An improved hydrodynamics formulation for multiphase flow lattice-boltzmann models

Lattice-Boltzmann (LB) models provide a systematic formulation of effective-field computational approaches to the calculation of multiphase flow by replacing the mathematical surface of separation between the vapor and liquid with a thin transition region, across which all magnitudes change continuously. Many existing multiphase models of this sort do not satisfy the rigorous hydrodynamic constitutive laws. Here, we extend the two-dimensional, seven-speed Swift et al. LB model¹ to rectangular grids (nine speeds) by using symbolic manipulation (Mathematica^™) and compare the LB model predictions with benchmark problems, in order to evaluate its merits. Particular emphasis is placed on the stress tensor formulation. Comparison with the two-phase analogue of the Couette flow and with a flow involving shear and advection of a droplet surrounded by its vapor reveals that additional terms have to be introduced in the definition of the stress tensor in order to satisfy the Navier-Stokes equation in regions of high density gradients. The use of Mathematica obviates many of the difficulties with the calculations "by-hand,allowing at the same time more flexibility to the computational analyst to experiment with geometrical and physical parameters of the formulation.