Modeled Lattice Boltzmann Equation and the Constant-Density Assumption

c = sound speed D = flow dimension, 2 for two-dimensional flow and 3 for three-dimensional flow f = particle distribution function f = equilibrium particle distribution function h = half-width of the channel upstream of the expansion in a sudden expansion geometry Kn = Knudsen number L = characteristic length M = Mach number p = static pressure Re = Reynolds number t = time u = velocity vector u = velocity component along x axis v = velocity component along y axis x, y = Cartesian coordinates xj = position vector = index for the lattice velocity = fluid viscosity = particle velocity vector = particle velocity along lattice = fluid density = relaxation time ij = viscous stress tensor o = reference relaxation time ! = vorticity