added color model formulation

docs/source/userGuide/models/color/index.rst

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