Flow rate-pressure drop relation for viscoelastic fluids in narrow and confined non-uniform geometries

Pressure-driven flows of viscoelastic polymer solutions in narrow non-uniform geometries are ubiquitous in nature and various applications. One of the key interests for such flows is understanding the relationship between the flow rate and pressure drop, which, to date, is studied primarily using numerical simulations. Here, we aim to rationalize the longstanding contradiction between simulations using the "simple" Oldroyd-B and FENE-CR models and experiments for the flow rate-pressure drop relation of viscoelastic fluids in some geometries. To this end, we provide a theoretical framework for calculating the flow rate-pressure drop relation in arbitrarily shaped, narrow channels. Through the combination of the lubrication approximation and the Lorentz reciprocal theorem, our theory allows deriving analytical expressions for the flow rate-pressure drop relation for a wide variety of continuum-level constitutive viscoelastic models in the weakly viscoelastic limit. Furthermore, we discuss the shortcomings of the Oldroyd-B and FENE-CR models and suggest their modification through accounting for additional microscopic features of polymer flows.