[example][1d3d] Add advection term in formula

examples/embedded_network_1d3d/README.md

@@ -43,7 +43,7 @@ We solve the following coupled, mixed-dimensional PDE system:
 ```math
 \begin{align}
 	 \frac{ \partial (\phi_\mathrm{T} \varrho_\mathrm{T} x_\mathrm{T})}{\partial t} - \nabla\cdot \left( \phi_\mathrm{T} D_{\text{app},\mathrm{T}} \varrho_\mathrm{T} \nabla x_\mathrm{T} \right) &= \hat{q} \Phi_\Lambda & \text{in} \quad \Omega, \\
-	 \frac{\partial (A_\mathrm{B} \varrho_\mathrm{B} x_\mathrm{B})}{\partial t} - \frac{\partial}{\partial s}  \left( A_\mathrm{B} D_{\text{app},A_\mathrm{B}} \varrho_\mathrm{B} \frac{ \partial x_\mathrm{B}}{\partial s} \right) &= -\hat{q} & \text{on} \quad \Lambda, \\
+	 \frac{\partial (A_\mathrm{B} \varrho_\mathrm{B} x_\mathrm{B})}{\partial t} + \frac{\partial}{\partial s}  \left(A_\mathrm{B}v_\mathrm{B}\varrho_\mathrm{B}x_\mathrm{B} - A_\mathrm{B} D_{\text{app},A_\mathrm{B}} \varrho_\mathrm{B} \frac{ \partial x_\mathrm{B}}{\partial s} \right) &= -\hat{q} & \text{on} \quad \Lambda, \\
      \hat{q} &= - \int_P C_M D \bar{\varrho} ( x_\mathrm{T} - \Pi x_\mathrm{B}) \mathrm{d}\zeta,
 \end{align}
 ```
@@ -52,6 +52,7 @@ where the subscript T and B denote the tissue and the network (blood flow) compa
 $`x`$ is the tracer mole fraction, $`\varrho`$ the molar density of the mixture, $`\phi`$ is the porosity,
 $`A_\mathrm{B}`$ denotes the network (vessel lumen) cross-sectional area, $`P`$ denotes the cross-sectional perimeter,
 $`D`$ is the free diffusion coefficient, $`D_{\text{app}}`$ apparent diffusion coefficients and $C_M$ a membrane diffusivity factor.
+$`Q_\mathrm{B} := A_\mathrm{B}v_\mathrm{B}`$ is a given blood flow field transporting the tracer by advection.
 Furthermore, isothermal conditions with a homogeneous temperature distribution of constant $`T=37^\circ C`$ are assumed.
 The 1D network PDE is formulated in terms of the local axial coordinate $`s`$.
 

examples/embedded_network_1d3d/doc/_intro.md

@@ -41,7 +41,7 @@ We solve the following coupled, mixed-dimensional PDE system:
 ```math
 \begin{align}
 	 \frac{ \partial (\phi_\mathrm{T} \varrho_\mathrm{T} x_\mathrm{T})}{\partial t} - \nabla\cdot \left( \phi_\mathrm{T} D_{\text{app},\mathrm{T}} \varrho_\mathrm{T} \nabla x_\mathrm{T} \right) &= \hat{q} \Phi_\Lambda & \text{in} \quad \Omega, \\
-	 \frac{\partial (A_\mathrm{B} \varrho_\mathrm{B} x_\mathrm{B})}{\partial t} - \frac{\partial}{\partial s}  \left( A_\mathrm{B} D_{\text{app},A_\mathrm{B}} \varrho_\mathrm{B} \frac{ \partial x_\mathrm{B}}{\partial s} \right) &= -\hat{q} & \text{on} \quad \Lambda, \\
+	 \frac{\partial (A_\mathrm{B} \varrho_\mathrm{B} x_\mathrm{B})}{\partial t} + \frac{\partial}{\partial s}  \left(A_\mathrm{B}v_\mathrm{B}\varrho_\mathrm{B}x_\mathrm{B} - A_\mathrm{B} D_{\text{app},A_\mathrm{B}} \varrho_\mathrm{B} \frac{ \partial x_\mathrm{B}}{\partial s} \right) &= -\hat{q} & \text{on} \quad \Lambda, \\
      \hat{q} &= - \int_P C_M D \bar{\varrho} ( x_\mathrm{T} - \Pi x_\mathrm{B}) \mathrm{d}\zeta,
 \end{align}
 ```
@@ -50,6 +50,7 @@ where the subscript T and B denote the tissue and the network (blood flow) compa
 $`x`$ is the tracer mole fraction, $`\varrho`$ the molar density of the mixture, $`\phi`$ is the porosity,
 $`A_\mathrm{B}`$ denotes the network (vessel lumen) cross-sectional area, $`P`$ denotes the cross-sectional perimeter,
 $`D`$ is the free diffusion coefficient, $`D_{\text{app}}`$ apparent diffusion coefficients and $C_M$ a membrane diffusivity factor.
+$`Q_\mathrm{B} := A_\mathrm{B}v_\mathrm{B}`$ is a given blood flow field transporting the tracer by advection.
 Furthermore, isothermal conditions with a homogeneous temperature distribution of constant $`T=37^\circ C`$ are assumed.
 The 1D network PDE is formulated in terms of the local axial coordinate $`s`$.