For the Chapman-Enskog expansion we expand the distribution function as a series in the Knudsen number $\epsilon$ as (I am following this paper):

$$f = f^{(0)} +\epsilon f^{(1)} + ...$$

and it is also necessary to do this to do a similar expansion for the time derivatives as follows :

$$\partial_t = \partial_{t_0} + \epsilon \partial_{t_1} + ...$$

But I am confused as to what it means to do an expansion in time derivatives? what are $t_0,t_1,...$? Does each $f^{(n)}$ where $n=0,1,...$ depend on the different "time scales"?

Does anyone have a reference for this? I am finding myself quite confused trying to understand this expansion.