A fluid representation of viscosity has been incorporated into a set of fluid equations that are maximally ordered in the ``short-radial-gradient-scale-length`` (srgsl) ordering that is appropriate for the edge of tokamak plasmas. The srgsl ordering raises viscous drifts and other viscous terms to leading order and fundamentally alters the character of the fluid equations. A leasing order viscous drift is identified. Viscous-driven radial particle and energy fluxes in the scrape-off layer and divertor channel are estimated to have an order unity effect in reducing radial peaking of energy fluxes transported along the field lines to divertor collector plates.