This study presents a new formalism for understanding the long-term evolution of binary orbits influenced by circumbinary disks, based on hydrodynamic simulations. The authors developed a Python package called "spindler" to model the interaction between a binary system and a circumbinary disk. Their results show that, under the assumed thin disk model with fixed thickness and viscosity, accretion onto the binary depletes the disk mass before significantly altering the orbital separation or mass ratio, unless the mass reservoir feeding the disk is comparable to the binary's mass. However, the interaction can excite the binary's eccentricity up to an equilibrium value and induce a statistical correlation between the mass ratio and eccentricity, provided the disk mass is at least a few percent of the binary's mass. The study considers the applicability of the model to various astrophysical scenarios, including star formation, evolved stellar binaries, triples, and supermassive black hole binaries. The authors discuss the theoretical and observational implications of their findings. They find that the long-term orbital evolution of binary systems is influenced by factors such as mass ratio, eccentricity, disk aspect ratio, viscosity, disk-orbit inclination, and cooling rate. The study highlights the importance of considering these factors when modeling binary-disk interactions. The authors compare their results with those from other studies, finding that while the three models broadly agree on the equilibrium eccentricity, there are differences in the behavior of systems with small eccentricities and the rate of orbital decay. The study also discusses the implications of these findings for astrophysical scenarios, including binary star formation, interacting binaries, post-common-envelope systems, post-AGB binaries, and supermassive black hole binaries. The results suggest that circumbinary disks may not be an efficient mechanism for shrinking binary orbits in most scenarios, but they can influence the eccentricity and mass ratio of binary systems. The study concludes that the long-term evolution of binary systems is complex and depends on a variety of factors, and that further research is needed to fully understand the role of circumbinary disks in binary evolution.This study presents a new formalism for understanding the long-term evolution of binary orbits influenced by circumbinary disks, based on hydrodynamic simulations. The authors developed a Python package called "spindler" to model the interaction between a binary system and a circumbinary disk. Their results show that, under the assumed thin disk model with fixed thickness and viscosity, accretion onto the binary depletes the disk mass before significantly altering the orbital separation or mass ratio, unless the mass reservoir feeding the disk is comparable to the binary's mass. However, the interaction can excite the binary's eccentricity up to an equilibrium value and induce a statistical correlation between the mass ratio and eccentricity, provided the disk mass is at least a few percent of the binary's mass. The study considers the applicability of the model to various astrophysical scenarios, including star formation, evolved stellar binaries, triples, and supermassive black hole binaries. The authors discuss the theoretical and observational implications of their findings. They find that the long-term orbital evolution of binary systems is influenced by factors such as mass ratio, eccentricity, disk aspect ratio, viscosity, disk-orbit inclination, and cooling rate. The study highlights the importance of considering these factors when modeling binary-disk interactions. The authors compare their results with those from other studies, finding that while the three models broadly agree on the equilibrium eccentricity, there are differences in the behavior of systems with small eccentricities and the rate of orbital decay. The study also discusses the implications of these findings for astrophysical scenarios, including binary star formation, interacting binaries, post-common-envelope systems, post-AGB binaries, and supermassive black hole binaries. The results suggest that circumbinary disks may not be an efficient mechanism for shrinking binary orbits in most scenarios, but they can influence the eccentricity and mass ratio of binary systems. The study concludes that the long-term evolution of binary systems is complex and depends on a variety of factors, and that further research is needed to fully understand the role of circumbinary disks in binary evolution.