High-Deborah-number flows of dilute polymer solutions

Abstract

A boundary layer approximation is developed for the steady flow of dilute polymer solutions at high Deborah numbers. In a class of such flows polymer molecules are most highly extended in narrow strands downstream of stagnation points, as only those molecules that pass close to a stagnation point reside in the flow long enough to become significantly extended. This structure can be seen from optical experiments {1–8} where the strands of extended polymer appear as bright birefringent lines while the rest of the fluid appears dark. High elastic stresses occur only within these elongated regions (here called ‘ birefringent strands’), and it is shown that these may be treated as line distributions of forces within an otherwise Newtonian fluid. This asymptotic approximation is used to analyse the flow past a spherical bubble; the flow in the exit channel of the cross-slot apparatus and the flow between co- and counter-rotating rollers. For flow past a spherical bubble we find quantitative agreement with a full numerical solution of the constitutive equations, and the results for the remaining flows agree closely with experiment.