Normalised basis¶
Description¶
Velocity-gradient tensor¶
\[\left(
\sum_j
\frac{1}{H_j}
\vec{\hat{E}}_j
\pder{}{X^j}
\right)
\otimes
\left(
\sum_i
\vec{\hat{E}}_i
\hat{U}^i
\right)
=
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\pder{\vec{\hat{E}}_i}{X^j}
\right)
\frac{1}{H_j}
\hat{U}^i
+
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\vec{\hat{E}}_i
\right)
\frac{1}{H_j}
\pder{\hat{U}^i}{X^j}.\]
By using the changes of the normalised basis vectors:
\[\dehatdx{i}{j}{k},\]
the first term yields
\[+
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\frac{1}{H_i}
\pder{H_j}{X^i}
\vec{\hat{E}}_j
\right)
\frac{1}{H_j}
\hat{U}^i
-
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\sum_k
\delta_{ij}
\frac{1}{H_k}
\pder{H_i}{X^k}
\vec{\hat{E}}_k
\right)
\frac{1}{H_j}
\hat{U}^i.\]
By doing arithmetic and simplifying (e.g. interchanging the free indices) the relation, this leads to
\[+
\sum_j
\left(
\vec{\hat{E}}_j
\otimes
\vec{\hat{E}}_j
\right)
\sum_k
\frac{1}{H_j}
\frac{1}{H_k}
\pder{H_j}{X^k}
\hat{U}^k
-
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\vec{\hat{E}}_i
\right)
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_j}{X^i}
\hat{U}^j.\]
Thus the conclusive relation is, written explicitly, as follows.
\[\begin{split}\newcommand{\normal}[2]{
\frac{1}{H_#2} \pder{\hat{U}^#1}{X^#2}
}
\newcommand{\diag}[2]{
+
\frac{1}{H_#1 H_#2} \pder{H_#2}{X^#1} \hat{U}^#1
}
\newcommand{\nondiag}[2]{
-
\frac{1}{H_#1 H_#2} \pder{H_#2}{X^#1} \hat{U}^#2
}
&
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\normal{1}{1} & \normal{2}{1} & \normal{3}{1} \\
\normal{1}{2} & \normal{2}{2} & \normal{3}{2} \\
\normal{1}{3} & \normal{2}{3} & \normal{3}{3} \\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix} \\
&
+ \\
&
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\diag{2}{1}
\diag{3}{1}
&
\nondiag{2}{1}
&
\nondiag{3}{1}
\\
\nondiag{1}{2}
&
\diag{1}{2}
\diag{3}{2}
&
\nondiag{3}{2}
\\
\nondiag{1}{3}
&
\nondiag{2}{3}
&
\diag{1}{3}
\diag{2}{3}
\\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix}.\end{split}\]
Strain-rate tensor¶
The strain-rate tensor is defined as the symmetric part of it:
\[\begin{split}\newcommand{\nondiag}[2]{
-
\frac{1}{2}
\frac{1}{H_#1 H_#2}
\pder{H_#2}{X^#1}
\hat{U}^#2
}
&
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\normal{1}{1} & \frac{1}{2} \normal{2}{1} + \frac{1}{2} \normal{1}{2} & \frac{1}{2} \normal{3}{1} + \frac{1}{2} \normal{1}{3} \\
sym. & \normal{2}{2} & \frac{1}{2} \normal{3}{2} + \frac{1}{2} \normal{2}{3} \\
sym. & sym. & \normal{3}{3} \\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix} \\
&
+ \\
&
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\diag{2}{1}
\diag{3}{1}
&
\nondiag{2}{1}
\nondiag{1}{2}
&
\nondiag{3}{1}
\nondiag{1}{3}
\\
sym.
&
\diag{1}{2}
\diag{3}{2}
&
\nondiag{3}{2}
\nondiag{2}{3}
\\
sym.
&
sym.
&
\diag{1}{3}
\diag{2}{3}
\\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix}.\end{split}\]
Example¶
In cylindrical coordinates, the velocity-gradient tensor is given by
\[\begin{split}\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\pder{\hat{U}^1}{X^1}
&
\pder{\hat{U}^2}{X^1}
&
\pder{\hat{U}^3}{X^1}
\\
\frac{1}{X^1} \pder{\hat{U}^1}{X^2} - \frac{\hat{U}^2}{X^1}
&
\frac{1}{X^1} \pder{\hat{U}^2}{X^2} + \frac{\hat{U}^1}{X^1}
&
\frac{1}{X^1} \pder{\hat{U}^3}{X^2}
\\
\pder{\hat{U}^1}{X^3}
&
\pder{\hat{U}^2}{X^3}
&
\pder{\hat{U}^3}{X^3}
\\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix}.\end{split}\]
The strain-rate tensor is given by
\[\begin{split}\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\pder{\hat{U}^1}{X^1}
&
\frac{1}{2}
\pder{\hat{U}^2}{X^1}
+
\frac{1}{2}
\frac{1}{X^1}
\pder{\hat{U}^1}{X^2}
-
\frac{1}{2}
\frac{\hat{U}^2}{X^1}
&
\frac{1}{2}
\pder{\hat{U}^3}{X^1}
+
\frac{1}{2}
\pder{\hat{U}^1}{X^3}
\\
sym.
&
\frac{1}{X^1}
\pder{\hat{U}^2}{X^2}
+
\frac{\hat{U}^1}{X^1}
&
\frac{1}{2}
\frac{1}{X^1}
\pder{\hat{U}^3}{X^2}
+
\frac{1}{2}
\pder{\hat{U}^2}{X^3}
\\
sym.
&
sym.
&
\pder{\hat{U}^3}{X^3}
\\
\end{pmatrix}
\begin{pmatrix}
\vec{\hat{E}}_1
\\
\vec{\hat{E}}_2
\\
\vec{\hat{E}}_3
\end{pmatrix}.\end{split}\]