Lattice Boltzmann simulation of a binary fluid with different phase viscosities and its application to fingering in two dimensions

A thermodynamically consistent lattice Boltzmann scheme for simulating the flow of a binary fluid is extended to allow the fluid components to have di fferent viscosities. The approach is tested for the shear and Poiseuille fl ow of layered immiscible fluids and for the dispersion relation and the dam ping of a capillary wave. We then consider the fingering that results when a fluid is displaced by a less viscous fluid in a two-dimensional channel. The finger widths obtained match the results of Reinelt and Saffman [1], bu t differ somewhat from those of Halpern and Gaver [2] for capillary numbers above 2. A limiting finger width close to 1/2 is obtained for high capilla ry numbers and high viscosity ratios.