Lattice Boltzmann model with fluid phase separation

The LB method of modeling fluid dynamics is actually a family [3] of models with varying degrees of faithfulness to the properties of real liquids. These methods are currently in a state of evolution as the models become better understood and corrected for various deficiencies. In this paper we utilize a version of LB proposed by Shan and Chen[1,2] that is particularly simple in form and adaptable to complex flow conditions like the presence of solid-fluid and fluid-fluid boundaries.

The approach of LB is to consider a typical volume element of fluid to be composed of a collection of particles that are represented in terms of a particle velocity distribution function at each point in space. The particle velocity distribution, $n^{i}_a({\bf x},t)$, is the number density of particles at node x, time t, and velocity, e_a , where (a=1,...,b) indicates the velocity direction and superscript i labels the fluid component. The time is counted in discrete time steps, and the fluid particles can collide with each other as they move under applied forces.

For this study we use the D3Q19 (3 Dimensional lattice with b=19)[4] lattice[2]. The microscopic velocity, e _a , equals all permutations of (±1, ±1, 0) for 1   a  12, (±1, 0, 0) for 13  a  18, and (0, 0, 0) for a = 19. The units of e_a are the lattice constant divided by the time step. Macroscopic quantities such as the density, n ⁱ (x, t ) , and the fluid velocity, uⁱ, of each fluid component, i, are obtained by taking suitable moment sums of $n^{i}_a({\bf x},t)$. Note that while the velocity distribution function is defined only over a discrete set of velocities, the actual macroscopic velocity field of the fluid is continuous.

The time evolution of the particle velocity distribution function satisfies the following LB equation:

$$n^{i}_a({\bf x}+{\bf e}_a,t+1)-n^{i}_a({\bf x},t) = \Omega^{i}_a({\bf x},t),$$ (1)

where $\Omega^{i}_a$ is the collision operator representing the rate of change of the particle distribution due to collisions. The collision operator is greatly simplified by use of the single time relaxation approximation[5,6]

$$\Omega^{i}_a({\bf x},t) =-\frac{1}{\tau}_{i} \left[n^{i}_a({\bf x},t)-n^{i (eq)}_a({\bf x},t)\right],$$ (2)

where $n^{i (eq)}_a({\bf x},t)$ is the equilibrium distribution at (x, t ) and $\tau_{i}$ _i is the relaxation time that controls the rate of approach to equilibrium. The equilibrium distribution can be represented in the following form for particles of each type[2,6]:

$$n^{i(eq)}_a({\bf x})=t_an^{i}({\bf x})\left[\frac{3}{2}(1-d_o... ...ac{3}{2}({3\bf e}_a{\bf e}_a:{\bf v}{\bf v}-{\bf v}^2)\right]$$ (3)

$$n^{i(eq)}_{19}({\bf x})=t_{19}n^{i}({\bf x})\left[3d_o- \frac{3}{2}{\bf v}^2\right],$$ (4)

where

$${\bf v}=\frac{\sum_{i}^{S}m^{i}\sum_{a} n^{i}_{a}{\bf e}_{a}/ \tau_i} {\sum_{i}^{S}m^{i} n^{i}({\bf x})/\tau_i},$$ (5)

and where mⁱ is the molecular mass of the ith component, and t_a=1/36 for 1 a 12, t_a =1/18 for 13 a 18 and t₁₉ =1/3 . The free parameter d_o can be related to an effective temperature, T, for the system by the following moment of the equilibrium distribution:

$$T({\bf x},t) = \frac{\sum_{a} n^{i(eq)}_{a}({\bf x},t) ({\bf e}_{a}-{\bf v})^2} {3n^{i}({\bf x},t)},$$ (6)

which results in T=(1-d_o)/2 (we take units such that the Boltzmann constant k_b=1).

It has been shown that the above formalism leads to a velocity field that is a solution of the Navier-Stokes[5] equation with the kinematic viscosity, $\nu=\frac{c^2}{6}(\sum_{i}^{S}c_{i} \tau_i -\frac{1}{2})$where c_i is the concentration of each component[6].