Hydrodynamic effects in internal shock of relativistic outflows

We study both analytically and numerically hydrodynamic effects of two colliding shells, the simplified models of internal shock in various relativistic outflows such as gamma-ray bursts and blazars. We pay particular attention to three interesting cases: a pair of shells with the same rest-mass density ("equal rest-mass density"), a pair of shells with the same rest mass ("equal mass"), and a pair of shells with the same bulk kinetic energy ("equal energy") measured in the interstellar medium frame. We find that the density profiles are significantly affected by the propagation of rarefaction waves. A split feature appears at the contact discontinuity of two shells for the equal-mass case, while no significant split appears for the equal-energy and equal rest-mass density cases. The shell spreading within a few 10% of the speed of light is also shown as a notable aspect caused by rarefaction waves. The conversion efficiency of bulk kinetic energy to internal energy is numerically evaluated. The time evolutions of the efficiency show deviations from the widely used inelastic two-point mass-collision model.