%0 Book Section %1 statphys23_0164 %A Falkovich, G. %A Afonso, M. Martins %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K batchelor exponent flow fluctuation fluid-particle lyapunov pair random regime relation separation statphys23 topic-5 %T Fluid-particle separation in the Batchelor regime with the telegraph-noise model %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=164 %X We study the statistics of the relative separation $$R$(t)$ between two fluid particles in a random flow. We confine ourselves to the Batchelor regime, i.e.~we only examine the evolution of distances smaller than the smallest active scale of the flow. Here, the latter is spatially smooth and the linearized equation reads $\mbox\boldmath $R$=\sigma(t)\cdot$R$$. The Lagrangian strain (scalar or matrix) $\sigma$ is assumed as given in its statistics and is modelled by a telegraph noise. This is a stationary random Markov process, which can only take two values with known transition probabilities. The presence of two independent parameters (intensity of velocity gradient and flow correlation time) allows the definition of their ratio as the Kubo number, whose infinitesimal and infinite limits describe the delta-correlated and quasi-deterministic cases, respectively. However, the simplicity of the model enables us to write closed equations for the interparticle distance in the presence of a finite-correlated, i.e.~coloured, noise. In 1D, the flow is locally compressible in every single realization, but the average `volume' must keep finite. This provides us with a mathematical constraint, which physically reflects in the fact that, in the Lagrangian frame, particles spend longer time in contracting regions than in expanding ones. Imposing this condition consistently, we are able to find analytically the long-time growth rate of the interparticle-distance moments and, consequently, the senior Lyapunov exponent, which coherently turns out to be negative. Analysing the large-deviation form of the joint probability distribution, we also show the exact expression of the Cramér function, which happens to satisfy the well-known Gallavotti--Cohen relation despite the time irreversibility of the strain statistics. The 2D incompressible isotropic case was also studied. The evolution of the linear and quadratic components was analysed thoroughly, while for higher moments, due to high computational cost, we focused on a restricted, though exact, dynamics. As a result, we obtained the moment asymptotic growth rates and the Lyapunov exponent (positive) in the two above-mentioned limits, together with the leading corrections. The quasi-deterministic limit turns out to be singular, while a perfect agreement was found with the already-known delta-correlated case.

@incollection{statphys23_0164, abstract = {We study the statistics of the relative separation $\mbox{\boldmath $R$}(t)$ between two fluid particles in a random flow. We confine ourselves to the Batchelor regime, i.e.~we only examine the evolution of distances smaller than the smallest active scale of the flow. Here, the latter is spatially smooth and the linearized equation reads $\dot{\mbox{\boldmath $R$}}=\sigma(t)\cdot\mbox{\boldmath $R$}$. The Lagrangian strain (scalar or matrix) $\sigma$ is assumed as given in its statistics and is modelled by a telegraph noise. This is a stationary random Markov process, which can only take two values with known transition probabilities. The presence of two independent parameters (intensity of velocity gradient and flow correlation time) allows the definition of their ratio as the Kubo number, whose infinitesimal and infinite limits describe the delta-correlated and quasi-deterministic cases, respectively. However, the simplicity of the model enables us to write closed equations for the interparticle distance in the presence of a finite-correlated, i.e.~coloured, noise. In 1D, the flow is locally compressible in every single realization, but the average `volume' must keep finite. This provides us with a mathematical constraint, which physically reflects in the fact that, in the Lagrangian frame, particles spend longer time in contracting regions than in expanding ones. Imposing this condition consistently, we are able to find analytically the long-time growth rate of the interparticle-distance moments and, consequently, the senior Lyapunov exponent, which coherently turns out to be negative. Analysing the large-deviation form of the joint probability distribution, we also show the exact expression of the Cram\'er function, which happens to satisfy the well-known Gallavotti--Cohen relation despite the time irreversibility of the strain statistics. The 2D incompressible isotropic case was also studied. The evolution of the linear and quadratic components was analysed thoroughly, while for higher moments, due to high computational cost, we focused on a restricted, though exact, dynamics. As a result, we obtained the moment asymptotic growth rates and the Lyapunov exponent (positive) in the two above-mentioned limits, together with the leading corrections. The quasi-deterministic limit turns out to be singular, while a perfect agreement was found with the already-known delta-correlated case.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Falkovich, G. and Afonso, M. Martins}, biburl = {https://www.bibsonomy.org/bibtex/2e09ed61da53f0294783aba9566a9324d/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {04ceafa0bfb1e3b9ce1bdd7e2103ef11}, intrahash = {e09ed61da53f0294783aba9566a9324d}, keywords = {batchelor exponent flow fluctuation fluid-particle lyapunov pair random regime relation separation statphys23 topic-5}, month = {9-13 July}, timestamp = {2007-06-20T10:16:13.000+0200}, title = {Fluid-particle separation in the Batchelor regime with the telegraph-noise model}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=164}, year = 2007 }