Appendix B. The Global Efficiency of Stretching

Let's suppose the flow to be 2D, steady-state, in a closed domain.

We will explain how we can calculate the time evolution of the global efficiency of stretching (see chapter 2 for the definition of this parameter).

For example, for the linear stretch, at time , this efficiency is:

(1)

To calculate such a parameter, the following steps are necessary:

  1. In the mixing files are calculated the stretching and the dissipation rate along trajectories.

  2. We have to define, in Polystat, a new property, the cumulated dissipation, which is the time integration of the dissipation rate:

    (2)

    This new parameter depends on the material point and on time.

  3. We perform a slicing on the time ( slices, every Δ seconds).

  4. We calculate the sum function of the stretching :

    (3)

  5. We calculate the sum function of the cumulated dissipation :

    (4)

  6. Finally, we divide the two sum functions to obtain the global efficiency of stretching:

    (5)