################ Normalised basis ################ *********** Description *********** ======================== Velocity-gradient tensor ======================== .. math:: \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 :ref:`the changes of the normalised basis vectors `: .. math:: \dehatdx{i}{j}{k}, the first term yields .. math:: + \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 .. math:: + \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. .. math:: \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}. ================== Strain-rate tensor ================== The strain-rate tensor is defined as the symmetric part of it: .. math:: \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}. ******* Example ******* In cylindrical coordinates, the velocity-gradient tensor is given by .. math:: \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}. The strain-rate tensor is given by .. math:: \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}.