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An Eulerian multidimensional model has been developed for computing the behavior of fuel sprays in direct injection internal combustion engines. The model involves a description of all basic processes that take place in two-phase flow with inter-phase exchanges of mass, momentum, and energy. Both the multi-component compressible gas-phase flow as well as the droplet-phase flow equations are solved in Eulerian coordinates. Basic laws of conservation are formulated on finite volumes with arbitrarily movable boundaries to facilitate the modeling of movable boundary problems. The model features a detailed description of droplet-phase accounting for droplet mass change due to evaporation and with possibility of incorporation of potential droplet breakup, collisions, and coalescence. The application chosen to demonstrate the predictive capabilities of the developed model is the injection of hollow-cone spray into high-density air in a cylindrical chamber with moving boundary.