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\Blatt{7}{WS 2004/2005, 14.12.2004}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[21.]
{\bf Wasserreservoir mit instationärer Strömung}
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\includegraphics[width=\textwidth]{Wasserreservoir.eps}
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An einem großen, oben offenen Wasserreservoir ist ein Fallrohr der
Länge $L$ mit Querschnitt $A_2$ senkrecht angebracht. Das Rohr wird
mit einem kleinen Ventil am Ende während der Schließzeit $T$ so
geschlossen, dass der Volumenstrom linear bis auf null abnimmt.
Berechnen Sie den zeitlichen Verlauf des Druckes $p_3(t)$ vor dem
Ventil und der Geschwindigkeit $c_4(t)$ der Strömung unter dem Ventil.

Hinweis: Verwenden Sie die Bernoulli'sche Gleichung für instationäre
Strömungen.
\end{minipage}


\item[22.]
{\bf Transporttheoreme}
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\begin{enumerate}
\item Zeigen Sie: Sei f eine beliebige differenzierbare Feldgröße
  (Funktion vom Ort), dann gilt zum Zeitpunkt $t=0$
  \begin{equation}
    \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{W_t} \varrho f \, \mathrm{d}V =
    \int\limits_W \varrho D_t f \, \mathrm{d}V
  \end{equation}
  und
\item 
  \begin{equation}
    \label{eq:Reynolds}
    \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{W_t} f \, \mathrm{d}V =
    \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_W f \, \mathrm{d}V +
    \int\limits_{\partial W} f \vek{v} \cdot \mathrm{d}\vek{s} \,.
  \end{equation}
  Gleichung \eqref{eq:Reynolds} heißt das Reynolds'sche
  Transporttheorem.
\item Interpretieren Sie \eqref{eq:Reynolds} anschaulich.
\item Zeigen Sie, dass, falls die Volumenkräfte $f(\vek{x},t) = \grad
  U(\vek{x})$ sind (stationäre Potentialkräfte), gilt
  \begin{equation}
    \label{eq:Energie}
    \int\limits_W \vek{v} \cdot \varrho \vek{f} \mathrm{d}V = 
    - \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{W_t} \varrho U \, \mathrm{d}V
  \end{equation}
\item Interpretieren Sie \eqref{eq:Energie} anschaulich.
\end{enumerate}


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\item[23.]
{\bf Wirbellinie}
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Zeigen Sie mit Hilfe von
\begin{equation}
  \vek{\omega}( \varphi_t(\vek{x}), t ) = 
  \partial \varphi_t(\vek{x}) \, \vek{\omega}(\vek{x},0)
\end{equation}
aus der Vorlesung, dass bei einer idealen inkompressiblen Flüssigkeit
eine Wirbelinie $C$ unter der Bewegung eine Wirbellinie bleibt, dass
also $C_t$ auch eine Wirbellinie ist.




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\footnotesize
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Besprechung der "Ubungen am Dienstag, den 21.12.2004 um 9.30 Uhr im Raum HS3
(Garching).\\
"Ubungsleitung: Moritz Franosch, mail@Franosch.org, http://www.physik.tu-muenchen.de/lehrstuehle/T35/.


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