AI 中文总结
本文针对自由悬挂粘弹性液体薄膜的轴对称孔洞扩张问题，采用Oldroyd-B模型推导薄膜方程，发现粘弹性应力会提高孔洞扩张的指数增长率，其结果与实验和纯粘性理论的差异相关。
AI 中文摘要
对高粘性聚合物薄膜的实验表明，被刺穿的孔洞会随时间呈指数扩张，且孔洞边缘不会持续积聚液体。这种响应与Taylor-Culick描述不同，Taylor-Culick描述中被 displaced（此处保留原词，指被排挤的）液体会积聚在以恒定速度移动的增长边缘。尽管这些差异最初被归因于粘弹性，但后来用纯粘性理论解释，使得粘弹性应力的作用仍未解决。我们分析了由Oldroyd-B模型描述的自由悬挂粘弹性液体薄膜中轴对称孔洞的扩张。利用孔洞半径与薄膜厚度之间的长度尺度分离，我们在孔洞尺度上推导了拉伸薄膜方程，并从尖端区域的渐近力平衡中得到有效边界条件。针对弱粘弹性（Weissenberg数Wi≪1）和超稀极限（聚合物粘度μ_p≪溶剂粘度μ_s），获得了解析解，其中Wi为Weissenberg数，μ_s和μ_p分别为溶剂和聚合物粘度。对于弱粘弹性，无量纲孔洞半径近似按e^((0.5+αWiβ_p)T)增长，其中α=(12−6log2−π)/21≈0.224，β_p=μ_p/(μ_s+μ_p)。在超稀极限中，半径近似按e^((0.5+α*β_p)T)增长，其中α*(Wi)>0为数值计算得到的值。在这两种情况下，与牛顿流体极限相比，粘弹性应力会提高指数增长率并引起薄膜厚度变化，在回缩边缘附近增厚。这种加速源于聚合物的周向拉伸和径向压缩，这会重新分布薄膜中的应力并改变尖端的应力平衡，从而产生更强的向外径向拉伸流动。
英文摘要
Experiments on highly viscous polymeric films show that punctured holes expand exponentially in time, without sustained accumulation of liquid near the rim. This response departs from the Taylor--Culick description, in which displaced liquid accumulates in a growing rim that moves at constant speed. Although these differences were initially attributed to viscoelasticity, they were later rationalized using a purely viscous theory, leaving the role of viscoelastic stresses unresolved. We analyze the expansion of an axisymmetric hole in a freely suspended viscoelastic liquid sheet described by the Oldroyd-B model. Exploiting the separation of length scales between hole radius and film thickness, we derive extensional thin-film equations on the scale of the hole and an effective boundary condition from an asymptotic force balance in the tip region. Analytical solutions are obtained for weak viscoelasticity, $Wi\ll 1$, and the ultra-dilute limit, $μ_p\llμ_s$, where $Wi$ is the Weissenberg number, while $μ_s$ and $μ_p$ are solvent and polymeric viscosities, respectively. For weak viscoelasticity, the dimensionless hole radius grows approximately as $e^{(0.5+αWi β_p)T}$, where $α=(12 - 6\log 2-π)/21\approx 0.224$ and $β_p=μ_p/(μ_s+μ_p)$. In the ultra-dilute limit, the radius grows approximately as $e^{(0.5+α^{*}β_p )T}$, where $α^{*}(Wi)>0$ is evaluated numerically. In both regimes, viscoelastic stresses increase the exponential growth rate relative to the Newtonian limit and induce film-thickness variations, with thickening near the retracting edge. This acceleration arises from azimuthal stretching and radial compression of the polymers, which redistribute stresses in the film and modify the stress balance at the tip, leading to a stronger outward radial extensional flow.
Comments30 pages, 7 figures
