Application of the finite-size Lyapunov exponent to particle tracking velocimetry in fluid mechanics experiments

A finite-size (or scale) Lyapunov exponent (FSLE), lambdaa(x), is presented in a statistical mechanical framework and employed to characterize mixing in a variety of laboratory and computational fluid mechanics experiments. The FSLE is the exponential rate at which two particles separate from a distance x to ax. Laboratory particle tracking experiments are used to study penetrative convection and flow in porous media while computational experiments are used to study Lévy processes and deterministic diffusion. The apparent scaling relation lambaa(x) approximately Cax(-beta(a)) of the FSLE holds over intermediate initial separations where the laboratory experiment data is most accurate and asymptotically for the computational experiments. The dependence of the exponent beta on a decreases with increasing a. In the matched index porous system, Ca is also a function of mean fluid velocity. The exponent beta is alpha when the Lévy process is alpha-stable and in this case beta is independent of a.