[docu][staggered] Correct equations

doc/docextra/staggered/staggeredgrid_dumux.tex

@@ -25,13 +25,13 @@
 \subsection{Mass balance equation}
 \begin{equation}
  \begin{alignedat}{3}
- \frac{\partial \varrho}{\partial t} &+ \nabla \cdot (\varrho \textbf{v}) &&- q_{\textup{p}} &= 0 \\[1em]
- \int_{\Omega} \frac{\partial \varrho}{\partial t} \text{d} \Omega &+ \int_{\Omega} \nabla \cdot (\varrho \textbf{v}) \text{d} \Omega &&- \int_{\Omega} q_{\textup{p}} \text{d} \Omega &= 0 \\[1em]
- \int_{\Omega} \frac{\partial \varrho}{\partial t} \text{d} \Omega &+ \int_{\partial \Omega} (\varrho \textbf{v}) \cdot \textbf{n} \text{d} \Gamma_{\Omega} &&- \int_{\Omega} q_{\textup{p}} \text{d} \Omega &= 0 \\[1em]
- \int_{\Omega} \frac{\partial \varrho}{\partial t} \text{d} \Omega &+ \int_{\partial \Omega} \begin{pmatrix}\varrho u \\ \varrho v\end{pmatrix} \cdot \begin{pmatrix}n_1 \\ n_2\end{pmatrix} \text{d} \Gamma_{\Omega} 
-   &&- \int_{\Omega} q_{\textup{p}} \text{d} \Omega &= 0 \\[1em]
- \int_{\Omega} \frac{\partial \varrho}{\partial t} \text{d} \Omega &+ \int_{\partial \Omega} (\varrho u n_1) + (\varrho v n_2) \text{d} \Gamma_{\Omega} 
-   &&- \int_{\Omega} q_{\textup{p}} \text{d} \Omega &= 0
+ \frac{\partial \varrho}{\partial t} &+ \nabla \cdot (\varrho \textbf{v}) &&- q &= 0 \\[1em]
+ \int_{V} \frac{\partial \varrho}{\partial t} \text{d} V &+ \int_{V} \nabla \cdot (\varrho \textbf{v}) \text{d} V &&- \int_{V} q \text{d} V &= 0 \\[1em]
+ \int_{V} \frac{\partial \varrho}{\partial t} \text{d} V &+ \int_{\partial V} (\varrho \textbf{v}) \cdot \textbf{n} \text{d} \Gamma_{V} &&- \int_{V} q \text{d} V &= 0 \\[1em]
+ \int_{V} \frac{\partial \varrho}{\partial t} \text{d} V &+ \int_{\partial V} \begin{pmatrix}\varrho u \\ \varrho v\end{pmatrix} \cdot \begin{pmatrix}n_1 \\ n_2\end{pmatrix} \text{d} \Gamma_{V}
+   &&- \int_{V} q \text{d} V &= 0 \\[1em]
+ \int_{V} \frac{\partial \varrho}{\partial t} \text{d} V &+ \int_{\partial V} [(\varrho u n_1) + (\varrho v n_2)] \text{d} \Gamma_{V}
+   &&- \int_{V} q \text{d} V &= 0
 \end{alignedat}
 \end{equation}
 
