On the separation angle of the free surface of a viscous fluid at a straight edge

Robert G. Owens

We rework some of the die swell singularity analysis for Stokes flow, originally by Ramalingam (1994) in Appendix A of his PhD thesis, in an attempt to demonstrate that for capillary numbers in the range $(0,\infty)$ the curvature may enter into the normal stress balance on the free surface and lead to separation angles exceeding $180^\text{o}$ and infinite curvature at the separation point. The singular coefficients in the asymptotic solution and the free surface shape in a neighbourhood of the separation point cannot be determined by a local analysis of the Michael type (see Michael (1958)) but must be found from matching with the solution valid away from the die edge.

The numerical method that we use in the truncated die-swell domain is a boundary element method adapted from an earlier method of Hansen and Kelmanson (1992, 1994) and suitable for the solution of creeping flow boundary value problems where the boundary conditions lead to singularities in the stresses.

Although there is some variation in the extrudate swell ratios at different capillary numbers reported in the numerical literature, our results for capillary numbers $Ca$ from $1$ to $1000$ are within the range of values published in earlier papers. The values of the separation angle that we have computed are in agreement with those presented in previous experimental and numerical papers, although there is, admittedly, a paucity of such results available in the literature.

This is joint work with Loïc Gobet.

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