A new approach to model the flow of polymer solutions is investigated. The flow behavior of these materials has been under investigation for decades without reaching a consistent explanation of the observed phenomena such as shear banding in shear flow and the spurt phenomena in pressure-driven pipe. In this investigation, we explore the coupling of the mass transport and the viscoelastic dynamics and its influence on the flow of polymer solutions in pipes. We implement a predictive constitutive model for polymer solutions that reproduces observed flow behaviors using scaling arguments to estimate material properties from a minimal number of measured quantities. The inherent existence of multiple solutions to the model's system of equations allows for the prediction of sudden increases of flow rate in a pressure-driven channel flow is reported due to a stress-induced depletion layer at the pipe wall of the size of the correlation length of the solution. This investigation bridges the gap between theoretical studies that focus on the interfacial physics and those that focus on the bulk flows to explain the spurt phenomena of polymer solutions.