@@ -39,13 +39,13 @@
 \begin{equation}
  \begin{alignedat}{6}
   \frac{\partial (\varrho \textbf{v})}{\partial t} &+ \nabla \cdot (\varrho \textbf{v} \textbf{v}^{\textup{T}}) &&- \nabla \cdot (\mu (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) \\
-    &+ \nabla p &&- \varrho \textbf{g} &&- q_{\textup{v}} &&= 0 \\[2em]
-  \int_{\Omega} \frac{\partial (\varrho \textbf{v})}{\partial t} \text{d} \Omega &+ \int_{\Omega} \nabla \cdot (\varrho \textbf{v} \textbf{v}^{\textup{T}}) \text{d} \Omega 
-    &&- \int_{\Omega} \nabla \cdot (\mu (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) \text{d} \Omega \\
-    &+ \int_{\Omega} \nabla p \text{d} \Omega &&- \int_{\Omega} \varrho \textbf{g} \text{d} \Omega &&- \int_{\Omega} q_{\textup{v}} \text{d} \Omega &&= 0 \\[2em]
-  \int_{\Omega} \frac{\partial (\varrho \textbf{v})}{\partial t} \text{d} \Omega &+ \int_{\partial \Omega} (\varrho \textbf{v} \textbf{v}^{\textup{T}}) \cdot \textbf{n} \text{d} \Gamma_{\Omega}
-    &&- \int_{\partial \Omega} (\mu (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) \cdot \textbf{n} \text{d} \Gamma_{\Omega} \\
-    &+ \int_{\partial \Omega} p \text{d} \Gamma_{\Omega} &&- \int_{\Omega} \varrho \textbf{g} \text{d} \Omega &&- \int_{\Omega} q_{\textup{v}} \text{d} \Omega &&= 0     
+    &+ \nabla p &&- \varrho \textbf{g} &&- \textbf{f} &&= 0 \\[2em]
+  \int_{V} \frac{\partial (\varrho \textbf{v})}{\partial t} \text{d} V &+ \int_{V} \nabla \cdot (\varrho \textbf{v} \textbf{v}^{\textup{T}}) \text{d} V
+    &&- \int_{V} \nabla \cdot (\mu (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) \text{d} V \\
+    &+ \int_{V} \nabla p \text{d} V &&- \int_{V} \varrho \textbf{g} \text{d} V &&- \int_{V} \textbf{f} \text{d} V &&= 0 \\[2em]
+  \int_{V} \frac{\partial (\varrho \textbf{v})}{\partial t} \text{d} V &+ \int_{\partial V} (\varrho \textbf{v} \textbf{v}^{\textup{T}}) \cdot \textbf{n} \text{d} \Gamma_{V}
+    &&- \int_{\partial V} (\mu (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) \cdot \textbf{n} \text{d} \Gamma_{V} \\
+    &+ \int_{\partial V} p \textbf{n} \text{d} \Gamma_{V} &&- \int_{V} \varrho \textbf{g} \text{d} V &&- \int_{V} \textbf{f} \text{d} V &&= 0
  \end{alignedat}
 \end{equation}
 
@@ -61,18 +61,18 @@
 x-direction:
 \begin{equation}
 \begin{split}
-  \int_{\Omega} \frac{\partial (\varrho u)}{\partial t} \text{d} \Omega + \int_{\partial \Omega} (\varrho u u)n_1 + (\varrho u v)n_2 \text{d} \Gamma_{\Omega} \\[1em]
-    - \int_{\partial \Omega} \left(\mu (2\frac{\partial u}{\partial x})n_1 + \mu (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})n_2 \right) \text{d} \Gamma_{\Omega} \\[1em]
-    + \int_{\partial \Omega} p \text{d} \Gamma_{\Omega} - \int_{\Omega} \varrho g_1 \text{d} \Omega - \int_{\Omega} q_{\textup{v}} \text{d} \Omega = 0     
+  \int_{V} \frac{\partial (\varrho u)}{\partial t} \text{d} V + \int_{\partial V} (\varrho u u)n_1 + (\varrho u v)n_2 \text{d} \Gamma_{V} \\[1em]
+    - \int_{\partial V} \left(\mu (2\frac{\partial u}{\partial x})n_1 + \mu (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})n_2 \right) \text{d} \Gamma_{V} \\[1em]
+    + \int_{\partial V} p n_1 \text{d} \Gamma_{V} - \int_{V} \varrho g_1 \text{d} V - \int_{V} f_1 \text{d} V = 0
 \end{split}
 \end{equation}
 
 y-direction:
 \begin{equation}
 \begin{split}
-  \int_{\Omega} \frac{\partial (\varrho v)}{\partial t} \text{d} \Omega + \int_{\partial \Omega} (\varrho v u)n_1 + (\varrho v v)n_2 \text{d} \Gamma_{\Omega} \\[1em]
-    - \int_{\partial \Omega} \left(\mu (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})n_1 + \mu (2\frac{\partial v}{\partial y})n_2\right) \text{d} \Gamma_{\Omega} \\[1em]
-    + \int_{\partial \Omega} p \text{d} \Gamma_{\Omega} - \int_{\Omega} \varrho g_2 \text{d} \Omega - \int_{\Omega} q_{\textup{v}} \text{d} \Omega = 0     
+  \int_{V} \frac{\partial (\varrho v)}{\partial t} \text{d} V + \int_{\partial V} (\varrho v u)n_1 + (\varrho v v)n_2 \text{d} \Gamma_{V} \\[1em]
+    - \int_{\partial V} \left(\mu (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})n_1 + \mu (2\frac{\partial v}{\partial y})n_2\right) \text{d} \Gamma_{V} \\[1em]
+    + \int_{\partial V} p n_2 \text{d} \Gamma_{V} - \int_{V} \varrho g_2 \text{d} V - \int_{V} f_2 \text{d} V = 0
 \end{split}
 \end{equation}