# 10.2.2: Compressible Flow Stream Function

- Page ID
- 777

The stream function can be defined also for the compressible flow substances and steady state. The continuity equation is used as the base for the derivations. The continuity equation for compressible substance is

\[
\label{if:eq:continutyRho}

\dfrac{\partial \rho\, \pmb{U}_x}{ dx} + \dfrac{\partial \rho\, \pmb{U}_y}{ dy} = 0

\]

\[
\label{if:eq:rhoStreamFunY}

\dfrac{\partial \psi }{ dy} = \dfrac{\rho\, U_x}{\rho_0}

\]
and

\[
\label{if:eq:rhoStreamFunX}

\dfrac{\partial \psi }{ dx} = -\dfrac{\rho\, U_y}{\rho_0}

\]
Where \(\rho_0\) is the density at a location or a reference density. Note that the new stream function is not identical to the previous definition and they cannot be combined. The stream function, as it was shown earlier, describes (constant) stream lines. Using the same argument in which equation (50) and equation (??) were developed leads to equation (53) and there is no difference between compressible flow and incompressible flow case. Substituting equations (60) and (61) into equation (53) yields

\[
\label{if:eq:streamUcompressible}

\left( \dfrac{\partial \psi}{\partial y} \,dy +

\dfrac{\partial \psi}{\partial x} \,dx \right)\, \dfrac{\rho_0}{\rho} =

\dfrac{\rho_0}{\rho} \, d\psi

\]
Equation suggests that the stream function should be redefined so that similar expressions to incompressible flow can be developed for the compressible flow as

\[
\label{if:eq:compressibleFlowStreamFun}

d\psi = \dfrac{\rho_0}{\rho} \, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d\ell

\]
With the new definition, the flow crossing the line \(1\) to \(2\), utilizing the new definition of (63) is

\[
\label{if:eq:mDOTcompressibleFlow}

\dot{m} = \int_1^2 \rho\, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d'\ell =

\rho_0 \int_1^2 d\psi = \rho_0 \left( \psi_2 -\psi_1 \right)

\]

## 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.