Hot, X-ray-emitting plasma permeates clusters of galaxies. The X-ray surface brightness often shows a peak near the centre of the cluster that is coincident with a drop in the entropy of the gas. This has been taken as evidence for a 'cooling flow', where the gas cools by radiating away its energy, and then falls to the centre. Searches for this cool gas have revealed significantly less than predicted, indicating that the mass deposition rate is much lower than expected. Most clusters with cooling flows, however, also host an active galactic nucleus at their centres. These active galactic nuclei can inflate large bubbles of hot plasma that subsequently rise through the cluster 'atmosphere', thus stirring the cooling gas and adding energy. Here we report highly resolved hydrodynamic simulations which show that buoyant bubbles increase the cooling time in the inner regions of clusters and significantly reduce the deposition of cold gas.