Elliptic regions

In elliptic regions, using again the incompressibility $\partial_y u_y = - \partial_x u_x$ and defining

$$\theta = \sqrt{\det ({\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$u$}}) }$$ (A.17)

the matrix ${\mbox{\boldmath$A$}}$ can be decomposed in a diagonal part and a pure rotational part:

$${\mbox{\boldmath$A$}} = {\mbox{\boldmath$N$}}({\mbox{\boldmath$D$}} + {\mbox{\boldmath$R$}}){\mbox{\boldmath$N$}}^{-1}$$ (A.18)

where

$${\mbox{\boldmath$D$}} + {\mbox{\boldmath$R$}} = \left( \be... ...u} & 0 \\ 2 \theta & 0 & -{2 \over \tau} \end{array}\right)$$ (A.19)

The eigenvectors matrix ${\mbox{\boldmath$N$}}= (\bar{u}_1,\bar{u}_2,\bar{u}_3 )$ is

$${\mbox{\boldmath$N$}} = \left( \begin{array}{ccc} \partial... ...artial_x u_y (\partial_y u_y - \theta) \\ \end{array}\right)$$ (A.20)

and the exponential matrix reads

$$e^{{\mbox{\boldmath$A$}}(t)dt} = e^{-{2 \over \tau} dt} {\m... ...os(2 \theta dt) \end{array}\right) {\mbox{\boldmath$N$}}^{-1}$$ (A.21)