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\Blatt{8}{WS 2004/2005, 4.1.2005}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[24.]
{\bf Differentiation von Linien-, Oberflächen-, und Volumenintegralen
  über mitbewegte Integrationsbereiche}
\begin{enumerate}
  Seien $\vek{u}(\vek{x},t)$ ein Vektorfeld, $f(\vek{x},t)$ ein
  skalares Feld, $C$ eine Kurve, $W$ ein Volumen und $C_t$ bzw. $W_t$
  deren zeitliche Entwicklung unter der Abbildung $\varphi_t(\vek{x})$
  mit $\frac{\mathrm{d}}{\mathrm{d}t} \varphi_t(\vek{x}) =
  \vek{v}(\vek{x},t)$.
\item Zeigen Sie, dass zur Zeit $t=0$ gilt
  \begin{equation}
    \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{C_t} \vek{u} \cdot \mathrm{d}\vek{s} = 
    \int\limits_C D_t \vek{u} \cdot \mathrm{d}\vek{s} + 
    \int\limits_C \vek{u} \cdot \partial \vek{v} \, \mathrm{d}\vek{s} \,.
  \end{equation}
\item
  Begründen Sie die symbolische Schreibweise
  \begin{equation}
    D_t \mathrm{d}\vek{s} = \partial \vek{v} \, \mathrm{d}\vek{s} \,.
  \end{equation}
  Was bedeutet $D_t \mathrm{d}\vek{s}$ anschaulich?
\item Wiederholen Sie aus der Vorlesung, dass zur Zeit $t=0$ gilt
  \begin{equation}
    \frac{\mathrm{d}}{\mathrm{d}t} \int\limits_{W_t} f(\vek{x},t) \, \mathrm{d}V = 
    \int\limits_W \left( D_t f + f \div \vek{v} \right) \,\mathrm{d}V \,.
\end{equation}
\item Begründen Sie die symbolische Schreibweise
  \begin{equation}
    D_t \, \mathrm{d}V = \div \vek{v} \, \mathrm{d}V \,.
  \end{equation}
  Was bedeutet $D_t \, \mathrm{d}V$ anschaulich?
\item Zeigen Sie, dass für die materielle Ableitung des
  Oberflächenelements gilt
  \begin{equation}
    D_t \, \mathrm{d}\vek{A} = \div \vek{v} \, \mathrm{d}\vek{A} - (\partial \vek{v})^T
    \mathrm{d}\vek{A} \,.
  \end{equation}
  Hinweis: Beachten Sie, dass $D_t \, \mathrm{d}V = D_t ( (\vek{s}_1
  \times \vek{s}_2) \cdot \mathrm{d}\vek{s} ) = D_t (
  \mathrm{d}\vek{A} \cdot \mathrm{d}\vek{s} )$ und bearbeiten Sie die
  rechte Seite mit den obigen Formeln.
\end{enumerate}


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\item[26.]
{\bf Flachwasserströmung}
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\begin{center}
  \includegraphics[width=0.4\textwidth]{Flachwasserstroemung.eps}
\end{center}
\begin{enumerate}
\item Berechnen Sie die Geschwindigkeits- und Druckverteilung der
  stationären Strömung, die sich längs einer schiefen Ebene aufgrund
  der Schwerkraft ausbildet. Die Höhe der Strömung senkrecht zur
  Platte sei $H$, der Neigungswinkel $\alpha$.
\item Welche Kraft wirkt die Wand auf das Fluid aus?
\item Der Volumendurchsatz pro Breite der Strömung sei $\dot{V}/b$.
  Berechnen Sie hieraus die Höhe $H$ der Strömung.
\end{enumerate}






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\footnotesize
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Besprechung der "Ubungen am Dienstag, den 11.1.2005 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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