Velocity potential

For an irrotational flow;

$$ curl(V) = 0$$ or $$ \nabla \times V = 0$$

It follows from vector algebra that there should be a potential such that

$$ V = \nabla \phi$$

$ \phi$ is called the Velocity potential. The velocity components are related to $\phi$ through the following relations.

$$ u = \frac{\partial \phi}{\partial x} , v = \frac{\partial \phi}{\partial y},  w = \frac{\partial \phi}{\partial z}$$

 

Velocity potential is a powerful tool in analyzing irrotational flows.

The velocity potential in the irrotationality condition is:

$$ \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y } = \frac{\partial}{\partial x}\frac{\partial \phi}{\partial y} - \frac{\partial}{\partial y}\frac{\partial \phi}{\partial x} = 0$$

 

Does the velocity potential satisfy the continuity equation?

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x}\frac{\partial \phi}{\partial x} - \frac{\partial}{\partial y}\frac{\partial \phi}{\partial y} = \frac{\partial^{2} \phi}{\partial x^{2}} + \frac{\partial ^{2} \phi}{\partial y^{2}} $$

 

In vector notation;

$$ \nabla ^{2} \phi = 0 $$

This equation is called the Laplace equation. A flow governed by this equation is called a Potential flow.

 

As with stream functions, lines along which potential $\phi$ is constant, these are called equipotential lines of the flow.