10.2.3.1: Existences of Stream Functions
- Page ID
- 780
The potential function in order to exist has to have demised vorticity. For two dimensional flow the vorticity, mathematically, is demised when
\[
\label{if:eq:zeroVortisity}
\dfrac{\partial U_x}{\partial y} -
\dfrac{\partial U_x}{\partial x} = 0
\]
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
\[
\label{canItBePotential:check}
\dfrac{\partial^2\psi}{\partial y^2} +
\dfrac{\partial^2\psi}{\partial x^2} \overset{?}{=} 0
\]
\[
\label{canItBePotential:theCheck}
0 \overset{?}{=} 0 + 60\,x^3
\]
Since \(x^3\) is only zero at \(x=0\) the requirement is fulfilled and therefore this function cannot be appropriate stream function.
Contributors and Attributions
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.