  Circumbinary disks are found in a variety of astrophysical scenarios,
spanning binary star formation to accreting supermassive black hole binaries.
The interaction with a circumbinary disk can yield opposite effects on the
binary orbit leading to circularization, or exciting the eccentricity, widening
the orbit or shrinking it and facilitating mergers. We present a new formalism
for the long-term evolution of the disk-binary interaction based on the results
of recent suites of hydrodynamic simulations, which resolve the complex
geometry of the gas in the vicinity of the binary and fully account for the
gravitational and accretion forces. We release a python package,
\texttt{spindler}, that implements our model. We show that, unless the mass
reservoir feeding the disk is comparable to the mass of the binary, accretion
onto the binary depletes the disk mass before inducing a significant change in
orbital separation or mass ratio. This finding implies that, in most scenarios,
interaction with a circumbinary disk is not an efficient mechanism to shrink
the orbit of the binary. However, as long as the mass of the disk is at least a
few percent of the mass of the binary, the interaction can excite the
eccentricity up to an equilibrium value, and induce a statistical correlation
between mass ratio and eccentricity. We consider the applicability of our model
to a variety of astrophysical scenarios: during star formation, in evolved
stellar binaries, triples and in supermassive black hole binaries. We discuss
the theoretical and observational implications of our predictions.
