The phenomenon of the ‘coil-stretch’ (C-S) transition, wherein a long-chain polymer initially in a coiled state undergoes a sudden configuration change to become fully stretched under steady elongational flows, has been widely recognized. This transition can display intricate hysteresis behaviours under specific critical conditions, giving rise to unique rheological characteristics in dilute polymer solutions. Historically, microscopic stochastic models and Brownian dynamics simulations have shed light on the underlying mechanisms of the transition by uncovering bistable configurations of polymer chains. Following the initial work by Cerf (J. Chem. Phys., vol. 20, 1952, pp. 395–402), we introduce a continuum model in this study to investigate the C-S transition in a constant uniaxial elongational flow. Our approach involves approximating the unfolding process of the polymer chain as an axisymmetric deformation of an elastic particle. We make the assumption that the particle possesses uniform material properties, which can be represented by a nonlinear, strain-hardening constitutive equation to replicate the finite extensibility of the polymer chain. Subsequently, we analytically solve for the steady-state deformation using a polarization method. By employing this reduced model, we effectively capture the C-S transition and establish its specific correlations with material and geometric properties. The hysteresis phenomena can be comprehended through a force-balance analysis, which involves comparing the externally applied viscous forces with the intrinsic elastic responsive forces. We demonstrate that our model, while simple, unveils rich elastohydrodynamics of the C-S transition.

中文翻译：

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稀聚合物在拉伸流动中的卷曲-拉伸转变的弹性颗粒模型

“卷曲-拉伸”（CS）转变现象已得到广泛认可，其中最初处于卷曲状态的长链聚合物经历突然的构型变化，在稳定的拉伸流下变得完全拉伸。这种转变可以在特定的临界条件下表现出复杂的滞后行为，从而在稀聚合物溶液中产生独特的流变特性。从历史上看，微观随机模型和布朗动力学模拟通过揭示聚合物链的双稳态构型，揭示了转变的潜在机制。继 Cerf 的初步工作之后（J.化学。物理。，卷。 20, 1952, pp. 395–402），我们在本研究中引入了连续介质模型来研究恒定单轴拉伸流中的 CS 转变。我们的方法涉及将聚合物链的展开过程近似为弹性颗粒的轴对称变形。我们假设颗粒具有均匀的材料特性，可以用非线性应变硬化本构方程来表示，以复制聚合物链的有限可延伸性。随后，我们使用极化方法分析求解稳态变形。通过采用这种简化模型，我们有效地捕获 CS 转变并建立其与材料和几何特性的特定相关性。滞后现象可以通过力平衡分析来理解，该分析涉及将外部施加的粘性力与内在弹性响应力进行比较。我们证明了我们的模型虽然简单，但却揭示了 CS 转变的丰富的弹性流体动力学。