SPH implementation of Multiple-phase Fluid Simulation

1. NSCH model

For $n$-phase fluid:

$$ \begin{cases} &\frac{Du}{Dt} = -\frac{1}{\rho}p + g + \nabla\cdot (\nu \nabla u) + SF\\ &\nabla\cdot u = 0\\ &\frac{Dc_k}{Dt} = \nabla\cdot (M\nabla \mu_k)\\ &\mu_k = \frac{\partial F}{\partial c_k} - \epsilon^2 \nabla^2c_k + \beta(c) \end{cases} $$

, where $\rho$ is the the aggregate density and $\nu$ is the aggregate viscosity.

$$ \rho = \frac{1}{\sum_k \frac{c_k}{\rho_k}} $$

$$ VBN_k = 14.534 \times \log(\log(\nu_k + 0.8)) + 10.975 $$

$$ VBN = \sum_k c_k VBN_k $$

$$ \nu = e^{e^{\frac{VBN - 10.975}{14.534}}} - 0.8 $$

$$ \beta(c) = -\frac{1}{n} \sum_k \frac{\partial F}{\partial c_k} $$

Free energy:

$$ F(c) = \alpha(c_1-s_1)^2(c_2-s_2)^2 $$

2. Discrelezation

$$ m_i = \frac{1}{\sum_k \frac{c_k}{m_{ik}}} $$

$$ \rho_i\approx \tilde{\rho}_i = m_i\sum_j W_{ij} $$

$$ p_i = k(\rho_i - \rho_0) $$

$$ \mathrm f_i^{\mathrm{pressure}} = -\sum_j m_j \frac{p_i+p_j}{2\rho_j}\nabla W^{\mathrm{pressure}}(r_i-r_j, h) $$

$$ \mathrm f_i^{\mathrm{viscosity}} = \mu\sum_j m_j \frac{v_j-v_i}{2\rho_j}\nabla^2 W^{\mathrm{viscosity}}(r_i-r_j, h) $$

$$ \nabla\mu_i = \rho_i\sum_j m_j\left(\frac{\mu_i}{\rho_i^2} + \frac{\mu_j}{\rho_j^2}\right)\nabla W_{ij} $$

$$ \nabla\cdot (M_i\nabla \mu_i) = \sum_j \frac{m_j}{\rho_j} (M_i+M_j)\mu_{ij}\frac{r_{ij}\cdot \nabla W_{ij}}{r_{ij}^2 + \eta^2} $$

$$ \nabla^2 c_i = 2\sum_j \frac{m_j}{\rho_j} c_{ij} \frac{r_{ij}\cdot \nabla W_{ij}}{r_{ij}^2 + \eta^2} $$

3. Time Integration

4. Original Paper

"Fast Multiple-fluid Simulation Using Helmholtz Free Energy"