# 10.2.3.1: Existences of Stream Functions

The potential function in order to exist has to have demised vorticity. For two dimensional flow the vorticity, mathematically, is demised when

$\label{if:eq:zeroVortisity} \dfrac{\partial U_x}{\partial y} - \dfrac{\partial U_x}{\partial x} = 0 \tag{80}$
The stream function can satisfy this condition when

Stream Function Requirements

$\label{if:eq:streamRequirement} \dfrac{\partial}{\partial y} \left( \dfrac{\partial \psi}{\partial y} \right) + \dfrac{\partial}{\partial x} \left( \dfrac{\partial \psi}{\partial x} \right) = 0 \Longrightarrow \dfrac{\partial^2\psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial x^2} = 0 \tag{81}$

Example 10.4

Is there a potential based on the following stream function
$\label{canItBePotential:streamFun} \psi = 3\,x^5 - 2\,y \tag{82}$

Solution 10.4

Equation (81) dictates what are the requirements on the stream function. According to this equation the following must be zero
$\label{canItBePotential:check} \dfrac{\partial^2\psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial x^2} \overset{?}{=} 0 \tag{83}$
In this case it is
$\label{canItBePotential:theCheck} 0 \overset{?}{=} 0 + 60\,x^3 \tag{84}$
Since $$x^3$$ is only zero at $$x=0$$ the requirement is fulfilled and therefore this function cannot be appropriate stream function.

### Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.