Consider a circle of radius $r_s$ that is tangent to two curves $r(\theta)$ and $R(\theta)$ at points $E_1, E_2$ respectively, defined in polar coordinates. Knowing the function $r(\theta)$, find the function $R(\theta)$, so that when rotating both function in opposite directions by some angle $d\phi$ around the center of the coordinate system, the circle remains on the line OS (where O is the center of the coordinate system and S being the center of the circle) and is tangent to both curves at one point respectively. Can be thought of as if the circle rolls on both surfaces without slipping.
In case where curve $r(\theta)$ is a circle with radius $r$, the other curve $R(\theta)$ is also a circle with the radius $R = r+2r_s$.
I can generate the points of function $R(\theta)$ numerically, but I am struggling to find a way to get an analytic solution for the curve $R(\theta)$. Any thoughts?
A drawing for (maybe) better understanding of the problem: problem
The fact that your curves are given in polar coordinates is less important than the fact that you need curves having a good representation in terms of curvilinear abscissa.
There are very few of them, set apart circles (the trivial case you have mentionned) and "Involutes of a Circle" (IC in short ; disclaimer : non standard abbreviation).
Precisely, these curves are used in the profile of gears : see the animation here displaying two opposite ICs rolling without slipping one on the other (see as well the graphical representation below).
Now, if you separate them by a distance $2R$, you can introduce a disk (circular galley) with radius $R$ between them : this disk will communicate the desired rolling-without-slipping movement from one IC to the other while staying in-place.
Fig. 1 : see explanation below.
One can have a good picture of an IC as a limit curve (like for a circle which is the limit of inscribed regular polygons). Consider the "spiral-looking" red curve in fig. 1 ; imagine a tethered goat in point A of an octogonal pillar discovering that its rope has in fact been rolled around the pillar ; the goat, by progressively unrolling the rope will be able to graze more and more land ; The frontier curve is an approximate IC ; replace now the octogon by any n-gon with $n \to \infty$ (always inscribed in the same circle) : you have now a correct idea of what an IC is.
Now, if we place a second similar IC in the position represented on fig. 1, the two ICs can roll one on the other because the addition of the resp. distances to their centers is a constant.
I haven't made a figure representing the circle that can be placed between the two ICs but it is not difficult to imagine it.
See many inspiring animations on this site.