Numerical geometric integration of ODE by Hairer E.

By Hairer E.

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2. Steady periodic shear yow with periodic cross sweep We now consider how tracer transport is in#uenced when a cross sweep w(t/ ) acting across the T shear #ow (see Eq. (48)) has periodic variations in time (with period ). We focus the case of T a steady periodic shear #ow v"v(x) in which the main features are all manifest. 1, but as the computations are a bit lengthier, we defer their presentation until after we discuss the results. usivity enhanced along the shear above its bare molecular value (1) by an amount KM .

J. R. usivity was found to decrease as a function of the temporal frequency . usivity in a shear #ow with periodic temporal variations, and is due to the diminishing persistence of tracer motion as the temporal oscillations become more rapid. ;t))sin(2 y) , 25 where ; is the phase speed. Note that the velocity "eld averaged over a temporal period still vanishes everywhere. Through some numerical experiments, it was found in [165] that tracer transport is faster in a travelling wave than in a standing wave.

Usivity KM . usion equation. J. R. Kramer / Physics Reports 314 (1999) 237}574 291 Eqs. (74) and (75) that for a steady, periodic shear #ow with constant or zero cross sweep, the mean and variance of the tracer displacement relax to their homogenized limits on an order unity time scale which is independent of Pe.

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