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We study the dynamics of a viscous protoplanetary disc hosting a populationof dust grains with a range of sizes. We compute steady-state solutions, andshow that the radial motion of both the gas and the dust can deviatesubstantially from those for a single-size dust population. Although theaerodynamic drag from the dust on the gas is weaker than in the case where allgrains are optimally coupled to the gas, the cumulative "back-reaction" of thedust particles can still alter the gas dynamics significantly. In typicalprotoplanetary discs, the net effect of the dust back-reaction decreases thegas accretion flow compared to the dust-free (viscous) case, even fordust-to-gas ratios of order $1\%$. In the outer disc, where dust grains aretypically less strongly coupled to the gas and settle towards the midplane, thedust back-reaction can even drive outward gas flow. Moreover, the radial inwarddrift of large grains is reduced below the gas motion in the inner discregions, while small dust grains follow the gas dynamics over all the discextent. The resulting dust and gas dynamics can give rise to observablestructures, such as gas and dust cavities. Our results show that the dustback-reaction can play a major role in both the dynamics and observationalappearance of protoplanetary discs, and cannot be ignored in models ofprotoplanetary disc evolution.

Gas and multi-species dust dynamics in viscous protoplanetary discs: the importance of the dust back-reaction