diff --git a/slides/coupled_ff-pm.md b/slides/coupled_ff-pm.md
index 50e299abd8d2f1765cd207616c024dadcb1ebe8c..7644cc7c17b871c3397ec81db2c5047e48e30609 100644
--- a/slides/coupled_ff-pm.md
+++ b/slides/coupled_ff-pm.md
@@ -6,52 +6,47 @@ title: Coupled Free-Flow and Porous Media Flow Models in DuMu^x^
## Environmental and Agricultural Issues
-{style="width: 60%; margin: auto; "}
-<figcaption align = "center">
-<font size = "2">
-Fig.1 - Evaporation of soil water (Heck et al. (2020))<sup>1</sup>
-</font>
-</figcaption>
-
+:::::: {.columns}
+::: {.column width=65%}
+<img src="img/FFPM_radiation.gif"/>
+<small>Fig.1 - Evaporation of soil water (Heck et al. (2020))<sup>1</sup></small>
+:::
+::: {.column width=35%}
* Evaporation of soil water
* Soil salinization
* Underground storage (e.g. CO2, atomic waste)
+:::
+::::::
## Technical Issues
-{style="width: 45%; align: left;"}
-<figcaption align = "center">
-<font size = "2">
-Fig.2 - Filter (Schneider et al. (2023))<sup>2</sup>
-</font>
-</figcaption>
-
-* Fuel cells
-* Filters (e.g. air)
-* Heat exchangers (e.g. CPU cooling)
+:::::: {.columns}
+::: {.column width=50%}
+ 
+ <small style="text-align: center;">Fig.2 - Filter (Schneider et al. (2023))<sup>2</sup></small>
+:::
+::: {.column width=50%}
+ * Fuel cells
+ * Filters (e.g. air)
+ * Heat exchangers (e.g. CPU cooling)
+:::
+::::::
## Biological Issues
-{style="width: 25%; align: left;"}
-<figcaption align = "center">
-<font size = "2">
-Fig.3 - Brain tissue (Koch et al. (2020))<sup>3</sup>
-</font>
-</figcaption>
-
+:::::: {.columns}
+::: {.column width=28%}
+<img src="img/FFPM_braintissue.png"/>
+<small>Fig.3 - Brain tissue (Koch et al. (2020))<sup>3</sup></small>
+:::
+::: {.column width=50%}
* Brain tissue
* Leaf structure
+:::
+::::::
# Model Overview
-## Conceptual Physical Model
-<img src=img/FFPM-PhysicalModelOverview.png width="80%">
-<figcaption align = "center">
-<font size = "2">
-Fig.4 - Coupled dynamics at the soil-atmosphere interface (Photo: Edward Coltman)
-</font>
-</figcaption>
-
## Conceptual Physical Model
{style="width: 80%; align: left;"}
<figcaption align = "center">
@@ -60,55 +55,112 @@ Fig.5 - Exchange processes at the free-flow porous-medium interface at different
</font>
</figcaption>
-## Mathematical Model: Overview
+## Mathematical Model
-{style="width: 15%; margin: auto; float: left;"}
-
-<font size = "6">
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=85%}
**Free Flow:**
+<font size=5.9>
+
* Stokes / Navier-Stokes / RANS
* 1-phase, n-components, non-isothermal
+</font>
+
**Interface conditions:**
-* no thickness, no storage
-* local thermodynamic equilibrium
-* continuity of fluxes
-* continuity of state variables
+<font size=5.9>
+
+* no thickness, no storage, local thermal equilibrium
+* continuity of fluxes and state variables
+
+</font>
**Porous media:**
+<font size=5.9>
+
* Darcy / Forchheimer
* m-phases, n-components, non-isothermal
</font>
-## Mathematical Model: Free Flow
-<img src=img/FFPM-freeflowsymbol.png width="40%">
+:::::
+::::::
-## Mathematical Model: Free Flow
-* Momentum balance
+## Numerical Model
+<img src=img/FFPM-numericalmodel.png width="25%">
+<figcaption align = "center">
+<font size = "2">
+Fig.6 - Discretization scheme (Fetzer (2018))<sup>4</sup>
+</font>
+</figcaption>
+
+# Exercises
+
+## Exercise Tasks
+
+1. __Interface__
+ - Change flow direction
+ - Introduce slip condition
+ - Change shape of interface
+2. __Porous Medium Model__
+ - Use 2-phase multicomponent model
+ - Investigate and export water loss and visualize it
