Neutral regions

In regions where the relative intensity of rotation and stretching are equal (e.g. in the case of shear flows) the determinant of the velocity gradients vanishes. In this case the matrix ${\mbox{\boldmath$A$}}$ has a single eigenvector with multiplicity $3$ and its exponential can be evaluated as:

$$e^{{\mbox{\boldmath$A$}}(t)dt} = e^{-{2 \over \tau} dt} \le... ...math$A$}}+{2 \over \tau}{\mbox{\boldmath$1$}})^2 dt^2 \right)$$ (A.22)

This numerical scheme allows to perform accurate simulations of the passive limit of Oldroyd-B model, which are non achievable with standard Eulerian codes.

In principle it is possible to include the feedback of polymers on the velocity field, by reconstructing at each time step the Eulerian conformation tensor field on a regular grid from the Lagrangian values obtained along $n$ simultaneously integrated fluid trajectories.

Unfortunately this mixed Eulerian-Lagrangian code does not solve the problem of numerical instabilities of Oldroyd-B model, because Lagrangian particles with different values of the conformation tensor are transported arbitrarily close to each other during the simulation, and thus the Eulerian field reconstructed from the Lagrangian one still have diverging gradients, which are involved in the feedback and cannot be resolved by the Eulerian part of the code.