Birefringent pipes: the steady flow of a dilute polymer solution near a stagnation point

Abstract

In steady flows at high Deborah numbers high polymer stresses are often concentrated within thin boundary layers along streamlines downstream of flow stagnation points where the polymer extension is large. The layers appear as birefringent lines in optical experiments. Detailed observations of the flow near a stagnation point have shown a complex sequence of birefringence structures, which appear as the flow rate increases, for polymer concentrations above some critical value. The first transition is from a solid birefringent line to a hollow birefringent cylinder or 'pipe'. In this paper we calculate the modification of the flow due to the presence of polymer for a FENE (finitely extensible non-linear elastic) dumbbell model with non-linear hydrodynamic friction, and demonstrate that the associated reduction in strain rate at the stagnation point can be sufficient to produce a pipe structure. The polymer concentrations required to produce this transition are found to be in qualitative agreement with experiment. We determine also the thickness of birefringent strands as a function of polymer concentration, molecular weight, flow rate and inertia. These results too are found to be in qualitative agreement with experiment. We show finally that for a FENE model with constant hydrodynamic friction birefringent strands are produced, but we do not find pipes at realistic values of the parameters.