Start-up of a strongly extensional flow of a dilute polymer solution

Abstract

We consider an impulsively started flow of a dilute monodisperse polymer solution for which the Reynolds number is low but the Weissenberg number may be large. The polymers are modelled as linear-locked dumbbells, and the aspect ratio of the flow geometry (a ‘cross-slot’) is taken to be sufficiently large that lubrication methods can be applied. In the limit in which the polymers are highly extended, the flow may be described asymptotically using a birefringent strand approximation (O.G. Harlen, J.M. Rallison and M.D. Chilcott, J. Non-Newtonian Fluid Mech., 34 (1990) 319-349) and we determine how such strands develop from rest. The flow is computed using a novel mixed Lagrangian-Eulerian formulation that is particularly well suited to the asymptotic method. The numerical method is checked by comparing its results with known expressions for steady extensional flows. We find that the time evolution takes place in a number of stages of which the first two or three are sharply delineated and in each of which the strand near the stagnation point suddenly becomes thicker, after which a front of increased strand width propagates downstream at constant velocity. The strain-rate at the stagnation point correspondingly first undershoots and then overshoots at each stage. Successive stages take progressively longer. These theoretical results are found to be in qualitative agreement with measurements of optical birefringence in a two-roll mill made by Geffroy and Leal [J. Non-Newtonian Fluid Mech., 35 (1990) 361-400] on the Boger fluid M1, and on a monodisperse polystyrene solution. [J. Polymer Sci., Part B, 30 (1992) 1329-1348].