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The influence of gravitational torques on the evolution of the common envelope of a binary star system consisting of a $3 \msun$ red giant and a $1 \msun$ companion star is studied. The numerical results of two dimensional hydrodynamical simulations of the binary in its orbital plane demonstrate that the matter in the immediate vicinity of the low mass companion is brought into corotation with the orbital motion. The degree of spin up is a function of orbital phase with the departure from corotation increasing for matter which experienced the previous companion passage the earliest. As a consequence of the spin up, a low density region is formed over a significant fraction of the orbit. This region is limited in extent and its width is approximately equal to the size of the tidal lobe of the low mass companion. Throughout the common envelope the matter rotates differentially. Exterior to the gap region, the specific angular momentum tends toward a spatially constant value such that the angular velocity, $\Omega$, is approximately proportional to $r^{-2}$ where $r$ is the distance from the red giant core. The gravitational interaction between the cores and the common envelope leads to a slow material outflow at velocities $\lapprox 5 \times 10^6$ cm s$^{-1}$. At the end of the calculation the time scale for the spiralling together of the two cores becomes long, and it is suggested that they will remain in orbit as a 1.15 day binary.
