added color model formulation
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James McClure
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@ -67,94 +67,107 @@ Model Formulation
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Two LBEs are constructed to model the mass transport, incorporating the anti-diffusion
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.. math::
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:nowrap:
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$$
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\\begin{eqnarray}
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A_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) &=& w_q N_a \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2}
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A_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) = w_q N_a \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2}
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+ \beta \frac{N_b}{N_a+N_b} \bm{n} \cdot \bm{\xi}_q\Big] \;
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\\
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B_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) &=&
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$$
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.. math::
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:nowrap:
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$$
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B_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) =
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w_q N_b \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2}
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- \beta \frac{N_a}{N_a+N_b} \bm{n} \cdot \bm{\xi}_q\Big]\;,
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\\end{eqnarray}
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$$
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The number density is obtained directly for each fluid as the sum of the mass transport distributions
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The number density for each fluid is obtained from the sum of the mass transport distributions
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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N_a = \sum_q A_q\;, \quad N_b = \sum_q B_q\;
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\\end{equation}
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$$
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The phase indicator field is then defined as
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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\phi = \frac{N_a-N_b}{N_a+N_b}
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\\end{equation}
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$$
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The local fluid viscosity and density are determined based on linear interpolation
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The fluid density and kinematic viscosity are determined based on linear interpolation
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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\nu = \frac{(1+\phi) \nu_n}{2}+\frac{(1-\phi) \nu_w}{2} \;,
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\\end{equation}
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$$
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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\rho_0 = \frac{(1+\phi) \rho_n}{2}+ \frac{(1-\phi) \rho_w}{2} \;,
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\\end{equation}
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$$
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.. math::
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:nowrap:
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$$
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\nu = \frac{(1+\phi) \nu_n}{2}+\frac{(1-\phi) \nu_w}{2} \;,
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$$
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where
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.. math::
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:nowrap:
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$$
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\nu_w = \frac{1}{3}\Big(\tau_w - \frac{1}{2} \Big) \;, \quad
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\nu_n = \frac{1}{3}\Big(\tau_n - \frac{1}{2} \Big) \;.
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$$
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These values are then used to model the momentum transport.
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The LBE governing momentum transport is defined based on a MRT relaxation process with additional
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terms to account for the interfacial stresses
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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f_q(\bm{x}_i + \bm{\xi}_q \delta t,t + \delta t) - f_q(\bm{x}_i,t) = \sum^{Q-1}_{k=0} M^{-1}_{qk} \lambda_{k} (m_k^{eq}-m_k) + t_q \bm{\xi}_q \cdot \frac{\bm{F}}{c_s^2} \;,
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\\end{equation}
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$$
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The moments are linearly indepdendent
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The moments are linearly indepdendent:
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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m_k = \sum_{q=0}^{18} M_{qk} f_q\;.
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\\end{equation}
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$$
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The relaxation parameters are determined from the relaxation time:
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.. math::
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:nowrap:
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$$
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\\begin{eqnarray}
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\lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu\;, \\
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\lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu\;,
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$$
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.. math::
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:nowrap:
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$$
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\lambda_{4}= \lambda_{6}= \lambda_{8} = \lambda_{16} = \lambda_{17} = \lambda_{18}= \frac{8(2-s_\nu)}{8-s_\nu} \;,
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\\end{eqnarray}
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$$
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The non-zero equilibrium moments are defined as
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@ -163,34 +176,60 @@ The non-zero equilibrium moments are defined as
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:nowrap:
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$$
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\\begin{eqnarray}
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m_1^{eq} &=& (j_x^2+j_y^2+j_z^2) - \alpha |\textbf{C}|, \\
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m_9^{eq} &=& (2j_x^2-j_y^2-j_z^2)+ \alpha \frac{|\textbf{C}|}{2}(2n_x^2-n_y^2-n_z^2), \\
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m_{11}^{eq} &=& (j_y^2-j_z^2) + \alpha \frac{|\textbf{C}|}{2}(n_y^2-n_z^2), \\
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m_{13}^{eq} &=& j_x j_y + \alpha \frac{|\textbf{C}|}{2} n_x n_y\;, \\
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m_{14}^{eq} &=& j_y j_z + \alpha \frac{|\textbf{C}|}{2} n_y n_z\;, \\
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m_{15}^{eq} &=& j_x j_z + \alpha \frac{|\textbf{C}|}{2} n_x n_z\;,
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\\end{eqnarray}
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m_1^{eq} = (j_x^2+j_y^2+j_z^2) - \alpha |\textbf{C}|, \\
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$$
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.. math::
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:nowrap:
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$$
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m_9^{eq} = (2j_x^2-j_y^2-j_z^2)+ \alpha \frac{|\textbf{C}|}{2}(2n_x^2-n_y^2-n_z^2), \\
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$$
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.. math::
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:nowrap:
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$$
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m_{11}^{eq} = (j_y^2-j_z^2) + \alpha \frac{|\textbf{C}|}{2}(n_y^2-n_z^2), \\
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$$
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.. math::
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:nowrap:
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$$
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m_{13}^{eq} = j_x j_y + \alpha \frac{|\textbf{C}|}{2} n_x n_y\;, \\
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$$
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.. math::
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:nowrap:
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$$
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m_{14}^{eq} = j_y j_z + \alpha \frac{|\textbf{C}|}{2} n_y n_z\;, \\
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$$
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.. math::
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:nowrap:
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$$
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m_{15}^{eq} = j_x j_z + \alpha \frac{|\textbf{C}|}{2} n_x n_z\;,
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$$
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where the color gradient is determined from the phase indicator field
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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\textbf{C}=\nabla \phi\;.
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\\end{equation}
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$$
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and the unit normal vector is
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.. math::
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:nowrap:
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$$
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\\begin{equation}
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\bm{n} = \frac{\textbf{C}}{|\textbf{C}|}\;.
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\\end{equation}
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$$
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****************************
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