Dissipative and dispersive behaviors of lattice-based models for hydrodynamics

摘要

Both dissipation and dispersion are present in many complex systems; their interactions through nonlinearity can lead to interesting features. We investigate in this paper the dissipation-dispersion interactions that exist in lattice-based kinetic models for hydrodynamics. The classical Chapman-Enskog expansion is used to derive the dispersion coefficients at third order of Knudsen number. Unlike the dissipation coefficient (viscosity) that is always positive, the dispersion coefficient can be either positive or negative. It would be interesting to know if there is any other physics in these models as compared with the traditional dispersionless Navier-Stokes dynamics. Traveling wave solutions in one dimension are studied and two different solutions have been found: (1) monotonic shack solutions and (2) oscillatory shock solutions, according to different conditions. In two- and three-dimensional systems, whether or not these oscillatory behaviors caused by the interactions between nonlinearity, dissipation, and dispersion have anything to do with vortex cascades (direct or inverse) would be an interesting question and we leave it for future studies. [References: 17]