+3. __Free-Flow region__
+ - Introduce a turbulence model
+ - Use symmetry boundary conditions
+ - Apply grid refinement towards interface
+
+# <small> Supplementary Material</small> </br>Model equations
+## Eqs - Free Flow
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=85%}
+
+* Momentum balance (Navier-Stokes equation)
$$
-\frac{\partial \rho_g \textbf{v}_g}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \nabla \cdot (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) +\nabla \cdot (p_g\textbf{I})- \rho_g \textbf{g} = 0
+\frac{\partial \left(\rho_g \textbf{v}_g\right)}{\partial t} + \nabla \cdot (\rho_g \textbf{v}_g \textbf{v}_g^T) - \nabla \cdot \mathbf{\tau}_g +\nabla \cdot (p_g\textbf{I})- \rho_g \textbf{g} = 0
$$
* Component mass balance
$$
\frac{\partial \left(\rho_g X^\kappa_g\right)}{\partial t} + \nabla \cdot \left( \rho_g X^\kappa_g \textbf{v}_g + \mathbf{j}_{\text{diff}}^\kappa\right) - q^\kappa = 0
$$
+:::::
+::::::
+
+## Eqs - Free Flow
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=85%}
* Energy balance
$$
-\frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) + \sum_{\kappa} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff},t}^\kappa)} - \nabla \cdot ( (\lambda_{g} + \lambda_{t}) \nabla T) = 0
+\begin{aligned}
+\frac{\partial (\rho_g u_g) }{\partial t} + \nabla \cdot (\rho_g h_g \textbf{v}_g) &+ \sum_{\kappa} {\nabla \cdot (h_g^\kappa \textbf{j}_{\text{diff}}^\kappa)} \\ &- \nabla \cdot (\lambda_{g} \nabla T) = 0
+\end{aligned}
$$
-## Mathematical Model: Porous Medium Flow
-<img src=img/FFPM-pmfsymbol.png width="40%">
+:::::
+::::::
-## Mathematical Model: Porous Medium Flow
+## Eqs - Porous Medium Flow
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=90%}
* Darcy velocity (momentum balance)
$$
@@ -120,71 +172,117 @@ $$
\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
$$
-* Energy balance
+:::::
+::::::
+
+## Eqs - Porous Medium Flow
+
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=90%}
+
+* Total energy balance
$$
-\sum\limits_{\alpha \in \{\text{l, g} \}}\left(\phi\frac{\partial \left(\rho_\alpha S_\alpha u_\alpha\right)}{\partial t} + \nabla \cdot \left(\rho_\alpha h_\alpha \textbf{v}_\alpha \right)\right) + \left(1-\phi\right) \frac{\partial \left(\rho_s c_{p,s}T\right)}{\partial t} - \nabla\cdot \left(\lambda_{pm} \nabla T \right) = 0
+\begin{aligned}
+\sum\limits_{\alpha \in \{\text{l, g} \}} &\left(\phi\frac{\partial \left(\rho_\alpha S_\alpha u_\alpha\right)}{\partial t} + \nabla \cdot \left(\rho_\alpha h_\alpha \textbf{v}_\alpha \right)\right) \\
+&+ \left(1-\phi\right) \frac{\partial \left(\rho_s c_{p,s}T\right)}{\partial t} - \nabla\cdot \left(\lambda_{pm} \nabla T \right) = 0
+\end{aligned}
$$
+:::::
+::::::
+
-## Mathematical Model: Coupling Conditions
+## Eqs - Coupling Conditions
-<img src=img/FFPM-couplingsymbol.png width="30%">
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=85%}
-* Total mass condition
+* Continuity of total mass flux
$$
[(\rho_g \textbf{v}_g) \cdot \textbf{n}]^{\text{ff}} = - [(\rho_g \textbf{v}_g + \rho_w \textbf{v}_w) \cdot \textbf{n}]^{\text{pm}}
$$
-## Mathematical Model: Coupling Conditions
-<img src=img/FFPM-BJS.png width="30%">
-
-* Momentum (tangential) condition
+* Continuity of component flux
$$
-\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\}\,
+\begin{aligned}
+ &\left[(\rho_g X_g^\kappa \textbf{v}_g + \textbf{j}_{\text{diff}^\kappa}) \cdot \textbf{n}\right]^{\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}}\,
