  In order to understand the processes that quench star formation within rich
clusters, we construct a library of subhalo orbits drawn from $\Lambda$CDM
cosmological N-body simulations of four rich clusters. The orbits are combined
with models of star formation followed by quenching in the cluster environment.
These are compared with observed bulge and disc colours and stellar absorption
linestrength indices of satellite galaxies. Models in which the bulge stellar
populations depend only on the galaxy subhalo mass while the disc quenching
depends on the cluster environment are acceptable fits to the data. An
exponential disc quenching timescale of 3 - 3.5 Gyr is preferred. Models with
short ($\lesssim 1$ Gyr) quenching timescales yield cluster-centric gradients
in disc colours and Balmer line indices that are too steep compared to
observations. We also examine models in which there is quenching in lower mass
groups prior to cluster infall ("pre-processing"), finding that such models are
a better fit to the data than models without pre-processing and require similar
quenching times. The data slightly prefer models where quenching occurs only
for galaxies falling within about 0.5 $r_{200}$. Finally, we have examined
models with short quenching timescales of 1 Gyr, but a long delay time of 3 Gyr
prior to quenching. All models with short quenching timescales, even such
"delayed-then-rapid" quenching models, produce excessively red galaxies near
the cluster core and are strongly disfavoured by the data. These results imply
that the environments of rich clusters must impact star formation rates of
infalling galaxies on relatively long timescales -- several times longer than a
typical halo spends within the virial radius of a cluster. This scenario
favours gentler quenching mechanisms such as slow "strangulation" over more
rapid ram-pressure stripping.
