Covariant basis

Description

Velocity-gradient tensor

The gradient of a velocity vector is

\[\left( \sum_j \vec{E}^j \pder{}{X^j} \right) \otimes \left( \sum_i \vec{E}_i U^i \right) = \sum_{ij} \left( \vec{E}^j \otimes \pder{\vec{E}_i}{X^j} \right) U^i + \sum_{ij} \left( \vec{E}^j \otimes \vec{E}_i \right) \pder{U^i}{X^j}.\]

By using the changes of the basis vectors:

\[\dedx{i}{j}{k},\]

I find that the vector gradient is given by the sum of the following elements.

\[\newcommand{\tmpa}[2]{ \frac{1}{H_{#2} H_{#2}} \frac{1}{H_{#1}} \pder{H_{#1}}{X^{#2}} U^{#1} } \sum_{ij} \left( \vec{E}_j \otimes \vec{E}_i \right) \tmpa{i}{j}.\]
\[\newcommand{\tmpb}[1]{ \frac{1}{H_{#1} H_{#1}} \sum_k \frac{1}{H_{#1}} \pder{H_{#1}}{X^k} U^k } \sum_j \left( \vec{E}_j \otimes \vec{E}_j \right) \frac{1}{H_j H_j} \frac{1}{H_j} \sum_k \pder{H_j}{X^k} U^k.\]
\[\begin{split}& - \sum_{ijk} \left( \vec{E}^j \otimes \vec{E}_k \right) \delta_{ij} \frac{H_j}{2 H_k H_k} \pder{H_i}{X^k} U^i - \sum_{ijk} \left( \vec{E}^j \otimes \vec{E}_k \right) \delta_{ij} \frac{H_i}{2 H_k H_k} \pder{H_j}{X^k} U^i \\ = & - \sum_{jk} \left( \vec{E}^j \otimes \vec{E}_k \right) \frac{H_j}{2 H_k H_k} \pder{H_j}{X^k} U^j - \sum_{ik} \left( \vec{E}^i \otimes \vec{E}_k \right) \frac{H_i}{2 H_k H_k} \pder{H_i}{X^k} U^i \\ = & - \sum_{ij} \left( \vec{E}_j \otimes \vec{E}_i \right) \frac{1}{H_j H_j} \frac{H_j}{H_i H_i} \pder{H_j}{X^i} U^j \\ = & - \sum_{ij} \left( \vec{E}_j \otimes \vec{E}_i \right) \tmpa{j}{i}.\end{split}\]
\[\newcommand{\tmpd}[2]{ \frac{1}{H_{#2} H_{#2}} \pder{U^{#1}}{X^{#2}} } \sum_{ij} \left( \vec{E}_j \otimes \vec{E}_i \right) \tmpd{i}{j}.\]

Note that the negative of the third element is the transpose of the first one.

In summary, the velocity gradient is given by the sum of

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \tmpd{1}{1} & \tmpd{2}{1} & \tmpd{3}{1} \\ \tmpd{1}{2} & \tmpd{2}{2} & \tmpd{3}{2} \\ \tmpd{1}{3} & \tmpd{2}{3} & \tmpd{3}{3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \tmpb{1} & 0 & 0 \\ 0 & \tmpb{2} & 0 \\ 0 & 0 & \tmpb{3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} 0 & \tmpa{2}{1} - \tmpa{1}{2} & \tmpa{3}{1} - \tmpa{1}{3} \\ \tmpa{1}{2} - \tmpa{2}{1} & 0 & \tmpa{3}{2} - \tmpa{2}{3} \\ \tmpa{1}{3} - \tmpa{3}{1} & \tmpa{2}{3} - \tmpa{3}{2} & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

Strain-rate tensor

The strain-rate tensor is defined as the symmetric part of it; namely the sum of

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \tmpd{1}{1} & \frac{1}{2} \tmpd{2}{1} + \frac{1}{2} \tmpd{1}{2} & \frac{1}{2} \tmpd{3}{1} + \frac{1}{2} \tmpd{1}{3} \\ sym. & \tmpd{2}{2} & \frac{1}{2} \tmpd{3}{2} + \frac{1}{2} \tmpd{2}{3} \\ sym. & sym. & \tmpd{3}{3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \tmpb{1} & 0 & 0 \\ sym. & \tmpb{2} & 0 \\ sym. & sym. & \tmpb{3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

Example

Cylindrical coordinates

Details

The velocity-gradient tensor is the sum of

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \pder{U^1}{X^1} & \pder{U^2}{X^1} & \pder{U^3}{X^1} \\ \frac{1}{X^1 X^1} \pder{U^1}{X^2} & \frac{1}{X^1 X^1} \pder{U^2}{X^2} & \frac{1}{X^1 X^1} \pder{U^3}{X^2} \\ \pder{U^1}{X^3} & \pder{U^2}{X^3} & \pder{U^3}{X^3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & \frac{1}{X^1 X^1} \frac{U^1}{X^1} & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} 0 & \frac{U^2}{X^1} & 0 \\ - \frac{U^2}{X^1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

