Appendix I Rate of strain tensor, rotation tensor

The decomposition of the jacobian of the velocity field into a symmetric and an antisymmetric part yields

${\displaystyle \frac{\partial v_i}{\partial r_j}}$

$=\underset{S_{ij}=S_{ji}}{\underbrace{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial v_i}{\partial r_j}}+{\displaystyle \frac{\partial v_j}{\partial r_i}}\right)}}+\underset{R_{ij}=-R_{ji}}{\underbrace{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial v_i}{\partial r_j}}-{\displaystyle \frac{\partial v_j}{\partial r_i}}\right)}}$

(I.1)

The symmetric part $S_{ij}$ is called rate of strain tensor and the antisymmetric part is called rotation tensor. $R_{ij}$ has only three independent components and can be expressed in terms of a (pseudo-) 3-vector.³⁰³⁰$S_{ij}$ has six independent components. Is it possible to find a representation in terms of two (pseudo-) 3-vectors? This vector is equivalent to the negative curl of the velocity field $𝝎=\nabla \times 𝒗$ as we can see by multiplying $R_{jk}$ with ${ϵ}_{ijk}$

	${\displaystyle \frac{1}{2}}{ϵ}_{ijk}{R}_{jk}$	$={\displaystyle \frac{1}{2}}{ϵ}_{ijk}\left({\displaystyle \frac{\partial v_j}{\partial r_k}}-{\displaystyle \frac{\partial v_k}{\partial r_j}}\right)={\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\displaystyle \frac{\partial v_j}{\partial r_k}}-{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\displaystyle \frac{\partial v_k}{\partial r_j}}$
		$={\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\displaystyle \frac{\partial v_j}{\partial r_k}}-{\displaystyle \frac{1}{2}}{ϵ}_{ikj}{\displaystyle \frac{\partial v_j}{\partial r_k}}={\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\displaystyle \frac{\partial v_j}{\partial r_k}}+{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\displaystyle \frac{\partial v_j}{\partial r_k}}$
		$={ϵ}_{ijk}{\displaystyle \frac{\partial v_j}{\partial r_k}}={ϵ}_{ikj}{\displaystyle \frac{\partial v_k}{\partial r_j}}=-{ϵ}_{ijk}{\displaystyle \frac{\partial v_k}{\partial r_j}}$
		$=-{\omega }_i$

Further one can show that

	$-{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\omega }_k$	$=-{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{ϵ}_{klm}{\displaystyle \frac{\partial v_m}{\partial r_l}}=-{\displaystyle \frac{1}{2}}{ϵ}_{kij}{ϵ}_{klm}{\displaystyle \frac{\partial v_m}{\partial r_l}}$
		$=-{\displaystyle \frac{1}{2}}\left({\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}\right){\displaystyle \frac{\partial v_m}{\partial r_l}}$
		$=-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial v_j}{\partial r_i}}-{\displaystyle \frac{\partial v_i}{\partial r_j}}\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial v_i}{\partial r_j}}-{\displaystyle \frac{\partial v_j}{\partial r_i}}\right)$
		$=R_{ij}$

and

	$R_{ij}{R}_{ij}$	$=-{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{\omega }_k\cdot -{\displaystyle \frac{1}{2}}{ϵ}_{ijl}{\omega }_l$		(I.2)
		$={\displaystyle \frac{1}{4}}{ϵ}_{ijk}{ϵ}_{ijl}{\omega }_k{\omega }_l={\displaystyle \frac{1}{4}}\cdot 2{\delta }_{kl}{\omega }_k{\omega }_l$		(I.3)
		$={\displaystyle \frac{1}{2}}{\omega }_k{\omega }_k$		(I.4)

Additionally the following relation between the contraction of rate of strain tensor and the contraction of the rotation tensor is useful

(I.5)