Appendix I Color fields

There are two possibilities to implement a color field $c$ in a fluid code. One can treat color like the density, i.e. the color variable obeys a conservation law like the density

${\displaystyle \frac{\partial }{\partial t}}c+{\displaystyle \frac{\partial }{\partial r_j}}(v_{j}c)$

$=0.$

(I.1)

In this case $c$ will exactly behave like density, if density and color have the same initial value.

On the other hand one can treat it like a specific quantity obeying a conservation law like

${\displaystyle \frac{\partial }{\partial t}}\rho c+{\displaystyle \frac{\partial }{\partial r_j}}(v_j\rho c)$

$=0.$

(I.2)

In this case we see that if $c$ is spatially constant at a time $t_0$, i.e. $c(t_0)=c_0$, $\frac{\partial c(t_0)}{\partial r_j}=0$, it will stay constant forever

	${\displaystyle \frac{\partial }{\partial t}}\rho c+{\displaystyle \frac{\partial }{\partial r_j}}(v_j\rho c)$	$=0$
	$\iff {\displaystyle \frac{\partial c}{\partial t}}+v_j{\displaystyle \frac{\partial c}{\partial r_j}}$	$=0$
	$\iff {\displaystyle \frac{d}{dt}}c$	$=0$
	$\Rightarrow c$	$=const.=c_0$