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.

A Novel Boundary Integral Method for Slow Free Surface Flows

“…Since that time there has been widespread disagreement, both in the experimental and numerical literature, about the character of the free surface (and therefore about the solution to the problem) in the neighbourhood of the channel orifice and, in particular, about the angle of separation. The values of the separation angles computed by Owens [14] for capillary numbers Ca ranging from 1 to 1,000 fell well within the interval of values published in previous experimental and numerical papers, although there is a paucity of such results available in the literature. His analysis showed that for non-zero surface tensions the normal stress and curvature of the free surface were unbounded at the point of separation, consistent with the analysis and results of Anderson and Davis [15], Schultz and Gervasio [16] and Salamon et al [17].…”

“…It is shown in Section 4.1 that for any given surface tension, the (positive) slope of the free surface attains a maximum near the separation point and that the second derivative grows dramatically as the separation point is approached along the free surface, the curvature at the separation point itself being infinite. In Section 4.2 we present results for the extrudate swell ratio over a range of capillary numbers and these are shown to be in excellent agreement with others in the numerical literature [14], [17], [19], [25]. However, recognising that the extrudate swell ratio is a somewhat crude measure of the correctness of the solution, in Section 4.3 we also post our calculated values for the separation angles α and leading order indices λ 1 appearing in the asymptotic forms of the flow variables in the immediate neighbourhood of the separation point.…”

“…The boundary conditions of Section 2.2.2 also allow A 2 , B 2 , C 2 and D 2 to be written in terms of C 1 . In agreement with the r exponent in the correction Caψ 1 to the leading order term in the perturbation expansion (2.13a) of the stream function in an article of Anderson and Davis [15], it may be shown [14] that λ 2 = 2λ 1 is necessary in order that the separation angle α be different from π. Thus, for i = 1, 2 we may write…”

“…The problem of the planar extrusion of a Newtonian fluid at vanishing Reynolds number Re was considered by Kelmanson [18] using a classical BIM but no account of the singularity in the stresses at the separation point C was taken, leading to the author's admission that it was to be expected that near this point the numerical results would be in error. Owens [14] has also used a BIM to solve the same problem, with the incorporation of the singular solution near the separation point. However, due to the non-integrability at C of ∂ω/∂n, appearing in the integral formulae for ψ and ω, he chose a boundary integration path that closely skirted the singular point, thus introducing a source of error.…”

“…Then, in Section 3.2 we show that it is not necessary to use integration by parts on the entire boundary when ∂ω/∂n is replaced by ∂p/∂s in the integral representations of ψ and ω, but that it may be confined to the intersection of the boundary with a small neighbourhood of the separation point. Since the integration path used in the present BIM passes directly through the separation point this is an important difference between the new method and that of Owens [14] who was obligated to skirt the singularity. In supplying details of the discretization of the new BIM in Section 3.2.1 it becomes clear that there may be fewer jump terms in the pressure to contend with than in Hansen and Kelmanson's BIM and that the boundary grid for the computation of p and the other flow variables becomes a staggered one.…”