Laplace equation and represents a form of

potential flow. The table below lists basic, plane potential functions that can be combined to form

differing flows.

Description of

Flow field

Velocity Potential

Stream Function

Velocity

Components

Uniform Flow

At an angle α

with the x-axis

φ = U(xcosα+ysinα)

ψ = U(ycosα-xsinα)

u = U cosα

v = U sinα

Source or Sink

m > 0 source

m< 0 sink

φ = (m/2π) ln r

ψ = (m/2π) θ

v_r = m/(2πr)

v_θ= 0

Free Vortex

Γ > 0 CCW motion

Γ < 0 CW motion

(Shown below)

φ = (Γ/2π) θ

ψ = - (Γ/2π) ln r

v_r = 0

v_θ= Γ/2πr

Doublet

(Shown below)

φ = (K cos θ)/r

ψ = - (K sin θ)/r

v_r = - (K cos θ)/r²

v_θ= - (K sin θ)/r²