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\begin{document}

\Blatt{11}{WS 2009/2010, 13.1.2010}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[29.]{\bf Dreieckiges Rohr}
\par\smallskip
\begin{center}
  \includegraphics[width=0.5\textwidth]{Dreieckiges_Rohr.eps}
\end{center}
Die Abbildung zeigt einen Schnitt durch ein dreieckiges Rohr, gefüllt
mit einem inkompressiblen Newton'schen Fluid. Die Rohrwände sind
parallel zur $z$-Achse. Die Länge des Rohres sei $H$. Der Druck in der
gezeigten Schnittebene ($x$-$y$-Ebene an der Stelle $z=0$) sei null,
der Druck am Ende des Rohres an der Stelle $z=H$ sei $\Delta p$.  Zwei
Wände bilden den Winkel $\alpha$. Die Koordinaten $(q_1,q_2,q_3)$,
$q_3$ entlang der $z$-Achse, bilden ein schiefwinkliges
Koordinatensystem, wie eingezeichnet.
\begin{enumerate}
\item Wie lauten die Kontinuitätsgleichung und die
  Navier-Stokes-Gleichung in den $q$-Ko\-ordi\-na\-ten?
\item Wir betrachten im Folgenden den Spezialfall, dass der
  Querschnitt des Rohres ein gleichseitiges Dreieck bildet mit
  Kantenlänge $L$ ($\alpha=60^\circ$). Wie lauten die Randbedingungen
  in $q$-Koordinaten?
\item Geben Sie eine stationäre Lösung der Navier-Stokes- und
  Kontinuitätsgleichungen (ohne Gravitation) in $q$-Koordinaten an,
  welche die Randbedingungen erfüllt.
\end{enumerate}


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\item[30.]{\bf Bewegte Kugel}
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In einer reibungsfreien, inkompressiblen Flüssigkeit bewegt sich eine
Kugel (Radius $a$) mit Geschwindigkeit $\vek{w}$. Der Mittelpunkt der
Kugel sei gerade am Ursprung. Prüfen Sie nach, dass sich im stationären
Fall in der Flüssigkeit ein Geschwindigkeitsfeld
\begin{equation}
  \vek{v}(\vek{r}) = - \grad \Phi(\vek{r}) \,,
\end{equation}
ausbildet, wobei 
\begin{equation}
  \Phi(\vek{r}) = \frac{a^3}{2 |\vek{r}|^3} \vek{w} \cdot \vek{r} \,.
\end{equation}
Hinweise: 
\begin{itemize}
\item Berücksichtigen Sie dabei auch, dass eine Lösung für den
Druck existieren muss.
\item Es ist 
\begin{equation}
  \nabla ( \vek{a} \cdot \vek{b} ) =
  \vek{a} \times (\nabla \times \vek{b}) + \vek{b} \times ( \nabla
  \times \vek{a} ) + (\vek{a} \cdot \nabla) \vek{b} + (\vek{b} \cdot
  \nabla) \vek{a} \,.
\end{equation}
\end{itemize}




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\footnotesize
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Besprechung der Übungen Montags 12-14 PH127, Mittwochs 8.30-10 HS2 und
Donnerstags 10-12 HS3. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.


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