diff --git a/examples/embedded_network_1d3d/README.md b/examples/embedded_network_1d3d/README.md index e100ab694451e75261ca329c2a38ce7f7df3d20b..599eb3565c134e4f65fcd8c161abc2d4761f927c 100644 --- a/examples/embedded_network_1d3d/README.md +++ b/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 $`s`$. diff --git a/examples/embedded_network_1d3d/doc/_intro.md b/examples/embedded_network_1d3d/doc/_intro.md index 8b0348d7849c90dd0756e4cb18740a867b96d265..32aaf2b9037763edb8f522524d208843793414d5 100644 --- a/examples/embedded_network_1d3d/doc/_intro.md +++ b/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 $`s`$.