Strain Rate and Rotation

Strain rate

It is the rate at which deformation occurs in all directions. It takes into changes in volume and shape. Strain rate tensor

$$S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$

For $i,j=1,2,3$ (Dimensions)

Rotation

It is the rate at which a fluid particle rotates about it’s own axis. The rotation of at a point is given by the Vorticity ($\vec{\omega }=\nabla \times \vec{u}$) of the flow (which defines the Rotation for a whole field) Rotational Tensor

$${\Omega }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$$

Derivation

Both Strain rate and rotation can be unpacked from the gradient of velocity.

$$\begin{array}{rl}\frac{\partial u_i}{\partial x_j}=& \stackrel{\text{Symmetric part}}{\overbrace{\frac{1}{2}\frac{\partial u_i}{\partial x_j}+\frac{1}{2}\frac{\partial u_j}{\partial u_i}}}+\stackrel{\text{Asymmetric part}}{\overbrace{\frac{1}{2}\frac{\partial u_i}{\partial u_j}-\frac{1}{2}\frac{\partial u_j}{\partial u_i}}}\\ & \\ =& S_{ij}+{\Omega }_{ij}\end{array}$$