# A Lattice Boltzmann Method and Asynchronous Model Coupling for Viscoelastic Fluids
## 3 The Coupled LBM of Viscoelastic Fluids
### Incompressible LBGK Model with the Extra Force  
In this study, we use the 2D incompressible Lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) model [25] based on the two-dimensional nine velocity (D2Q9) model to recover the Navier-Stokes equations. The particle velocity \(\mathbf{\mathrm{e}}_{\kappa}\) may be written as  
\[\mathbf{\mathrm{e}}_{\alpha}=\begin{cases}(0,0)&\alpha=0\\ c(\cos[(\alpha-1)\frac{\pi}{2}],\sin[(\alpha-1)\frac{\pi}{2}])&\alpha=1,\;2, \;3,\;4,\\ \sqrt{2}c\left(\cos[(2\alpha-1)\frac{\pi}{4}],\sin[(2\alpha-1)\frac{\pi}{4}] \right)&\alpha=5,\;6,\;7,\;8\end{cases} \tag{4}\]  
here, \(\delta x\), \(\delta t\) are, respectively, the lattice grid spacing and the time step, \(c=\frac{\delta x}{\delta t}\) represents the particle velocity.  
The lattice Boltzmann equation (LBE) reads  
\[f_{\alpha}(\mathbf{\mathrm{x}}+\mathbf{\mathrm{e}}_{\alpha}\delta t,t+\delta t)-f_{ \alpha}(\mathbf{\mathrm{x}},t)=\Omega_{\alpha}(f(\mathbf{\mathrm{x}},t)), \tag{5}\]  
where \(f_{\alpha}(\mathbf{\mathrm{x}},t)\) denotes the distribution function at the node of \(\mathbf{\mathrm{x}}\) at the time \(t\), and the BGK collision model, \(\Omega_{\alpha}(f(\mathbf{\mathrm{x}},t))\), is given by  
\[\Omega_{\alpha}(f(\mathbf{\mathrm{x}},t))=-\frac{1}{\lambda}[f_{\alpha}(\mathbf{ \mathrm{x}},t)-f_{\alpha}^{(\alpha)}(\mathbf{\mathrm{x}},t)], \tag{6}\]  
where \(\lambda~{}=~{}\lambda_{0}/\delta t\) represents the dimensionless relaxation time, \(\lambda_{0}\) is relaxation time. The equilibrium distribution function (EDF), \(f_{\alpha}^{(\alpha)}\), is given by  
\[f_{\alpha}^{(\alpha)}(\mathbf{\mathrm{x}},t)\equiv E_{\alpha}(\mathbf{ \mathrm{u}})\] \[=\left\{\begin{array}{ll}\rho_{0}\big{[}1-(1-\alpha_{0})\frac{p} {\rho_{0}c_{s}^{2}}+s_{0}(\mathbf{\mathrm{u}})\big{]},&\alpha=0\\ \rho_{0}\big{[}\omega_{\alpha}\frac{p}{\rho_{0}c_{s}^{2}}+s_{\alpha}(\mathbf{ \mathrm{u}})\big{]},&\alpha=1,\cdots,8\end{array}\right., \tag{7}\]  
where  
\[s_{\alpha}(\mathbf{\mathrm{u}})=\omega_{\alpha}[\frac{\mathbf{\mathrm{e}}_{\alpha}\cdot \mathbf{\mathrm{u}}}{c_{s}^{2}}+\frac{(\mathbf{\mathrm{e}}_{\alpha}\cdot\mathbf{\mathrm{u}}) ^{2}}{2c_{s}^{4}}-\frac{\mathbf{\mathrm{u}}\cdot\mathbf{\mathrm{u}}}{2c_{s}^{2}}], \tag{8}\]\(\mathbf{u}\) and \(p\) are the velocity and the fluid pressure, respectively. The weight coefficients are  
\[\omega_{0}=\frac{4}{9},\ \ \omega_{1},\ldots,\omega_{4}=\frac{1}{9},\ \ \omega_{5},\ldots,\omega_{8}=\frac{1}{36},\ \ c_{s}^{2}=\frac{c^{2}}{3}, \tag{9}\]  
here \(c_{s}=c/\sqrt{3}\) represents the acoustic speed, \(\rho_{0}\) is a constant of density. The pressure and velocity of flow are deduced from the ensemble average of distribution functions  
\[\rho_{0}=\sum_{\alpha=1}^{8}f_{\alpha}(\mathbf{x}),\ \ \rho_{0}\mathbf{u}=\sum_{ \alpha=1}^{8}\mathbf{e}_{\alpha}f_{\alpha}(\mathbf{x}), \tag{10}\]  
\[p=\frac{\rho_{0}c_{s}^{2}}{1-\omega_{0}}\Bigg{(}\sum_{\alpha=1}^{8}f_{\alpha}( \mathbf{x})+s_{0}(\mathbf{u})\Bigg{)}, \tag{11}\]  
In the simulation of viscoelastic fluids, the treatment of the elastic force may be crucial for the coupled lattice Boltzmann method. Here, we incorporate the extra forces \(\mathbf{F}=\
abla\cdot\boldsymbol{\tau}\) into lattice Boltzmann equation, and treat the elastic force of polymers as an extra forced of the lattice Boltzmann BGK [26]. The incompressible LBGK Equation (5) with the additional elastic forced term is updated by  
\[\begin{array}{l}f_{\alpha}(\mathbf{x}+\mathbf{e}_{\alpha}\delta t,t+\delta t )-f_{\alpha}(\mathbf{x},t)\\ =-\frac{1}{\lambda}\big{[}f_{\alpha}(\mathbf{x},t)-\widehat{f}_{\alpha}^{(eq )}\big{]}+\delta tF_{\alpha}\end