Chromo-dynamic multi-component lattice Boltzmann equation scheme for axial symmetry

We validate the chromo-dynamic multi-component lattice Boltzmann equation (MCLBE) simulation for immiscible fluids with a density contrast against analytical results for complex flow geometries, with particular emphasis on the fundamentals of the method, i.e. compliance with inter-facial boundary conditions of continuum hydrodynamics. To achieve the necessary regimes for the chosen validations, we develop, from a three-dimensional, axially-symmetric flow formulation, a novel, two-dimensional, pseudo Cartesian, MCLBE scheme. This requires the inclusion in lattice Boltzmann methodology of a continuously distributed source and a velocity-dependent force density (here, the metric force terms of the cylindrical Navier-Stokes equations). Specifically, we apply our model to the problem of flow past a spherical liquid drop in Re=$0$, Ca$\rightarrow 0$ regime and, also, flow past a lightly deformed drop. The resulting simulation data, once corrected for the simulation's inter-facial micro-current (using a method we also advance herein, based on freezing the phase field) show good agreement with theory over a small range of density contrasts. In particular, our data extend verified compliance with the kinematic condition from flat \cite{Burgin} to the case of curved fluid-fluid interfaces. More generally, our results indicate a route to eliminate the influence of the inter-facial micro-current.