Merge branch 'fix/ff-pm-slides' into 'master'

Fix/ff pm slides

See merge request !215

slides/coupled_ff-pm.md

 ---
-title: Coupled Free-Flow and Porous Media Flow Models in DuMu<sup>X</sup>
+title: Coupled Free-Flow and Porous Media Flow Models in DuMu^x^
 ---
 
 # Motivation
@@ -15,7 +15,7 @@ Fig.1 - Evaporation of soil water (Heck et al. (2020))<sup>1</sup>
 
 * Evaporation of soil water
 * Soil salinization
-* Underground storage (e.g. CO2, atmoic waste)
+* Underground storage (e.g. CO2, atomic waste)
 
 ## Technical Issues
 
@@ -66,12 +66,12 @@ Fig.5 - Exchange processes at the free-flow porous-medium interface at different
 
 <font size = "6">
 
-**Freeflow:**
+**Free Flow:**
 
 * Stokes / Navier-Stokes / RANS
-* 1-phase, n-components, non-equilibrium
+* 1-phase, n-components, non-isothermal
 
-**Interface condtions:**
+**Interface conditions:**
 
 * no thickness, no storage
 * local thermodynamic equilibrium
@@ -80,29 +80,29 @@ Fig.5 - Exchange processes at the free-flow porous-medium interface at different
 
 **Porous media:**
 
-* Darcy/ Forchheimer / Richards
+* Darcy / Forchheimer
 * m-phases, n-components, non-isothermal
 
 </font>
 
-## Mathematical Model: Freeflow
+## Mathematical Model: Free Flow
 <img src=img/FFPM-freeflowsymbol.png width="40%">
 
-## Mathematical Model: Freeflow
+## Mathematical Model: Free Flow
 
-* Total mass balance
+* Momentum balance
 $$
 \frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \nabla \cdot (\mathbf{\tau}_g + \mathbf{\tau}_{g,t})  +\nabla \cdot (p_g\textbf{I})- \rho_g \textbf{g} = 0
 $$
 
-* Momentum balance
+* Component mass balance
 $$
-\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t} + \nabla \cdot \left( \rho_g \textbf{v}_g X^\kappa_g - \mathbf{j}_{\text{diff}}^\kappa\right) - q^\kappa = 0
+\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t} + \nabla \cdot \left( \rho_g X^\kappa_g \textbf{v}_g + \mathbf{j}_{\text{diff}}^\kappa\right) - q^\kappa = 0
 $$
 
-* Component mass balance
+* Energy balance
 $$
-\frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) + \sum_{i} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \nabla \cdot ( (\lambda_{g} + \lambda_{t}) \nabla T) = 0
+\frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) + \sum_{\kappa} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \nabla \cdot ( (\lambda_{g} + \lambda_{t}) \nabla T) = 0
 $$
 
 ## Mathematical Model: Porous Medium Flow
@@ -110,14 +110,14 @@ $$
 
 ## Mathematical Model: Porous Medium Flow
 
-* Component mass balance
+* Darcy velocity (momentum balance)
 $$
-\sum\limits_{\alpha \in \{\text{l, g} \}} \left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \textbf{v}_\alpha\rho_\alpha X_\alpha^\kappa + \sum_\kappa \nabla \cdot \left(  \textbf{D}_{pm,\alpha}^\kappa\rho_\alpha\nabla X_\alpha^\kappa \right)\right) = 0
+\textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right)
 $$
 
-* Darcy velocity
+* Component mass balance
 $$
-\textbf{v}_\alpha = - \frac{k_{r,\alpha}}{\mu_\alpha} K \left(\nabla p_\alpha - \rho_\alpha \textbf{g}\right)
+\sum\limits_{\alpha \in \{\text{l, g} \}} \left(\phi \frac{\partial \left(\rho_\alpha S_\alpha X_\alpha^\kappa\right)}{\partial t } + \nabla \cdot \rho_\alpha X_\alpha^\kappa \textbf{v}_\alpha + \nabla \cdot \mathbf{j}_{\text{diff}}^\kappa\right) = 0
 $$
 
 * Energy balance
@@ -138,9 +138,9 @@ $$
 ## Mathematical Model: Coupling Conditions
 <img src=img/FFPM-BJS.png width="30%">
 
-* Momentum (tangential)condition
+* Momentum (tangential) condition
 $$
-\left[\left(- \textbf{v}_g - \frac{\sqrt{(\textbf{K}\textbf{t}_i)\cdot \textbf{t}_i}}{\alpha_{BJ}} (\nabla \textbf{v}_g + \nabla \textbf{v}_g^T)\textbf{n} \right) \cdot \textbf{t}_i \right]^{\text{ff}} = 0\, , \quad i \in \{1, .. ,\, d-1\}\,
+\left[\left(- \textbf{v}_g - \frac{\sqrt{(\textbf{K}\textbf{t}_i)\cdot \textbf{t}_i}}{\alpha_{\mathrm{BJ}}} (\nabla \textbf{v}_g + \nabla \textbf{v}_g^T)\textbf{n} \right) \cdot \textbf{t}_i \right]^{\text{ff}} = 0\, , \quad i \in \{1, .. ,\, d-1\}\,
 $$
 
 ## Mathematical Model: Coupling Conditions
@@ -152,12 +152,12 @@ $$
 
 * Component mass condition
 $$
-[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}, t}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\,
+[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}^\kappa}) \cdot \textbf{n}]^{\text{ff}} = - \left[\left( \sum_{\alpha} (\rho_{\alpha} X_{\alpha}^\kappa \textbf{v}_\alpha + \textbf{j}^\kappa_{\text{diff}, \alpha})\right) \cdot \textbf{n}\right]^{\text{pm}}\,
 $$
 
 * Energy condition
 $$
-\left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa + \lambda_{g}\nabla T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_i h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\nabla T\right)\cdot \textbf{n}\right]^{\text{pm}}\,
+\left[\left(\rho_g h_g \textbf{v}_g + \sum_i h_g^\kappa \textbf{j}_{\text{diff},g}^\kappa - (\lambda_{g} + \lambda_{t})\nabla T\right)\cdot \textbf{n}\right]^{\text{ff}} = - \left[\left( \sum_\alpha (\rho_\alpha h_\alpha \textbf{v}_\alpha + \sum_\kappa h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\nabla T\right)\cdot \textbf{n}\right]^{\text{pm}}\,
 $$
 
 ## Numerical Model: Coupled Model