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\Blatt{9}{WS 2009/2010, 17.12.2009}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[25.]
{\bf Schwingende Platte}
\par\smallskip
Eine sehr große ebene Platte bewege sich mit Amplitude $a$ in ihrer
eigenen Ebene harmonisch mit Kreisfrequenz $\omega$ in einem
inkompressiblen Newton'schen Fluid (kinematische Viskosität $\nu$) hin
und her.  Welche Strömungsgeschwindigkeit $\vek{v}(x,y,z,t)$ bildet
sich im Fluid aus? Die Platte befinde sich in der $x,y$-Ebene und
bewege sich parallel zur $x$-Achse. Vernachlässigen Sie die
Schwerkraft.


\item[26.]
{\bf Navier-Stokes-Gleichungen in Zylinderkoordinaten}
\par\smallskip
Schreiben Sie die Navier-Stokes-Gleichungen für eine inkompressible
Flüssigkeit
\begin{eqnarray*}
  \div \vek{v} & = & 0 \\
  \partial_t \vek{v} + ( \vek{v} \circ \nabla ) \vek{v} & = & \vek{f} -
  \frac{1}{\varrho} \nabla p + \nu \Delta \vek{v}
\end{eqnarray*}
in Zylinderkoordinaten $\vek{v}(r,\varphi,h)=(v^r,v^\varphi,v^h)$,
$\vek{f}(r,\varphi,h)=(f^r,f^\varphi,f^h)$, $\varrho(r,\varphi,h)$,
$p(r,\varphi,h)$ (vgl. Aufgabe 27).
\\ Hinweis: Verwenden Sie zur Berechnung des Vektor-Laplace-Operators
$\Delta \vek{v}$ in Zylinderkoordinaten ein Computeralgebra-Programm
oder geben Sie nur einen Ansatz zur Berechnung an.

Ergebnis:

\begin{eqnarray*}
  & & \frac{1}{r} v^r + \partial_r v^r + \partial_\varphi v^\varphi +
  \partial_h v^h = 0 \\
  & & \partial_t
  \left(
  \begin{array}{c}
    v^r \\ v^\varphi \\ v^h
  \end{array}
  \right)
  +
  (v^r \partial_r + v^\varphi \partial_\varphi + v^h \partial_h )
  \left(
  \begin{array}{c}
    v^r \\ v^\varphi \\ v^h
  \end{array}
  \right)
  +
  \left(
  \begin{array}{c}
    -r (v^\varphi)^2 \\ \frac{2}{r} v^r v^\varphi \\ 0
  \end{array}
  \right)
  = \\
  & & =
  \left(
  \begin{array}{c}
    f^r \\ f^\varphi \\ f^h
  \end{array}
  \right)
  + 
  \frac{1}{\varrho} \left(
  \begin{array}{c}
    \partial_r p \\ \frac{1}{r^2} \partial_\varphi p \\ \partial_h p
  \end{array}
  \right)
  +
  \nu \left(
  \begin{array}{c}
    \frac{1}{r} \partial_r v^r + \partial_r^2 v^r - \frac{1}{r^2} v^r +
    \frac{1}{r^2} \partial_\varphi^2 v^r - \frac{2}{r}
    \partial_\varphi v^\varphi + \partial_h^2 v^r \\
    \frac{3}{r} \partial_r v^\varphi + \partial_r^2 v^\varphi +
    \frac{2}{r^3} \partial_\varphi v^r + \frac{1}{r^2}
      \partial_\varphi^2 v^\varphi + \partial_h^2 v^\varphi \\
      \frac{1}{r} \partial_r v^h + \partial_r^2 v^h + \partial_h^2 v^h
      + \frac{1}{r^2} \partial_\varphi^2 v^h
  \end{array}
  \right)
\end{eqnarray*}


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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
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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