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In hyperbolic regions, where the predominant effect
of velocity gradients is the stretching,
using the incompressibility
and defining
![\begin{displaymath}
\lambda = \sqrt{- \det ({\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$u$}}) }
\end{displaymath}](img717.png) |
(A.12) |
the matrix
can be diagonalized as
![\begin{displaymath}
{\mbox{\boldmath$A$}} = {\mbox{\boldmath$N$}}{\mbox{\boldmath$D$}}{\mbox{\boldmath$N$}}^{-1}
\end{displaymath}](img718.png) |
(A.13) |
where
![\begin{displaymath}
{\mbox{\boldmath$D$}} =
\left(
\begin{array}{ccc}
-{2 \ove...
... 0 \\
0 & 0 & -{2 \over \tau} - 2 \lambda
\end{array}\right)
\end{displaymath}](img719.png) |
(A.14) |
and the eigenvectors matrix
is
![\begin{displaymath}
{\mbox{\boldmath$N$}} =
\left(
\begin{array}{ccc}
\partial...
...rtial_x u_y (\partial_y u_y - \lambda) \\
\end{array}\right)
\end{displaymath}](img721.png) |
(A.15) |
The exponential matrix is thus valued as
![\begin{displaymath}
e^{{\mbox{\boldmath$A$}}(t)dt} =
e^{- {2 \over \tau} dt}
{\...
...- 2 \lambda dt}
\end{array}\right)
{\mbox{\boldmath$N$}}^{-1}
\end{displaymath}](img722.png) |
(A.16) |
Stefano Musacchio
2004-01-09