  We do extensive simulations of a simple model of shear-driven jamming in two
dimensions to analyze the velocity distribution at different densities $\phi$
around the jamming density $\phi_J$ and at different low shear strain rates,
$\dot\gamma$. We then find that the velocity distribution is made up of two
parts which are related to two different physical processes which we call the
slow process and the fast process as they are dominated by the slower and the
faster particles, respectively. Earlier scaling analyses have shown that the
shear viscosity $\eta$, which diverges as the jamming density is approached
from below, consists of two different terms and we present strong evidence that
these terms are related to the two different processes: the leading divergence
is due to the fast process whereas the correction-to-scaling term is due to the
slow process. The analysis of the slow process is possible thanks to the
observation that the velocity distribution for different $\dot\gamma$ and
$\phi$ at and around the shear-driven jamming transition, has a peak at low
velocities and that the distribution has a constant shape up to and slightly
above this peak. We then find that it is possible to express the contribution
to the shear viscosity due to the slow process in terms of height and position
of the peak in the velocity distribution and find that this contribution
matches the correction-to-scaling term, determined through a standard critical
scaling analysis. A further observation is that the collective particle motion
is dominated by the slow process. In contrast to the usual picture in critical
phenomena with a direct link between the diverging correlation length and a
diverging order parameter, we find that correlations and shear viscosity
decouple since they are controlled by different sets of particles and that
shear-driven jamming is thus an unusual kind of critical phenomenon.
