# 10.2.3: The Connection Between the Stream Function and the Potential Function

- Page ID
- 779

For this discussion, the situation of two dimensional incompressible is assumed. It was shown that

\[
\label{if:eq:UxpotentianlSteam}

\pmb{U}_x = \dfrac{\partial \phi}{\partial x}

= \dfrac{\partial \psi}{\partial y}

\]

and

\[
\label{if:eq:UypotentianlSteam}

\pmb{U}_y = \dfrac{\partial \phi}{\partial y}

= - \dfrac{\partial \psi}{\partial x}

\]

These equations (70) and (71) are referred to Definition of the potential function is based on the gradient operator as \(\pmb{U} = \boldsymbol{\nabla}\phi\) thus derivative in arbitrary direction can be written as

\[
\label{if:eq:arbitraryPotential}

\dfrac{d\phi}{ds} = \boldsymbol{\nabla}\phi \boldsymbol{\cdot} \widehat{s} =

\pmb{U} \boldsymbol{\cdot} \widehat{s}

\]

\[
\label{if:eq:velocitystreamPotential}

{U} = \dfrac{d\phi}{ds}

\]
If the derivative of the stream function is chosen in the direction of the flow then as in was shown in equation (58). It summarized as

\[
\label{if:eq:streamFpotentialFDerivative}

\dfrac{d\phi}{ds} = \dfrac{d\psi}{dn}

\]

Example 10.3

Solution 10.3

\[
\label{streamTOpotential:derivativeY}

\dfrac{\partial \phi}{\partial x} =

\dfrac{\partial \psi}{\partial y} = - 2\, y

\]

\[
\label{streamTOpotential:integralX}

\phi = - 2\,x\,y + f(y)

\]
where \(f(y)\) is arbitrary function of \(y\). Utilizing the other relationship (??) leads

\[
\label{streamTOpotential:eq:derivativeX}

\dfrac{\partial \phi}{\partial y} = - 2\, x + \dfrac{d\,f(y) }{dy} =

- \dfrac{\partial \psi}{\partial x} = - 4\,x^3

\]
Therefore

\[
\label{streamTOpotential:eq:potentialODE}

\dfrac{d\,f(y) }{dy} = 2\,x - 4\, x^3

\]
After the integration the function \(\phi\) is

\[
\label{streamTOpotential:phiIntegration}

\phi = \left( 2\,x - 4\, x^3 \right)\, y + c

\]
The results are shown in Figure

## 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.