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

Abstract

We rework some of the die-swell singularity analysis for Stokes flow, originally by Ramalingam (Ramalingam, 1994 Fiber spinning and rheology of liquid-crystalline polymers, PhD thesis, Massachusetts Institute of Technology) 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 $180^{\circ }$ 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 (Michael, Mathematika, vol. 5, 1958, pp. 82–84) 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 incorporating the singular solution near the separation point. 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 computed separation angles at different values of $Ca$ agree well with the range of separation angles to be found in experimental and numerical papers. The separation angle appears to converge to a value different from $180^{\circ }$ as $Ca$ increases, leading us to conclude that the case of zero surface tension ( $Ca=\infty$ , with corresponding separation angle of $180^{\circ }$ ), is a singular limit.