Reciprocal theorem for calculating the flow rate–pressure drop relation for complex fluids in narrow geometries

Abstract

A key aspect in understanding pressure-driven flows of non-Newtonian fluids in narrow and confined geometries is the relationship between the flow rate and pressure drop. Using the Lorentz reciprocal theorem, we derive a closed-form expression for the flow rate-pressure drop relation of complex fluids in narrow channels of arbitrary shape, which holds for a wide class of viscoelastic and shear-thinning constitutive models. For the weakly non-Newtonian limit, our theory provides the first-order non-Newtonian correction for the flow rate-pressure drop relation solely using the corresponding Newtonian solution, eliminating the need to solve the non-Newtonian flow problem.