We study the evolution of the expanding solar wind using a one-dimensional and a two-dimensional expanding box model <Grappin et al., 1993> implemented here within a hybrid code <Liewer et al., 2001>. We first consider a plasma with protons and 5% of alpha particles, without drift between the protons and alphas, considering successively the low-beta and high-beta cases. Then we consider a strong drift between protons and alphas, again separately the low-beta and high-beta case. Without drift, the evolution of the low-beta plasma is adiabatic. In the high-beta plasma without drift, the fire hose instabilities (both parallel and oblique) disrupt the adiabatic evolution. Finally, with a drift, the adiabatic evolution is stopped by the oblique Alfv¿n instability and the parallel magnetosonic instability for low-beta plasma and for high-beta plasma, respectively. The two instabilities slow down thealphas and heat alphas and protons.