  We study the dynamics of a viscous protoplanetary disc hosting a population
of dust grains with a range of sizes. We compute steady-state solutions, and
show that the radial motion of both the gas and the dust can deviate
substantially from those for a single-size dust population. Although the
aerodynamic drag from the dust on the gas is weaker than in the case where all
grains are optimally coupled to the gas, the cumulative "back-reaction" of the
dust particles can still alter the gas dynamics significantly. In typical
protoplanetary discs, the net effect of the dust back-reaction decreases the
gas accretion flow compared to the dust-free (viscous) case, even for
dust-to-gas ratios of order $1\%$. In the outer disc, where dust grains are
typically less strongly coupled to the gas and settle towards the midplane, the
dust back-reaction can even drive outward gas flow. Moreover, the radial inward
drift of large grains is reduced below the gas motion in the inner disc
regions, while small dust grains follow the gas dynamics over all the disc
extent. The resulting dust and gas dynamics can give rise to observable
structures, such as gas and dust cavities. Our results show that the dust
back-reaction can play a major role in both the dynamics and observational
appearance of protoplanetary discs, and cannot be ignored in models of
protoplanetary disc evolution.
