Flow rate-pressure drop relation for viscoelastic fluids in narrow geometries

Pressure-driven flows of viscoelastic polymer solutions in narrow non-uniform geometries are ubiquitous in nature and various applications. For such flows, one of the key interests is understanding the relationship between the flow rate and pressure drop, which, to date, is studied primarily using numerical simulations. Here, we provide a theoretical framework for calculating the flow rate-pressure drop relation for viscoelastic flows in arbitrarily shaped, narrow channels. We apply lubrication theory and derive analytical expressions for the flow rate-pressure drop relation for the Oldroyd-B model in the weakly viscoelastic limit. Furthermore, we apply the Lorentz reciprocal theorem and show that the flow rate-pressure drop at higher orders can be determined only using the velocity and stress fields at the previous orders. We compare our analytical results with numerical simulations and find excellent agreement. Given the inability of numerical simulations using the Oldroyd-B and FENE-CR models to predict the experimental flow rate-pressure drop behavior of viscoelastic fluids in some cases, we believe that our approach may be important in providing insight into the cause of this disagreement and in resolving it by accounting for additional microscopic features of polymer flows.