# Speed potential

The **velocity** potential is introduced for eddy-free , two- and three-dimensional flows of fluid dynamics . With it, the calculations are simplified and you gain a deeper mathematical-physical understanding. The velocity potential of fluid dynamics corresponds *mathematically to* the electrostatic or gravitational potential .

This article deals with the *two-dimensional case* - the three- *dimensional case* is presented in the article Potential flow.

Solving the equation gives the equipotential lines of the flow field.

In addition, the current function is introduced , the clear meaning of which is that the solutions of the equation represent the streamlines of the velocity potential.

The **complex speed** potential is formed from the speed potential and the current function .

## Basics

For an eddy-free two-dimensional flow field , the rotation is equal to 0:

Similar to the case of the electrostatic potential, the velocity potential is now introduced . The gradient of this potential is precisely the flow field:

Because of this , the flow field is automatically free of eddies.

Furthermore, the continuity equation also applies to the velocity field in the case of an incompressible flow :

If you insert the definition of the velocity potential into it, you can see that the Laplace equation (as a special case of the Poisson equation ) satisfies:

## The current function

The speed potential was introduced in such a way that freedom from eddies is automatically fulfilled. However, the fulfillment of the continuity equation or the Laplace equation had to be explicitly required.

Now we introduce the current function , which is defined by:

From this definition one can see that the continuity equation is automatically fulfilled:

However, the freedom of rotation must be explicitly required:

The current function also fulfills the Laplace equation in eddy-free flows.

## Complex speed potential

With the definitions of speed potential and current function, we get:

This is exactly of the form of the Cauchy-Riemann differential equations for a holomorphic function , with a real part and an imaginary part . Thus one introduces the **complex**** speed** potential:
** **

The complex velocity potential thus also fulfills the Laplace equation:

## literature

- Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 3-540-00760-1 .
- M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 .