  With hydrodynamical simulations we examine the evolution of a protoplanetary
disc around $\alpha$ Centauri B including the effect of the eccentric orbit
binary companion $\alpha$ Centauri A. The initially circular orbit disc
undergoes two types of eccentricity growth. First, the eccentricity oscillates
on the orbital period of the binary, $P_{\rm orb}$, due to the eccentricity of
the binary orbit. Secondly, for a sufficiently small disc aspect ratio, the
disc undergoes global forced eccentricity oscillations on a time-scale of
around $20\,P_{\rm orb}$. These oscillations damp out through viscous
dissipation leaving a quasi-steady eccentricity profile for the disc that
oscillates only on the binary orbital period. The time-averaged global
eccentricity is in the range 0.05-0.1, with no precession in the steady state.
The periastrons of the gas particles are aligned to one another. The higher the
disc viscosity, the higher the disc eccentricity. With $N$-body simulations we
examine the evolution of a disc of planetesimals that forms with the orbital
properties of the quasi-steady protoplanetary disc. We find that the average
magnitude of the eccentricity of particles increases and their periastrons
become misaligned to each other once they decouple from the gas disc. The low
planetesimal collision velocity required for planet formation suggests that for
planet formation to have occurred in a disc of planetesimals formed from a
protoplanetary disc around $\alpha$ Centauri B, said disc's viscosity must be
have been small and planet formation must have occurred at orbital radii
smaller than about $2.5\,\rm au$. Planet formation may be easier with the
presence of gas.
