Three ways to lattice Boltzmann: A unified time-marching picture

It is shown that the lattice Boltzmann equation LBE corresponds to an explicit Verlet time-marching scheme for a continuum generalized Boltzmann equation with a memory delay equal to a half time step. This proves second-order accuracy of LBE with respect to this generalized equation, with no need of resorting to any implicit time-marching procedure Crank-Nicholson and associated nonlinear variable transformations. It is also shown, and numerically demonstrated, that this equivalence is not only formal, but it also translates into a complete equivalence of the corresponding computational schemes with respect to the hydrodynamic equa- tions. Second-order accuracy with respect to the continuum kinetic equation is also numerically demonstrated for the case of the Taylor-Green vortex. It is pointed out that the equivalence is however broken for the case in which mass and/or momentum are not conserved, such as for chemically reactive flows and mixtures. For such flows, the time-centered implicit formulation may indeed offer a better numerical accuracy

A note on the lattice Boltzmann method beyond the Chapman Enskog limits

A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the Lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to $Kn\sim {\cal O}(1)$. This may help the interpretation of recent Lattice Boltzmann simulations of microflows, which show satisfactory agreement with continuum kinetic theory in the moderate-Knudsen regime.Comment: 7 PAGES, 1 FIGUR

Dynamics of the spontaneous breakdown of superhydrophobicity

Drops deposited on rough and hydrophobic surfaces can stay suspended with gas pockets underneath the liquid, then showing very low hydrodynamic resistance. When this superhydrophobic state breaks down, the subsequent wetting process can show different dynamical properties. A suitable choice of the geometry can make the wetting front propagate in a stepwise manner leading to {\it square-shaped} wetted area: the front propagation is slow and the patterned surface fills by rows through a {\it zipping} mechanism. The multiple time scale scenario of this wetting process is experimentally characterized and compared to numerical simulations.Comment: 7 pages, 5 figure