  We use the \eagle\ simulations to study the connection between the quenching
timescale, $\tau_{\rm Q}$, and the physical mechanisms that transform
star-forming galaxies into passive galaxies. By quantifying $\tau_{\rm Q}$ in
two complementary ways - as the time over which (i) galaxies traverse the green
valley on the colour-mass diagram, or (ii) leave the main sequence of star
formation and subsequently arrive on the passive cloud in specific star
formation rate (SSFR)-mass space - we find that the $\tau_{\rm Q}$ distribution
of high-mass centrals, low-mass centrals and satellites are divergent. In the
low stellar mass regime where $M_{\star}<10^{9.6}M_{\odot}$, centrals exhibit
systematically longer quenching timescales than satellites ($\approx 4$~Gyr
compared to $\approx 2$~Gyr). Satellites with low stellar mass relative to
their halo mass cause this disparity, with ram pressure stripping quenching
these galaxies rapidly. Low mass centrals are quenched as a result of stellar
feedback, associated with long $\tau_{\rm Q}\gtrsim 3$~Gyr. At intermediate
stellar masses where $10^{9.7}\,\rm M_{\odot}<M_{\star}<10^{10.3}\,\rm
M_{\odot}$, $\tau_{\rm Q}$ are the longest for both centrals and satellites,
particularly for galaxies with higher gas fractions. At $M_{\star}\gtrsim
10^{10.3}\,\rm M_{\odot}$, galaxy merger counts and black hole activity
increase steeply for all galaxies. Quenching timescales for centrals and
satellites decrease with stellar mass in this regime to $\tau_{\rm
Q}\lesssim2$~Gyr. In anticipation of new intermediate redshift observational
galaxy surveys, we analyse the passive and star-forming fractions of galaxies
across redshift, and find that the $\tau_{\rm Q}$ peak at intermediate stellar
masses is responsible for a peak (inflection point) in the fraction of green
valley central (satellite) galaxies at $z\approx 0.5-0.7$.
