By Despres B.

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Kramer / Physics Reports 314 (1999) 237}574 287 Due to the spatially decoupled dynamics induced by a shear #ow, the stochastic trajectory equations (66) can be integrated successively by quadrature: (67a) X(t)"x #Pe wN t#(2= (t) , V R >(t)"y #Pe v(X(s)) ds#(2= (t) . usion. y 2 , 7 5 and the variance of its location, (68) (t"x ,y ),1(X(t)! (t))2 , 6 6 5 (69) (t"x ,y ),1(>(t)! (t))2 . 7 7 5 The brackets 1 ) 2 denote an averaging over the Brownian motion statistics. y )2" (t"x , y )#( (t"x , y )) .

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.

Kramer / Physics Reports 314 (1999) 237}574 281 Fig. 6. 9 (from VV [210]). 2, 2)R. Also shown is a line with slope 2 for reference. usivity of the #ow. usivity computed for this #ow in the absence of any mean #ow in Fig. ) The second case, in which the mean sweep is directed orthogonally to the shearing direction, exhibits instead two distinct scaling crossovers in the plot of ln KM vs. ln Pe. For small Pe, the curve has slope 2, a general consequence VV of the Stieltjes integral representation (39) for arbitrary periodic #ows [12].

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