# 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

$\label{if:eq:zeroVortisity} \dfrac{\partial U_x}{\partial y} - \dfrac{\partial U_x}{\partial x} = 0$

The stream function can satisfy this condition when

Stream Function Requirements

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

Example 10.4

Is there a potential based on the following stream function

$\label{canItBePotential:streamFun} \psi = 3\,x^5 - 2\,y$

Solution 10.4

Equation (81) dictates what are the requirements on the stream function. According to this equation the following must be zero

$\label{canItBePotential:check} \dfrac{\partial^2\psi}{\partial y^2} + \dfrac{\partial^2\psi}{\partial x^2} \overset{?}{=} 0$

In this case it is

$\label{canItBePotential:theCheck} 0 \overset{?}{=} 0 + 60\,x^3$ Since $$x^3$$ is only zero at $$x=0$$ the requirement is fulfilled and therefore this function cannot be appropriate stream function.