Colonies of Atta cephalotes (Myrmicinae: Formicidae) construct cleared paths between their nest and the vegetation sources at which they harvest leaf tissue. Here, we employ ideas from traffic engineering to study streams of laden and unladen ants on these paths. The relationship between average traffic speed and the concentration of workers on the road surface follows a relationship similar to what is expected by analogy to fluid dynamics. Although the traffic is composed of eusocial organisms with a common interest in group success, the coarse-grained behavior of Atta traffic displays little more coordination than a moving fluid. The relationship between speed and concentration implies that maximum flow rates (which are likely to be closely tied to colony-level rates of resource acquisition) occur at a relatively high concentration that keeps individual speeds well below their "free flow" maximum. We predict that this optimal concentration will characterize peak traffic throughout a trail network, and we propose a simple behavioral mechanism that would allow trails to be cleared to the correct width to provide the optimal concentration. Collisions (including encounters for antennation) are common in leaf-cutting ant traffic because traffic is not segregated into unidirectional streams. Nonetheless, we find a counterintuitive suggestion that flow rates (with concentration differences statistically removed) are higher when traffic is near a 50:50 mix of outbound and returning ants than when it contains majority flows in a single direction. Mixed-direction traffic may help disperse laden ants with reduced agility, thereby preventing inhomogeneities in the traffic stream that could clog the trail.