Energy diffusion and dissipationΒΆ
The divergence of a second-order tensor is given by the sum of
\[\begin{split}\newcommand{\tmpa}[2]{
\frac{1}{H_{#1}}
\pder{H_{#1}}{X^{#2}}
\left(
S^{#1 #2}
+
S^{#2 #1}
\right)
}
\newcommand{\tmpb}[2]{
\frac{H_{#2}}{H_{#1} H_{#1}}
\pder{H_{#2}}{X^{#1}}
S^{#2 #2}
}
\newcommand{\tmpc}[2]{
\frac{1}{J}
\pder{}{X^{#2}}
\left(
J
S^{#1 #2}
\right)
}
\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmpc{1}{1}
+
\tmpc{1}{2}
+
\tmpc{1}{3}
\\
\tmpc{2}{1}
+
\tmpc{2}{2}
+
\tmpc{2}{3}
\\
\tmpc{3}{1}
+
\tmpc{3}{2}
+
\tmpc{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmpa{1}{1}
+
\tmpa{1}{2}
+
\tmpa{1}{3}
\\
\tmpa{2}{1}
+
\tmpa{2}{2}
+
\tmpa{2}{3}
\\
\tmpa{3}{1}
+
\tmpa{3}{2}
+
\tmpa{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
-
\tmpb{1}{1}
-
\tmpb{1}{2}
-
\tmpb{1}{3}
\\
-
\tmpb{2}{1}
-
\tmpb{2}{2}
-
\tmpb{2}{3}
\\
-
\tmpb{3}{1}
-
\tmpb{3}{2}
-
\tmpb{3}{3}
\end{pmatrix}.\end{split}\]
I consider the inner product of this vector and the velocity vector:
\[\vec{u}
=
\sum_i
U^i
\vec{E}_i.\]
The first part yields
\[\frac{1}{J}
\pder{}{X^1}
\left(
J
U^1
S^{1 1}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
U^1
S^{1 2}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
U^1
S^{1 3}
\right)
-
\pder{U^1}{X^1}
S^{1 1}
-
\pder{U^1}{X^2}
S^{1 2}
-
\pder{U^1}{X^3}
S^{1 3}\]
\[\frac{1}{J}
\pder{}{X^1}
\left(
J
U^2
S^{2 1}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
U^2
S^{2 2}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
U^2
S^{2 3}
\right)
-
\pder{U^2}{X^1}
S^{2 1}
-
\pder{U^2}{X^2}
S^{2 2}
-
\pder{U^2}{X^3}
S^{2 3}\]
\[\frac{1}{J}
\pder{}{X^1}
\left(
J
U^3
S^{3 1}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
U^3
S^{3 2}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
U^3
S^{3 3}
\right)
-
\pder{U^3}{X^1}
S^{3 1}
-
\pder{U^3}{X^2}
S^{3 2}
-
\pder{U^3}{X^3}
S^{3 3}\]
In each term, the first-three terms describe the diffusion of the energy, while the last-three terms are the dissipation of the energy.
The rest of the vector yields the sum of
\[S^{1 1}
\left(
\frac{2}{H_1}
\pder{H_1}{X^1}
U^1
-
\frac{H_1}{H_1 H_1}
\pder{H_1}{X^1}
U^1
-
\frac{H_1}{H_2 H_2}
\pder{H_1}{X^2}
U^2
-
\frac{H_1}{H_3 H_3}
\pder{H_1}{X^3}
U^3
\right)\]
\[S^{1 2}
\left(
\frac{1}{H_1}
\pder{H_1}{X^2}
U^1
+
\frac{1}{H_2}
\pder{H_2}{X^1}
U^2
\right)\]
\[S^{1 3}
\left(
\frac{1}{H_1}
\pder{H_1}{X^3}
U^1
+
\frac{1}{H_3}
\pder{H_3}{X^1}
U^3
\right)\]
\[S^{2 1}
\left(
\frac{1}{H_1}
\pder{H_1}{X^2}
U^1
+
\frac{1}{H_2}
\pder{H_2}{X^1}
U^2
\right)\]
\[S^{2 2}
\left(
\frac{2}{H_2}
\pder{H_2}{X^2}
U^2
-
\frac{H_2}{H_1 H_1}
\pder{H_2}{X^1}
U^1
-
\frac{H_2}{H_2 H_2}
\pder{H_2}{X^2}
U^2
-
\frac{H_2}{H_3 H_3}
\pder{H_2}{X^3}
U^3
\right)\]
\[S^{2 3}
\left(
\frac{1}{H_3}
\pder{H_3}{X^2}
U^3
+
\frac{1}{H_2}
\pder{H_2}{X^3}
U^2
\right)\]
\[S^{3 1}
\left(
\frac{1}{H_1}
\pder{H_1}{X^3}
U^1
+
\frac{1}{H_3}
\pder{H_3}{X^1}
U^3
\right)\]
\[S^{3 2}
\left(
\frac{1}{H_2}
\pder{H_2}{X^3}
U^2
+
\frac{1}{H_3}
\pder{H_3}{X^2}
U^3
\right)\]
\[S^{3 3}
\left(
\frac{2}{H_3}
\pder{H_3}{X^3}
U^3
-
\frac{H_3}{H_1 H_1}
\pder{H_3}{X^1}
U^1
-
\frac{H_3}{H_2 H_2}
\pder{H_3}{X^2}
U^2
-
\frac{H_3}{H_3 H_3}
\pder{H_3}{X^3}
U^3
\right)\]