+\end{aligned}
$$
-## Mathematical Model: Coupling Conditions
+:::::
+::::::
+
+
+## Eqs - Coupling Conditions
+
+:::::: {.columns}
+::::: {.column width=15%}
+ <img src="img/FFPM-couplingsymbol.png">
+ <figure>
+ <img src="img/FFPM-BJS.png" alt="BJS Symbol">
+ <figcaption style="font-size: small; text-align: left;">Beaver-Joseph slip condition</figcaption>
+ </figure>
+:::::
+::::: {.column width=85%}
-* Momentum (normal) condition
+* Momentum condition in normal direction
$$
-[((\rho_g \textbf{v}_g \textbf{v}_g^T - (\mathbf{\tau}_g + \mathbf{\tau}_{g,t}) + p_g\textbf{I}) \textbf{n} )]^{\text{ff}} = [(p_g\textbf{I})\textbf{n}]^{\text{pm}}\,
+\left[((\rho_g \textbf{v}_g \textbf{v}_g^T - \mathbf{\tau}_g + p_g\textbf{I}) \textbf{n} )\right]^{\text{ff}} = \left[(p_g\textbf{I})\textbf{n}\right]^{\text{pm}}\,
$$
-* Component mass condition
+* Momentum condition in tangential direction
$$
-[(\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}}\,
+\begin{aligned}
+\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\}\,
+\end{aligned}
$$
-* Energy condition
+:::::
+::::::
+
+## Eqs - Coupling Conditions
+:::::: {.columns}
+::::: {.column width=15%}
+ 
+:::::
+::::: {.column width=85%}
+* Continuity of energy fluxes
+<font size = "5">
$$
-\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}}\,
+\begin{aligned}
+\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_\kappa h_\alpha^\kappa \textbf{j}_{\text{diff},\alpha}^\kappa) - \lambda_{\text{pm}}\nabla T\right)\cdot \textbf{n}\right]^{\text{pm}}\,
+\end{aligned}
$$
-
-## Numerical Model: Coupled Model
-<img src=img/FFPM-numericalmodel.png width="25%">
-<figcaption align = "center">
-<font size = "2">
-Fig.6 - Discretization scheme (Fetzer (2018))<sup>4</sup>
</font>
-</figcaption>
-
-# Example: Soil-Water Evaporation
+:::::
+::::::
-## Soil-Water Evaporation: Soil-Water Evaporation
-
-<img src=img/FFPM-TurbulentBoundaryLayer.png width="40%">
+# <small> Supplementary Material</small> </br>Example: Soil-Water Evaporation
## Example: Soil-Water Evaporation
-<img src=img/FFPM-SoilWaterEvapField.png width="40%">
+:::::: {.columns}
+::::: {.column width=50%}
+ <img src=img/FFPM-TurbulentBoundaryLayer.png width="80%">
+:::::
+::::: {.column width=50%}
+<img src=img/FFPM-SoilWaterEvapField.png width="80%">
<figcaption align = "center">
<font size = "2">
Fig.7 - Evaporation in the water cycle (Shahraeeni et al. (2012))<sup>5</sup>
</font>
</figcaption>
+:::::
+::::::
## Example: Soil-Water Evaporation
-
<img src=img/FFPM-evapStages.png width="60%">
<figcaption align = "center">
@@ -231,32 +329,6 @@ Fig.10 - Results: Evaporation from a simple setup (Fetzer (2018))<sup>4</sup>
</font>
</figcaption>
-# Exercises
-
-## Exercise: Interface
-
-_Tasks_
-
-- Change flow direction for a tangential flow as opposed to the original-normal flow
-- Introduce the Beavers-Joseph-tangential-flow interface condition
-- Redevelop the grid and introduce an undulating interface
-- Change the inflow boundary condition to a velocity profile
-
-## Exercise: Models
-
-_Tasks_
-
-- Modify the model to use a 2-phase multicomponent model in the porous medium
-- Experiment with various data output types: `.csv` and `.json`
-- Visualize with various visualization tools: `gnuplot` and `matplotlib`
-
-## Exercise: Turbulence
-
-_Tasks_
-
-- Introduce a turbulence model to the free-flow domain
-- Reduce the free-flow domain by using a symmetry condition at the upper domain boundary
-- Vary grid resolution and perform a qualitative grid convergence test
# References