  A self-similar spherical collapse model predicts a dark matter (DM) splashback and accretion shock in the outskirts of galaxy clusters while misses a key ingredient of structure formation - processes associated with mergers. To fill this gap, we perform simulations of merging self-similar clusters and investigate their DM and gas evolution in an idealized cosmological context. Our simulations show that the cluster rapidly contracts during the major merger and the splashback radius $r_{\rm sp}$ decreases, approaching the virial radius $r_{\rm vir}$. While $r_{\rm sp}$ correlates with a smooth mass accretion rate (MAR) parameter $\Gamma_{\rm s}$ in the self-similar model, our simulations show a similar trend with the total MAR $\Gamma_{\rm vir}$ (includes both mergers and $\Gamma_{\rm s}$). The scatter of the $\Gamma_{\rm vir}-r_{\rm sp}/r_{\rm vir}$ relation indicates a generally low $\Gamma_{\rm s}\sim1$ in clusters in cosmological simulations. In contrast to the DM, the hot gaseous atmospheres significantly expand by the merger-accelerated (MA-) shocks formed when the runaway merger shocks overtake the outer accretion shock. After a major merger, the MA-shock radius is larger than $r_{\rm sp}$ by a factor of up to $\sim1.7$ for $\Gamma_{\rm s}\lesssim1$ and is $\sim r_{\rm sp}$ for $\Gamma_{\rm s}\gtrsim3$. This implies that (1) mergers could easily generate the MA-shock-splashback offset measured in cosmological simulations, and (2) the smooth MAR is small in regions away from filaments where MA-shocks reside. We further discuss various shocks and contact discontinuities formed at different epochs of the merger, the ram pressure stripping in cluster outskirts, and the dependence of member galaxies' splashback feature on their orbital parameters. 
Evolution of Splashback Boundaries and Gaseous Outskirts: Insights from   Mergers of Self-similar Galaxy Clusters