We develop a mathematical model of direct-flow filtration operating at constant flux to understand the pressure-time signature. We combine the effects of fluid flow with those of membrane fouling and caking to explain the gradual increase in driving pressure that is often reported. We model the periodic backflushes used to clean such membranes and show that the elasticity of the membrane may explain the limited effectiveness of these backflushes. We also consider strategies for the operation of direct-flow filtration and show that tuning the flux and rate of backflushing, as well as the membrane material structure, may allow for improvements in membrane performance.