Lattice Boltzmann method for fractional Cahn-Hilliard equation

Abstract

Fractional phase field models have been reported to suitably describe the anomalous two-phase transport in heterogeneous porous media, evolution of structural damage, and image inpainting process. It is commonly different to derive their analytical solutions, and the numerical solution to these fractional models is an attractive work. As one of the popular fractional phase-field models, in this paper we propose a fresh lattice Boltzmann (LB) method for the fractional Cahn-Hilliard equation. To this end, we first transform the fractional Cahn-Hilliard equation into the standard one based on the Caputo derivative. Then the modified equilibrium distribution function and proper source term are incorporated into the LB method in order to recover the targeting equation. Several numerical experiments, including the circular disk, quadrate interface, droplet coalescence and spinodal decomposition, are carried out to validate the present LB method. It is shown that the numerical results at different fractional orders agree well with the analytical solution or some available results. Besides, it is found that increasing the fractional order promotes a faster evolution of phase interface in accordance with its physical definition, and also the system energy predicted by the present LB method conforms to the energy dissipation law.