Strain rate tensor

Remarks

The remaining part $W = \tfrac{1}{2}(L - L^{T})$ is the spin tensor, and $W_{ij} = -\tfrac{1}{2} \sum_k \varepsilon_{ijk} \omega_k$, where $\boldsymbol{\omega}$ is the vorticity and $\varepsilon_{ijk}$ is the Levi-Civita symbol. The trace of $E$ is the divergence $\nabla \cdot \mathbf{u}$. Example: for the shear flow $\mathbf{u} = (y, 0, 0)$, $L$ has the single nonzero entry $L_{12} = 1$, so $E_{12} = E_{21} = 1/2$ and all other entries of $E$ are $0$. Non-example: the velocity gradient $L$ of this shear flow is not the strain rate tensor, since $L$ is not symmetric. A rigid motion $\mathbf{u}(x) = \mathbf{a} + \mathbf{b} \times x$ has $E = 0$. It is a tensor field of rank two; other names are rate-of-strain tensor and rate-of-deformation tensor.