The strain-rate tensor is

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \pder{U^1}{X^1} & \frac{1}{2} \pder{U^2}{X^1} + \frac{1}{2} \frac{1}{X^1 X^1} \pder{U^1}{X^2} & \frac{1}{2} \pder{U^3}{X^1} + \frac{1}{2} \pder{U^1}{X^3} \\ sym. & \frac{1}{X^1 X^1} \pder{U^2}{X^2} + \frac{1}{X^1 X^1} \frac{U^1}{X^1} & \frac{1}{2} \frac{1}{X^1 X^1} \pder{U^3}{X^2} + \frac{1}{2} \pder{U^2}{X^3} \\ sym. & sym. & \pder{U^3}{X^3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

Rectilinear coordinates

Details

The velocity-gradient tensor is the sum of

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \frac{1}{H_1 H_1} \pder{U^1}{X^1} & \frac{1}{H_1 H_1} \pder{U^2}{X^1} & \frac{1}{H_1 H_1} \pder{U^3}{X^1} \\ \frac{1}{H_2 H_2} \pder{U^1}{X^2} & \frac{1}{H_2 H_2} \pder{U^2}{X^2} & \frac{1}{H_2 H_2} \pder{U^3}{X^2} \\ \frac{1}{H_3 H_3} \pder{U^1}{X^3} & \frac{1}{H_3 H_3} \pder{U^2}{X^3} & \frac{1}{H_3 H_3} \pder{U^3}{X^3} \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \frac{1}{H_1 H_1} \sum_k \frac{1}{H_1} \pder{H_1}{X^k} U^k & 0 & 0 \\ 0 & \frac{1}{H_2 H_2} \sum_k \frac{1}{H_2} \pder{H_2}{X^k} U^k & 0 \\ 0 & 0 & \frac{1}{H_3 H_3} \sum_k \frac{1}{H_3} \pder{H_3}{X^k} U^k \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

The strain-rate tensor is

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \frac{1}{H_1 H_1} \pder{U^1}{X^1} + \frac{1}{H_1 H_1} \sum_k \frac{1}{H_1} \pder{H_1}{X^k} U^k & \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^2}{X^1} + \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^2}{X^1} & \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^3}{X^1} + \frac{1}{2} \frac{1}{H_3 H_3} \pder{U^1}{X^3} \\ sym. & \frac{1}{H_2 H_2} \pder{U^2}{X^2} + \frac{1}{H_2 H_2} \sum_k \frac{1}{H_2} \pder{H_2}{X^k} U^k & \frac{1}{2} \frac{1}{H_2 H_2} \pder{U^3}{X^2} + \frac{1}{2} \frac{1}{H_3 H_3} \pder{U^2}{X^3} \\ sym. & sym. & \frac{1}{H_3 H_3} \pder{U^3}{X^3} + \frac{1}{H_3 H_3} \sum_k \frac{1}{H_3} \pder{H_3}{X^k} U^k \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

Application

Details

The velocity-gradient tensor is the sum of

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \tmpd{1}{1} & \tmpd{2}{1} & \tmpd{3}{1} \\ \tmpd{1}{2} & \tmpd{2}{2} & \tmpd{3}{2} \\ \tmpd{1}{3} & \tmpd{2}{3} & \tmpd{3}{3} \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & \frac{1}{H_2 H_2} \frac{1}{H_2} \pder{H_2}{X^1} U^1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} 0 & \tmpa{2}{1} & 0 \\ - \tmpa{2}{1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]

The strain-rate tensor is

\[\begin{split}\begin{pmatrix} \vec{E}_1 & \vec{E}_2 & \vec{E}_3 \end{pmatrix} \begin{pmatrix} \frac{1}{H_1 H_1} \pder{U^1}{X^1} + \frac{1}{H_1 H_1} \frac{1}{H_1} \pder{H_1}{X^1} U^1 & \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^2}{X^1} + \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^2}{X^1} & \frac{1}{2} \frac{1}{H_1 H_1} \pder{U^3}{X^1} + \frac{1}{2} \frac{1}{H_3 H_3} \pder{U^1}{X^3} \\ sym. & \frac{1}{H_2 H_2} \pder{U^2}{X^2} + \frac{1}{H_2 H_2} \frac{1}{H_2} \pder{H_2}{X^1} U^1 & \frac{1}{2} \frac{1}{H_2 H_2} \pder{U^3}{X^2} + \frac{1}{2} \frac{1}{H_3 H_3} \pder{U^2}{X^3} \\ sym. & sym. & \frac{1}{H_3 H_3} \pder{U^3}{X^3} \end{pmatrix} \begin{pmatrix} \vec{E}_1 \\ \vec{E}_2 \\ \vec{E}_3 \end{pmatrix}.\end{split}\]