doc equations updated

doc/2fluids.tex

@@ -93,7 +93,7 @@
 \newcommand{\nb}{\bm{\nabla}}
 \newcommand{\ub}{\bm{u}_k}
 \newcommand{\ib}{\bm{I}}
-\newcommand{\sigmab}{\bm{\sigma}_k}
+\newcommand{\taub}{\bm{\tau}_k}
 \newcommand{\gb}{\bm{g}}
 \newcommand{\dpartial}[2]{\dfrac{\partial#1}{\partial#2}}
 
@@ -105,7 +105,7 @@ Balance equations resulting from applying Navier-Stokes equations to a control v
 N_f\times\left\lbrace\begin{aligned}
 \rho_k\dpartial{\ub}{t}+\rho_k\nb\cdot\left(\dfrac{\ub\ub}{q_k}\right)\\
 \dpartial{q}{t}+\nb\cdot\ub+\varepsilon\nb\cdot \bm{r}_k\end{aligned}\right.&\begin{aligned}
-&=\nb\cdot\sigmab q_k+q_k\rho_k \gb\phantom{\dpartial{\ub}{t}}\\
+&=-q_k\nb p+\nb\cdot\taub +q_k\rho_k \gb\phantom{\dpartial{\ub}{t}}\\
 &=0\end{aligned}&\begin{aligned}\phantom{\dpartial{\ub}{t}}\text{phase momentum equations}\\
 \phantom{\dpartial{q}{t}}\text{phase mass equations}\end{aligned}\\
 1-c-\sum_{k=1}^{N_f}q_k&\begin{aligned}&=0\phantom{\dpartial{q}{t}}\end{aligned}&\begin{aligned}\text{global volume conservation equation}\end{aligned}
@@ -113,11 +113,11 @@ N_f\times\left\lbrace\begin{aligned}
 
 where $\bm{r}_k$ is the residual of the momentum equation used to stabilised the $\mathbb{P}_1-\mathbb{P}_1$ finite element formulation and the Cauchy stress tensor of the fluid phase $k$ can be written:
 
-\[\sigmab=-p\ib+\mu_k\left(\nb\dfrac{\ub}{q}+\left(\nb\dfrac{\ub}{q}\right)^T\right)\]
+\[\taub=\mu_k\left(\nb\dfrac{\ub}{q}+\left(\nb\dfrac{\ub}{q}\right)^T\right)\]
 
 It is important to note that the velocity appearing in the above equations is the mean velocity of the fluid phase under consideration and can be written as the product of the point phase velocity and the volume fraction of the fluid phase. The resiudal stabilisation introduces an unknown parameter $\varepsilon$ that could change the solution. The Pressure Stabilisation Petrov-Galerkin (PSPG) is the simplest way to stabilise the finite element formulation by linking all the unknown together to prevent the development of high frequency pressure modes. PSPG method of order one consists in only keeping the normal stress part of the Cauchy stress tensor:
 
-\[\bm{r}_k=-\nb\cdot q_k p\ib \]
+\[\bm{r}_k=-q_k\nb p+q_k\rho_k\bm{g} \]
 
 The weak form of these equations is obtained by integrating it multiplied by some test function over the whole domain:
 
@@ -125,21 +125,26 @@ The weak form of these equations is obtained by integrating it multiplied by som
 N_f\times\left\lbrace\begin{aligned}
 \langlC\rho_k\dpartial{\ub}{t}\psi_i\ranglC+\langlC\rho_k\nb\cdot\left(\dfrac{\ub\ub}{q_k}\right)\psi_i\ranglC\\
 \langlC\dpartial{q}{t}\psi_i\ranglC+\langlC\nb\cdot\ub\psi_i\ranglC+\langlC\varepsilon\nb\cdot\bm{r}_k\ranglC\end{aligned}\right.&\begin{aligned}
-&=\langlC\nb\cdot\sigmab q_k\psi_i\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
+&=-\langlC q_k\nb p \psi_i\ranglC+\langlC\nb\cdot\taub q_k\psi_i\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
 &=0\phantom{\dpartial{q}{t}}\end{aligned}\\
 \langlC \left(1-c-\sum_{k=1}^{N_f}q_k\right)\psi_i\ranglC&\begin{aligned}&=0\end{aligned}
 \end{align*}
 
