Appendix K Stress tensor in cartesian coordinates for 1d, 2d and 3d

For a so called newtonian fluid it can be shown, that the stress tensor ${\sigma }_{ij}^{\prime }$ in cartesian coordinates in $n$ dimensions is of the form

${\sigma }_{ij}^{\prime }=2\eta S_{ij}^{*}+\zeta {\delta }_{ij}{\displaystyle \frac{\partial }{\partial r_k}}{v}_k$

(K.1)

with $S_{ij}^{*}$ being the symmetric tracefree part of the tensor $\frac{\partial }{\partial x_j}{v}_i$

$S_{ij}^{*}=S_{ij}-{\displaystyle \frac{1}{n}}{\delta }_{ij}{\displaystyle \frac{\partial }{\partial r_k}}{v}_k$

(K.2)

and $S_{ij}$ being the so called rate-of-strain tensor

${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_j}}{v}_i+{\displaystyle \frac{\partial }{\partial r_i}}{v}_j\right)$

(K.3)

In 1d cartesian coordinates the component of the symmetric tracefree part of the stress tensor $S_{ij}^{*}$ is

$S_{xx}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_x\right)-{\displaystyle \frac{1}{1}}{\displaystyle \frac{\partial }{\partial r_x}}{v}_x=0.$

(K.4)

In 2d cartesian coordinates the components of $S_{ij}^{*}$ are

	$S_{xx}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_x\right)-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x+{\displaystyle \frac{\partial }{\partial r_y}}{v}_y\right)$		(K.5)
		$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x-{\displaystyle \frac{\partial }{\partial r_y}}{v}_y\right),$		(K.6)
	$S_{xy}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_y}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_y\right),$		(K.7)
	$S_{yx}$	$=S_{xy},$		(K.8)
	$S_{yy}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_y}}{v}_y-{\displaystyle \frac{\partial }{\partial r_x}}{v}_x\right).$		(K.9)

In 3d cartesian coordinates the components of $S_{ij}^{*}$ are

	$S_{xx}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_x\right)-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\partial }{\partial r_x}}{v}_x+{\displaystyle \frac{\partial }{\partial r_y}}{v}_y+{\displaystyle \frac{\partial }{\partial r_z}}{v}_z\right)$		(K.10)
		$={\displaystyle \frac{1}{3}}\left(2{\displaystyle \frac{\partial }{\partial r_x}}{v}_x-{\displaystyle \frac{\partial }{\partial r_y}}{v}_y-{\displaystyle \frac{\partial }{\partial r_z}}{v}_z\right),$		(K.11)
	$S_{xy}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_y}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_y\right),$		(K.12)
	$S_{xz}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_z}}{v}_x+{\displaystyle \frac{\partial }{\partial r_x}}{v}_z\right),$		(K.13)
	$S_{yx}$	$=S_{xy},$		(K.14)
	$S_{yy}$	$={\displaystyle \frac{1}{3}}\left(2{\displaystyle \frac{\partial }{\partial r_y}}{v}_y-{\displaystyle \frac{\partial }{\partial r_x}}{v}_x-{\displaystyle \frac{\partial }{\partial r_z}}{v}_z\right),$		(K.15)
	$S_{yz}$	$={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_z}}{v}_y+{\displaystyle \frac{\partial }{\partial r_y}}{v}_z\right),$		(K.16)
	$S_{zx}$	$=S_{xz},$		(K.17)
	$S_{zy}$	$=S_{yz},$		(K.18)
	$S_{zz}$	$={\displaystyle \frac{1}{3}}\left(2{\displaystyle \frac{\partial }{\partial r_z}}{v}_z-{\displaystyle \frac{\partial }{\partial r_x}}{v}_x-{\displaystyle \frac{\partial }{\partial r_y}}{v}_y\right).$		(K.19)