A Novel Lattice Boltzmann Model for Fourth Order Nonlinear Partial Differential Equations

Abstract

In this paper, a novel lattice Boltzmann (LB) equation model is proposed to solve the fourth order nonlinear partial differential equation (NPDE). Different from existing LB models, a source distribution function is introduced to remove some unwanted terms in the nonlinear part of the equation. Hereby, the equilibrium distribution function is designed to follow the rule of Chapman–Enskog (C–E) analysis. Through the C–E procedure, the fourth order NPDE can be recovered perfectly from the proposed LB model. A series of numerical experiments have been carried out to solve some widely studied fourth order NPDEs, including the Kuramoto–Sivashinsky equation, Cahn–Hilliard equation with double-well potential and a fourth order diffuse interface model with Peng–Robinson equation of state. Numerical results show that the performance of the present LB model is much better than other existing LB models.

Appendix

The definition of parameters a(T) and b are given by the following mixing rules

$$\begin{aligned} a(T)= & {} \sum _{i=1}^{M}\sum _{j=1}^{M}y_{i}y_{j}({a_{i}a_{j}})^{1/2}(1-k_{ij}),\\ b= & {} \sum _{i=1}^{M}y_{i}b_{i}, \end{aligned}$$

where $y_{i}=n_{i}/n$ is the mole fraction of component i, $k_{ij}$ is the binary interaction coefficient of Peng–Robinson EOS, which is usually computed from experimental correlation. The Peng–Robinson parameters $a_i$ and $b_i$ for pure-substance component i can be derived from the critical properties of the particular species as follows:

$$\begin{aligned} a_{i}\left( T\right) =0.45724\frac{R^{2}T_{c_{i}}^{2}}{P_{c_{i}}}\left( 1+m_{i}\left( 1-\sqrt{\frac{T}{T_{c_{i}}}}\right) \right) ^{2}, \quad b_{i}=0.07780\frac{RT_{c_{i}}}{P_{c_{i}}}. \end{aligned}$$

Here, $T_{c_i}$ and $P_{c_i}$ represent the critical temperature and the critical pressure of a pure substance, respectively. The parameter $m_i$ has the following relations with the acentric parameter $\omega _i$:

$$\begin{aligned} m_i= & {} 0.37464 + 1.54226\omega _i - 0.26992{\omega _i ^2},\omega _i \le 0.49,\\ m_i= & {} 0.379642 + 1.485030\omega _i - 0.164423{\omega _i ^2} + 0.016666{\omega _i ^3},\omega _i > 0.49. \end{aligned}$$

The acentric parameter $\omega _i$ can be computed by using critical temperature $T_{c_{i}}$, critical pressure $P_{c_{i}}$ and the normal boiling point $T_{b_{i}}$:

$$\begin{aligned} {\omega _i} = \frac{3}{7}\left( \frac{{{{\log }_{10}}\left( \frac{{{P_{{c_i}}}}}{{14.695\mathrm{{PSI}}}}\right) }}{{\frac{{{T_{{c_i}}}}}{{{T_{{b_i}}}}} - 1}}\right) - 1. \end{aligned}$$

The cross influence parameter $c_{ij}$ can be obtained by using the modified geometric mean rule

$$\begin{aligned} c_{ij}=(1-\beta _{ij})\sqrt{c_{i}c_{j}}, \end{aligned}$$

where $\beta _{ij}$ represents the binary interaction coefficient for the influence parameter. Its value is usually assumed to be zero in most engineering practice. $c_i$ is the pure component influence parameter, which is related to the Peng–Robinson parameters $a_i$ and $b_i$ by [4]

$$\begin{aligned} c_i = a_i{b_i^{2/3}}\left( m_{1,i}^c\left( 1 - \frac{T}{{{T_{c_i}}}}\right) + m_{2,i}^c\right) . \end{aligned}$$

Here, $m_{1,i}^c$ and $m_{2,i}^c$ denote the coefficients which can be related to the acentric factor $\omega _i$ by

$$\begin{aligned} m_{1,i}^c = - \frac{{{{10}^{ - 16}}}}{{1.2326 + 1.3757\omega _i }},\quad m_{2,i}^c = \frac{{{{10}^{ - 16}}}}{{0.9051 + 1.5410\omega _i }}. \end{aligned}$$