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In elliptic regions, using again the incompressibility
and defining
![\begin{displaymath}
\theta = \sqrt{\det ({\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$u$}}) }
\end{displaymath}](img724.png) |
(A.17) |
the matrix
can be decomposed in a diagonal part
and a pure rotational part:
![\begin{displaymath}
{\mbox{\boldmath$A$}} = {\mbox{\boldmath$N$}}({\mbox{\boldmath$D$}} + {\mbox{\boldmath$R$}}){\mbox{\boldmath$N$}}^{-1}
\end{displaymath}](img725.png) |
(A.18) |
where
![\begin{displaymath}
{\mbox{\boldmath$D$}} + {\mbox{\boldmath$R$}} =
\left(
\be...
...u} & 0 \\
2 \theta & 0 & -{2 \over \tau}
\end{array}\right)
\end{displaymath}](img726.png) |
(A.19) |
The eigenvectors matrix
is
![\begin{displaymath}
{\mbox{\boldmath$N$}} =
\left(
\begin{array}{ccc}
\partial...
...artial_x u_y (\partial_y u_y - \theta) \\
\end{array}\right)
\end{displaymath}](img727.png) |
(A.20) |
and the exponential matrix reads
![\begin{displaymath}
e^{{\mbox{\boldmath$A$}}(t)dt} =
e^{-{2 \over \tau} dt}
{\m...
...os(2 \theta dt)
\end{array}\right)
{\mbox{\boldmath$N$}}^{-1}
\end{displaymath}](img728.png) |
(A.21) |
Stefano Musacchio
2004-01-09