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We present numerical simulations investigating the interaction of active galactic nucleus jets with galaxy clusters, for the first time taking into account the dynamic nature of the cluster gas and detailed cluster physics. The simulations successfully reproduce the observed morphologies of radio sources in clusters. We find that cluster inhomogeneities and large-scale flows have a significant impact on the morphology of the radio source and cannot be ignored a priori when investigating radio source dynamics. Morphological comparison suggests that the gas in the centres of clusters like Virgo and Abell 4059 shows significant shear and/or rotation. We find that shear and rotation in the intracluster medium move large amounts of cold material back into the path of the jet, ensuring that subsequent jet outbursts encounter a sufficient column density of gas to couple with the inner cluster gas, thus alleviating the problem of evacuated channels discussed in the recent literature. The same effects redistribute the excess energy Δ E deposited by the jet, making the distribution of Δ E at late times consistent with being isotropic. [ABSTRACT FROM AUTHOR]