 These equations can be simplified by integrating by part and applying the divergence theorem to the flux terms:
 
+\begin{scriptsize}
 \begin{align*}
 N_f\times\left\lbrace\begin{aligned}
 \langlC\rho_k\dpartial{\ub}{t}\psi_i\ranglC+\langlC\rho_k\nb\cdot\left(\dfrac{\ub\ub}{q_k}\right)\psi_i\ranglC\\
+\\
 \langlC\dpartial{q}{t}\psi_i\ranglC+\langlC\nb\cdot\ub\psi_i\ranglC\end{aligned}\right.&\begin{aligned}
-&=-\langlC\sigmab\cdot\nb q_k\psi_i\ranglC+\langlC\langlC\sigmab q_k\psi_i\cdot\bm{n}\ranglC\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
+&=\langlC p\nb q_k\psi_i\ranglC-\langlC\langlC pq_k\psi_i\bm{n}\ranglC\ranglC-\langlC\taub\cdot\nb q_k\psi_i\ranglC+\langlC\langlC\taub q_k\psi_i\cdot\bm{n}\ranglC\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
+\\
 &=\langlC\varepsilon\bm{r}_k\cdot\nb\psi_i\ranglC+\langlC\langlC\varepsilon\bm{r}_k\psi_i\cdot\bm{n}\ranglC\ranglC\phantom{\dpartial{q}{t}}\end{aligned}\\
+\\
 \langlC \left(1-c-\sum_{k=1}^{N_f}q_k\right)\psi_i\ranglC&\begin{aligned}&=0\end{aligned}
 \end{align*}
+\end{scriptsize}
 
 The fully developed 2D equations considering that the vertical velocity is $v$ while the horizontal velocity is $u$ are given by:
 
@@ -157,12 +162,12 @@ $\langlC\bm{f}_{q_p}^0,\psi_i\ranglC$&&&$=0$\\[1em]
 
 
 \begin{align*}
-\bm{f}_{\bm{u}}^0&=\rho_k\left( \ub+ \ub\dfrac{\nb\cdot\ub}{q}+\dfrac{\bm{A}_k\cdot\ub}{q}\right)-p\ib\cdot\nb q_k+\textcolor{red}{\mu\dfrac{\bm{B}_k\cdot\nb q_k}{q_k}}-q_k\rho_k\bm{g}\\[1em]
+\bm{f}_{\bm{u}}^0&=\rho_k\left( \ub+ \ub\dfrac{\nb\cdot\ub}{q}+\dfrac{\bm{A}_k\cdot\ub}{q}\right)-p\ib\cdot\nb q_k+\mu\dfrac{\bm{B}_k\cdot\nb q_k}{q_k}-q_k\rho_k\bm{g}\\[1em]
 \bm{f}_{\bm{u}}^1&= -q_kp\ib+\mu \bm{B}_k\\[1em]
 \bm{f}_{\bm{ub}}&=\bm{B}_k\cdot\bm{n}_k\\[1em]
 \bm{f}_{q_k}^0&=\dpartial{q}{t}+\nb\cdot\ub\\[1em]
-\bm{f}_{q_k}^1&=\varepsilon\left(\textcolor{green!70!black}{p\ib\cdot\nb q_k}+q_k\nb p\cdot\ib\right)\\[1em]
-\bm{f}_{q_k b}&=\varepsilon\left(p\ib\cdot\nb q_k+q_k\nb p\cdot\ib\right)\cdot\bm{n}_k\\[1em]
+\bm{f}_{q_k}^1&=\varepsilon\left( q_k\nb p\cdot\ib-q_k\rho_k \bm{g}\right)\\[1em]
+\bm{f}_{q_k b}&=\varepsilon \left(q_k\nb p\cdot\ib\cdot\bm{n}_k-q_k\rho_k\bm{g}\right)\\[1em]
 \bm{f}_{q_p}^0&=1-c-\sum_{k=1}^{N_f}q_k
 \end{align*}