Mathematical Sciences: Modelling, Analysis and Computation in Viscoelasticity

在高弹性和粘性很强的流体中，如聚合物溶液和熔体，实验中观察到了有趣的现象。这种非牛顿材料可以被数学建模为具有衰退记忆的粘弹性流体，其表现出介于纯弹性材料的非线性双曲响应和粘性牛顿流体的强扩散抛物线响应之间的行为。在某些流动状态下，这些流体表现出严重破坏聚合物加工的不稳定性。实验室观察发现，在压力梯度驱动的流动中存在“喷射”不稳定性（Vinogradov et al., 1972），在固定体积流速下的流动中存在持续振荡（Lim & Schowalter, 1989），在阶跃剪切应变实验中存在异常（Morrison & Larson, 1991）。许多研究人员将观察结果归因于“滑移”或“明显滑移”，即流体与壁面的附着力丧失。这个项目涉及对这些现象的另一种解释的调查。我们的假设是，这三种材料都有一个共同的起源，在散装材料的性质，而不是粘合剂的性质。为了验证这一假设，对狭缝模具内相应的一维剪切流、压力驱动流和活塞驱动流以及Couette流进行了建模。所采用的流体模型的特征是稳态剪切应力与应变速率之间的非单调关系。分析和数值模拟表明，聚合物体系在靠近壁的薄层中发生状态变化，形成滑移层。非单调模型的初/边值问题的解结构非常丰富，能够模拟这些流动问题的数值方法的发展本身就很有意义。相同的基本系统的时间相关，拟线性偏微分方程被用来模拟所有三个实验；在这三种情况下，采用了不同的强迫项、边界条件和初始条件。在两种意义上，控制系统在时间上是全局适定的：关于由光滑初始数据产生的经典解，以及关于包含应力和应变速率不连续的“几乎经典”解。计算结果与实验结果在定性上是一致的，在压力驱动的流动中表现出喷射，在活塞驱动的流动中表现出持续的振荡，并且在阶跃应变后松弛模量出现异常。在简化的近似系统中，利用相平面技术可以分析压力驱动流动中的喷射及其相关现象。由于系统是无限维的，活塞驱动和阶梯应变实验的分析比较困难。例如，数值模拟强烈表明，在活塞驱动的流动中存在Hopf分岔，导致观察到的持续振荡。为了将预测与实验进行比较，确定极限环的幅度和频率如何取决于物理参数是至关重要的。这需要对控制方程进行详细的分析。这在每个实验中都是必需的，这也是本研究的中心焦点。该项目的主题一直是并将继续是使用数值模拟来指导对控制方程及其近似值的分析，以确定模型预测的时间尺度、幅度和其他特征。

Intriguing phenomena have been observed experimentally in highly elastic and very viscous fluids, such as polymer solutions and melts. Such non-Newtonian materials can be modeled mathematically as viscoelastic fluids with fading memory, which exhibit behavior intermediate between the nonlinear hyperbolic response of purely elastic materials and the strongly diffusive, parabolic response of viscous Newtonian fluids. In certain flow regimes, these fluids exhibit instabilities that severely disrupt polymer processing. Laboratory observations have found "spurt " instabilities in pressure-gradient driven flows (Vinogradov et al., 1972), persistent oscillations in flow at fixed volumetric flow rate (Lim & Schowalter, 1989), and anomalies in step shear strain experiments (Morrison & Larson, 1991). Many researchers attribute the observations to "slip" or "apparent slip, " i.e., loss of adhesion of the fluid to the wall. This project involves the investigation of an alternative explanation for these phenomena. The hypothesis is that all three have a common origin in bulk material properties, rather than adhesive properties. To test this hypothesis, the corresponding one-dimensional shear flows pressure-driven and piston-driven flow in a slit die, and Couette flow are modeled. The characteristic feature of the fluid models employed is a non-monotone relation between steady shear stress and strain rate. Analysis and numerical simulations show that the polymer system changes state in a thin layer near the wall, giving the appearance of a slip layer. The structure of solutions of initial/boundary-value problems for non-monotone models is remarkably rich, and the development of numerical methods capable of simulating these flow problems has interest in its own right. The same basic system of time-dependent, quasilinear partial differential equations is used to model all three experiments; different forcing terms, as well as boundary and initial conditions, are used in the three cases. The governing system is globally well-posed in time, in two senses: with respect to classical solutions arising from smooth initial data, and with respect to "almost classical" solutions, containing discontinuities in the stress and strain rate. Calculated solutions are in good qualitative agreement with the experiments, showing spurt in pressure-driven flow, persistent oscillations in piston-driven flow, and the development of anomalies in the relaxation modulus after a step strain. In terms of a reduced approximating system, spurt and its related phenomena in pressure-driven flow can be analyzed by using phase-plane techniques. The analysis for piston-driven and step strain experiments is more difficult because the system is infinite-dimensional. For example, numerical simulations strongly suggest that there is a Hopf bifurcation in piston-driven flow, leading to the observed persistent oscillations. In order to compare predictions to experiment, it is crucial to determine how the amplitude and frequency of the limit cycle depends on physical parameters. This requires detailed analysis of the governing equations. This is required in each of the experiments, and that is the central focus of this research. The theme of this project has been and will continue to be the use of numerical simulation to guide analysis of the governing equations and approximations to them, in order to identify the time-scales, amplitudes, and other characteristic features of the predictions of the model.