START-UP OF A STRONGLY EXTENSIONAL FLOW OF A DILUTE POLYMER-SOLUTION

We consider an impulsively started flow of a dilute monodisperse polym er solution for which the Reynolds number is low but the Weissenberg n umber may be large. The polymers are modelled as linear-locked dumbbel ls, and the aspect ratio of the flow geometry (a 'cross-slot') is take n 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 Mec h., 34 (1990) 319-349) and we determine how such strands develop from rest. The flow is computed using a novel mixed Lagrangian - Eulerian f ormulation that is particularly well suited to the asymptotic method. The numerical method is checked by comparing its results with known ex pressions for steady extensional flows. We find that the time evolutio n takes place in a number of stages of which the first two or three ar e sharply delineated and in each of which the strand near the stagnati on point suddenly becomes thicker, after which a front of increased st rand width propagates downstream at constant velocity. The strain-rate at the stagnation point correspondingly first undershoots and then ov ershoots at each stage. Successive stages take progressively longer. T hese theoretical results are found to be in qualitative agreement with measurements of optical birefringence in a two-roll mill made by Geff roy and Leal [J. Non-Newtonian Fluid Mech., 35 (1990) 361 400] on the Boger fluid MI, and on a monodisperse polystyrene solution.