[ex][basic][slides] Define darcys law as a velocity

slides/problem.md

@@ -14,8 +14,8 @@ Mass balance equations for two fluid phases:
 $$
 \begin{aligned}
 \frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t}
- -
- \nabla \cdot \boldsymbol{v}_\alpha
+ +
+ \nabla \cdot \left(\varrho_\alpha \boldsymbol{v}_\alpha \right)
  -
  q_\alpha = 0, \quad \alpha \in \lbrace w, n \rbrace.
 \end{aligned}
@@ -24,7 +24,7 @@ $$
 Momentum balance equations (multiphase-phase Darcy's law):
 $$
 \begin{aligned}
-\boldsymbol{v}_\alpha = \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\nabla\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right), \quad \alpha \in \lbrace w, n \rbrace.
+\boldsymbol{v}_\alpha = -\frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\nabla\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right), \quad \alpha \in \lbrace w, n \rbrace.
 \end{aligned}
 $$
 
@@ -59,7 +59,7 @@ $$
 $$
 
 * Constitutive relations: $p_n := p_w + p_c$, $p_c := p_c(S_w)$, $k_{r\alpha}$ = $k_{r\alpha}(S_w)$
-* Physical constraint (no free space): $S_w + S_n = 1$
+* Physical constraint: $S_w + S_n = 1$
 * Primary variables: $p_w$, $S_n$ (wetting phase pressure, non-wetting phase saturation)