  We study the evolution of satellite galaxies in clusters of the C-EAGLE
simulations, a suite of 30 high-resolution cosmological hydrodynamical zoom-in
simulations based on the EAGLE code. We find that the majority of galaxies that
are quenched at $z=0$ ($\gtrsim$ 80$\%$) reached this state in a dense
environment (log$_{10}$M$_{200}$[M$_{\odot}$]$\geq$13.5). At low redshift,
regardless of the final cluster mass, galaxies appear to reach their quenching
state in low mass clusters. Moreover, galaxies quenched inside the cluster that
they reside in at $z=0$ are the dominant population in low-mass clusters, while
galaxies quenched in a different halo dominate in the most massive clusters.
When looking at clusters at $z>0.5$, their in-situ quenched population
dominates at all cluster masses. This suggests that galaxies are quenched
inside the first cluster they fall into. After galaxies cross the cluster's
$r_{200}$ they rapidly become quenched ($\lesssim$ 1Gyr). Just a small fraction
of galaxies ($\lesssim 15\%$) is capable of retaining their gas for a longer
period of time, but after 4Gyr, almost all galaxies are quenched. This
phenomenon is related to ram pressure stripping and is produced when the
density of the intracluster medium reaches a threshold of $\rho_{\rm ICM}$
$\sim 3 \times 10 ^{-5}$ n$_{\rm H}$ (cm$^{-3}$). These results suggest that
galaxies start a rapid-quenching phase shortly after their first infall inside
$r_{200}$ and that, by the time they reach $r_{500}$, most of them are already
quenched.
