The recently proposed finite element (FE) formulation for viscoelastic flows that allows the use of equal order linear interpolants for all variables and simultaneously does not suffer from the high Weissenberg number problem, is extended to free surface flows. The coupling of this Petrov-Galerkin stabilized FE formulation with the quasi-elliptic mesh generator allows us to obtain stable numerical solutions in highly deformed meshes and for very high values of the Weissenberg number (Wi). We present benchmark solutions in three free surface flows: the axisymmetric filament stretching, the elastocapillary-driven formation of bead-on-a-string, and the 2-dimensional, planar extrudate swell flow. In all cases, we attain converged solutions for values of Wi that have never been reached before by FE. The accuracy and robustness of the proposed numerical scheme are illustrated by achieving mesh and time step convergence under extreme mesh deformation conditions such as the bead-on-a-string (BOAS) formation during filament stretching. The formulation is enriched further with a discontinuity capturing scheme that enhances the quality of the solution around singularities dramatically. Finally, our simulations reveal for the first time the existence of lip-vortices in the steady planar extrudate-swell flow of Oldroyd-B fluids, which converge with mesh refinement.

最近提出的用于粘弹性流的有限元（FE）公式扩展到自由表面流，该公式允许对所有变量使用等阶线性插值，并且同时不存在高Weissenberg数问题。Petrov-Galerkin稳定的FE配方与准椭圆网格生成器的耦合使我们能够在高度变形的网格中获得稳定的数值解，并获得很高的Weissenberg数（Wi）值。我们提供了三种自由表面流动的基准解决方案：轴对称长丝拉伸，弹力驱动的串珠串形成以及二维平面挤出物溶胀流动。在所有情况下，我们都能获得Wi值的融合解决方案FE从未达到过的目标。通过在极端网格变形条件下（例如在长丝拉伸过程中串珠（BOAS）形成）下实现网格和时间步收敛，说明了所提出数值方案的准确性和鲁棒性。通过不连续性捕获方案进一步丰富了制剂，该方案可极大地提高奇异点周围溶液的质量。最后，我们的模拟首次揭示了Oldroyd-B流体在稳定的平面挤出物-溶胀流中存在唇涡，并随着网格的细化而收敛。