Skip to main content
Engineering LibreTexts

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.