From 56a69ce13ac8bb7117a102cfa095cbc94b23b7ee Mon Sep 17 00:00:00 2001
From: Thomas Fischer <thomas.fischer@ufz.de>
Date: Thu, 10 Oct 2019 10:51:50 +0200
Subject: [PATCH] [web] Corrections in heat conduction neumann docu.
---
.../heatconduction/heatconduction-neumann.pandoc | 14 +++++---------
1 file changed, 5 insertions(+), 9 deletions(-)
diff --git a/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc b/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc
index 8b7dd7aa2f8..570d50b144c 100644
--- a/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc
+++ b/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc
@@ -25,32 +25,28 @@ $$
\eqalign{
T(x, t=0) = T_0,\cr
T(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr
-k{\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
+\lambda {\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
}
$$
where $T$ could be temperature, the subscripts $D$ and $N$ denote the Dirichlet- and Neumann-type boundary conditions, $n$ is the normal vector pointing outside of $\Omega$, and $\Gamma = \Gamma_D \cup \Gamma_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$.
## Problem specification and analytical solution
-We solve the Parabolic equation on a line domain $[60\times 1]$ with $k = 3.2$ and $\rho C_\textrm{p} = 2.5e10^6$ w.r.t. the specific boundary conditions:
+We solve the parabolic equation on a line domain $[60\times 1]$ with $\lambda = 3.2$ and $\rho C_\textrm{p} = 2.5 \times 10^6$ w.r.t. the specific boundary conditions:
$$
\eqalign{
-k {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr
-k {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N.
+\lambda {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr
+\lambda {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N.
}
$$
The solution of this problem is
-
-TODO: is not rendered correct
-
$$
\begin{equation}
-T(x,t) = \frac{2q}{\lambda}\left((\frac{\alpha t}{\pi})^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}( \frac{x}{2\sqrt{\alpha t}})\right),
+T(x,t) = \frac{2q}{\lambda}\left(\left(\frac{\alpha t}{\pi}\right)^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}\left( \frac{x}{2\sqrt{\alpha t}}\right)\right),
\end{equation}
$$
-
where $T_\textrm{b}$ is the boundary temperature, $\textrm{erfc}$ is the complementary error function and $\alpha = \lambda/(C_p \rho)$ is the thermal diffusivity.
## Input files
--
GitLab