One-dimensional flow with friction: Difference between revisions

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</math>
</math>


===== The ideal gas law =====
==== The ideal gas law ====
\noindent For a perfect gas the ideal gas law reads\\


\begin{equation}
For a perfect gas the ideal gas law reads
 
<math display="block">
p=\rho R T
p=\rho R T
\label{eqn:gaslaw:a}
</math>
\end{equation}\\


\noindent Differentiating (\ref{eqn:gaslaw:a}) gives:\\
Differentiating (\ref{eqn:gaslaw:a}) gives:


\begin{equation}
<math display="block">
dp=\rho R dT+RT d\rho
dp=\rho R dT+RT d\rho
\label{eqn:gaslaw:b}
</math>
\end{equation}\\


\noindent If $p\neq0.$, we can divide (\ref{eqn:gaslaw:b}) by $p$ which gives\\
If <math>p\neq0.</math>, we can divide (\ref{eqn:gaslaw:b}) by <math>p</math> which gives


\begin{equation}
<math display="block">
\frac{dp}{p}=\frac{dT}{T}+\frac{d\rho}{\rho}
\frac{dp}{p}=\frac{dT}{T}+\frac{d\rho}{\rho}
\label{eqn:gaslaw:c}
</math>
\end{equation}\\


\noindent which can be rearranged according to\\
which can be rearranged according to


\begin{equation}
<math display="block">
\left[\frac{dp}{p}-\frac{d\rho}{\rho}\right]=\frac{dT}{T}
\left[\frac{dp}{p}-\frac{d\rho}{\rho}\right]=\frac{dT}{T}
\label{eqn:gaslaw:d}
</math>
\end{equation}\\


\noindent Now, inserting $dT/T$ from equation (\ref{eqn:ttot}) gives\\
Now, inserting <math>dT/T</math> from equation (\ref{eqn:ttot}) gives


\begin{equation}
<math display="block">
\left[\frac{dp}{p}-\frac{d\rho}{\rho}\right]+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\left[\frac{dp}{p}-\frac{d\rho}{\rho}\right]+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\label{eqn:gaslaw:b}
</math>
\end{equation}\\


\noindent The $d\rho/\rho$-term can be replaced using equation (\ref{eqn:cont})\\
The <math>d\rho/\rho</math>-term can be replaced using equation (\ref{eqn:cont})


\begin{equation}
<math display="block">
\frac{dp}{p}+\frac{d(u^2)}{2u^2}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\frac{dp}{p}+\frac{d(u^2)}{2u^2}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\label{eqn:gaslaw:c}
</math>
\end{equation}\\


\noindent Collect terms and rewrite gives\\
Collect terms and rewrite gives


\begin{equation}
<math display="block">
\frac{dp}{p}+\left[\frac{1+(\gamma-1)M^2}{2}\right]\frac{d(u^2)}{u^2}=0.
\frac{dp}{p}+\left[\frac{1+(\gamma-1)M^2}{2}\right]\frac{d(u^2)}{u^2}=0.
\label{eqn:gaslaw}
</math>
\end{equation}\\


==== Momentum equation ====
==== Momentum equation ====
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