Fluid Interaction

An interaction force F ⁱ for each fluid is needed to drive the phase separation process. We use a simple interaction, suggested by Shan and Chen (the physical basis of this forcing is given in the Appendix A) that depends on the density of each fluid:

$${\bf F}^i=-n^i({\bf x})\sum_{i'}^ {S}\sum_{a}G_{ii'}^{a} n^{i'}({\bf x}+{\bf e}_a) {\bf e}_a$$ (10)

where G ^a_{ii '} = 2G, G, and 0 for the cases $\vert{\bf e}^a\vert = 1$, $\vert{\bf e}^a\vert = \sqrt{2}$ and i = i' respectively. G is a coupling constant controlling the interaction strength. This term is analogous to a nearest-neighbor interaction in lattice models of interacting fluids. The forcing term has been shown to drive the phase separation and to produce an interfacial surface tension effect consistent with the Laplace law [5], which states that there is a pressure drop proportional to the local curvature at the interface boundary between two fluids.

In the LB model of Shan and Chen, phase separation takes place when the mutual diffusivity of the binary mixture becomes negative, providing a condition determining the critical coupling G_c for phase separation. An analytical expression for the mutual diffusivity has been determined [25]. For a viscosity matched binary mixture in which the particle masses are also matched ("symmetric fluid mixture"), phase separation occurs when the critical coupling equals,

$$G_c=\frac{T}{48}\frac{-c_1n^1-c_2n^2+\sqrt{(c_1n^1+c_2n^2)^2+8n^1n^2}}{n^1n^2}.$$ (11)

It is not ordinarily possible to exactly calculate the critical coupling for phase separation in three-dimensional liquids and this condition for the critical coupling, G_c, is evidently a clue to the nature of the phase separation process. Since the LB method neglects thermal fluctuations that renormalize the critical coupling constant G, this method is a mean-field model of fluid mixtures. This observation, which has basic ramifications for the applicability of the model in comparison with real fluid mixtures, is established numerically in the next section where the critical properties are examined to establish the nature of the model.

Once the forcing is described, it must be properly incorporated into a LB model. Shan-Chen introduced the forcing by modifying the equilibrium velocity v [2]:

$$n^i({\bf x}){\bf v}^{'}({\bf x})=n^i {\bf v}({\bf x})+ \tau_i \frac{\bf {F}^i}{\rho^i}$$ (12)

where ${\bf v}'$ is the new velocity used in equations 5 and 6. This approach introduces a momentum transfer between fluids that preserves momentum globally. The main criticism of the Shan-Chen model is that when shifting the velocity in the equilibrium distribution, additional corrections in the pressure tensor will appear which are of order F ².

Instead of shifting the velocity in the equilibrium distribution as in the Shan Chen model, the forcing between two fluids can be directly included in the body-force term of the Boltzmann equation. In the continuum Boltzmann equation the body-force term is written $B^i=\frac{{\bf F}^i}{\rho^i} \cdot \nabla_e n^i( {\bf x},{\bf e})$, where $\frac{{\bf F}^i}{\rho^i}$ is an acceleration field due to a body-force. An expression of this body-force term, to second order in Hermite polynomials [7], in the discrete velocity space of the D3Q19 lattice is given by

$$B^i_a=-3t_an^i({\bf x})\left[({\bf e_a-v})\cdot\frac{ { \bf F... ... \cdot v})( {\bf e_a} \cdot \frac{ {\bf F}^i} {\rho^i})\right]$$ (13)

One can think of this acceleration field as being due to a "mean-field" produced by the surrounding molecules (see Appendix A). The main difference between this approach and the Shan Chen model is that it avoids terms of order F ² that result from the shift of the velocity in the equilibrium distribution, so that the linearity of the forcing is preserved. The effect of this modification of the LB model is investigated below.

Finally, Equation 10 can be modified to mimic an interaction between the solid surface and fluid [5]. Here n ⁱ ' (x + e_a) is given the value 1 or 0 depending on whether x + e_a resides on a point in the solid or fluid, respectively, and the value of G ^a_ii' is then set to allow the solid to attract a fluid (wetting) or to repulse a fluid (nonwetting).