A system consists of two adjacent regions in the coordinate system $(r,z)$, as shown in the following figure. 
Region $\color{red}{\rm{B}}$ is the top domain of a small circle cut by the horizontal axis $r$. Note that region $\color{red}{\rm{B}}$ is not a half circle, whose center locates in the negative half $z$-axis. Region $\color{blue}{\rm{A}}$ is the top half of the bigger $\color{blue}{\rm{circle}}$, whose center are $O (0,0)$, minus the region $\color{red}{\rm{B}}$ (smaller circle cap).
To render the governing equations in the two domains dimensionless, I want to introduce separate set of scales for spatial variables in the two regions. For example, in the region $\color{red}{\rm{B}}$,
$$R=\frac{r}{2}, \quad Z=\frac{z}{0.2},$$ which means the $r$-coordinate is shrunk by half, and $z$-coordinate is stretched by 5 times. While in the region $\color{blue}{\rm{A}}$ both coordinates are enlarged by five times. $$(R,Z)=\frac{(r,z)}{0.2}.$$
In my system, there are boundary conditions applied on the $\color{red}{\rm{arc}}$ of the small circle $\color{red}{\rm{B}}$. My question is: with abovementioned geometric feature, whether or not I can using different scalings for spatial variables in the two adjacent regions?
Thanks for your help. Any comments are welcome.