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We investigate the effects of neutrino heating and alpha-particle recombination on the hydrodynamics of core-collapse supernovae. Our focus is on the non-linear dynamics of the shock wave that forms in the collapse, and the assembly of positive energy material below it. To this end, we perform time-dependent hydrodynamic simulations with FLASH2.5 in spherical and axial symmetry. These generalize our previous calculations by allowing for bulk neutrino heating and for nuclear statistical equilibrium between n, p and alpha. The heating rate is freely tunable, as is the starting radius of the shock relative to the recombination radius of alpha-particles. An explosion in spherical symmetry involves the excitation of an overstable mode, which may be viewed as the L=0 version of the `Standing Accretion Shock Instability'. In 2D simulations, non-spherical deformations of the shock are driven by plumes of material with positive Bernoulli parameter, which are concentrated well outside the zone of strong neutrino heating. The non-spherical modes of the shock reach a large amplitude only when the heating rate is also high enough to excite convection below the shock. The critical heating rate that causes an explosion depends sensitively on the initial position of the shock relative to the recombination radius. Weaker heating is required to drive an explosion in 2D than in 1D, but the difference also depends on the size of the shock. Forcing the infalling heavy nuclei to break up into n and p below the shock only causes a slight increase in the critical heating rate, except when the shock starts out at a large radius. This shows that heating by neutrinos (or some other mechanism) must play a significant role in pushing the shock far enough out that recombination heating takes over.