This work aims to present our current best physical understanding of common-envelope evolution (CEE). We highlight areas of consensus and disagreement, and stress ideas which should point the way forward for progress in this important but long-standing and largely unconquered problem. Unusually for CEE-related work, we mostly try to avoid relying on results from population synthesis or observations, in order to avoid potentially being misled by previous misunderstandings. As far as possible we debate all the relevant issues starting from physics alone, all the way from the evolution of the binary system immediately before CEE begins to the processes which might occur just after the ejection of the envelope. In particular, we include extensive discussion about the energy sources and sinks operating in CEE, and hence examine the foundations of the standard energy formalism. Special attention is also given to comparing the results of hydrodynamic simulations from different groups and to discussing the potential effect of initial conditions on the differences in the outcomes. We compare current numerical techniques for the problem of CEE and also whether more appropriate tools could and should be produced (including new formulations of computational hydrodynamics, and attempts to include 3D processes within 1D codes). Finally we explore new ways to link CEE with observations. We compare previous simulations of CEE to the recent outburst from V1309 Sco, and discuss to what extent post-common-envelope binaries and nebulae can provide information, e.g. from binary eccentricities, which is not currently being fully exploited.

The hydrodynamic evolution of the common envelope phase of a low mass binary composed of a 1.05 Msun red giant and a 0.6 Msun companion has been followed for five orbits of the system using a high resolution method in three spatial dimensions. During the rapid inspiral phase, the interaction of the companion with the red giant's extended atmosphere causes about 25% of the common envelope to be ejected from the system, with mass continuing to be lost at the end of the simulation at a rate ~ 2 Msun/yr. In the process the resulting loss of angular momentum and energy reduces the orbital separation by a factor of seven. After this inspiral phase the eccentricity of the orbit rapidly decreases with time. The gravitational drag dominates hydrodynamic drag at all times in the evolution, and the commonly-used Bondi-Hoyle-Lyttleton prescription for estimating the accretion rate onto the companion significantly overestimates the true rate. On scales comparable to the orbital separation, the gas flow in the orbital plane in the vicinity of the two cores is subsonic with the gas nearly corotating with the red giant core and circulating about the red giant companion. On larger scales, 90% of the outflow is contained within 30 degrees of the orbital plane, and the spiral shocks in this material leave an imprint on the density and velocity structure. Of the energy released by the inspiral of the cores, only about 25% goes toward ejection of the envelope.