X-ray observations of clusters of galaxies reveal the presence of edges in surface brightness and temperature, known as "cold fronts." In relaxed clusters with cool cores, these commonly observed edges have been interpreted as evidence for the "sloshing" of the core gas in the cluster's gravitational potential. Such sloshing may provide a source of heat to the cluster core by mixing hot gas from the cluster outskirts with the cool-core gas. Using high-resolution N-body/Eulerian hydrodynamic simulations, we model gas sloshing in galaxy clusters initiated by mergers with subclusters. The simulations include merger scenarios with gas-filled and gasless subclusters. The effect of changing the viscosity of the intracluster medium is also explored, but heat conduction is assumed to be negligible. We find that sloshing can facilitate heat inflow to the cluster core, provided that there is a strong enough disturbance. Additionally, sloshing redistributes the gas in the cluster core, causing the gas to expand and decreasing the efficiency of radiative cooling. In adiabatic simulations, we find that sloshing can raise the entropy floor of the cluster core by nearly an order of magnitude in the strongest cases. If the ICM is viscous, the mixing of gases with different entropies is decreased and consequently the heat flux to the core is diminished. In simulations where radiative cooling is included, we find that although eventually a cooling flow develops, sloshing can prevent the significant buildup of cool gas in the core for times on the order of a Gyr for small disturbances and a few Gyr for large ones. If repeated encounters with merging subclusters sustain the sloshing of the central core gas, as is observed, this process can provide a relatively steady source of heat to the core, which can help prevent a significant cooling flow.