Mathematical Sciences: Modelling, Analysis and Computation in Viscoelasticity

数学科学：粘弹性建模、分析和计算

• 批准号: 9216153
• 负责人: David Malkus
• 金额: $ 17.95万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-01-15 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9216153&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Modelling; Analysis; Computation

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.

有趣的现象已经在实验中观察到， 高弹性和非常粘稠的流体，如聚合物溶液 并融化。这样的非牛顿材料可以用 在数学上，作为具有衰减记忆的粘弹性流体， 表现出介于非线性双曲线 纯弹性材料的响应和强扩散， 粘性牛顿流体的抛物线响应。在一定流量下 这些流体表现出不稳定性， 聚合物加工实验室观察发现“井喷式“ 压力梯度驱动流的不稳定性（Vinogradov等人 例如，1972年），在固定的体积流量持续振荡 流速（Lim Schowalter，1989）和阶跃剪切中的异常 菌株实验（莫里森拉森，1991）。许多研究人员 将观察结果归因于“滑动”或“明显滑动“，即， 流体对壁的粘附力丧失。这个项目涉及 对这些问题的替代解释的调查 现象。假设这三者有共同的起源 在散装材料的性能，而不是粘合剂的性能。 为了验证这一假设，相应的一维 剪切流在狭缝模头中压力驱动和活塞驱动流动， 和库埃特流的模型。的特性特征 所采用的流体模型是稳态之间的非单调关系， 剪切应力和应变率。分析和数值模拟 表明聚合物体系在接近100 ℃时在薄层中改变状态， 墙壁，给人一种滑层的感觉。的结构 非单调初边值问题的解 模型是非常丰富的，和发展的数值 能够模拟这些流动问题的方法， 自己的权利。同样的时间依赖的基本系统， 拟线性偏微分方程用于建模所有 三个实验;不同的强迫项，以及边界 和初始条件，用于这三种情况。的 治理系统在时间上是全局适定的，在两个意义上： 对于光滑初始条件下的经典解， 数据，并就“几乎经典”的解决方案， 包含应力和应变率的不连续性。 计算的解决方案是在良好的定性协议与 实验表明，在压力驱动的流动中， 活塞驱动流中的振荡，以及 在一个步骤应变后的松弛模量异常。方面 一个简化的近似系统，喷射和其相关的 压力驱动流中的现象可以用 相平面技术活塞驱动步进式液压缸的分析 应变实验更困难，因为系统 无限维的例如，数值模拟强烈 表明活塞驱动流动存在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，至关重要的是要确定 极限环的振幅和频率取决于 物理参数。这就需要详细分析 控制方程这是每个实验都需要的， 这也是本研究的中心焦点。的主题 项目已经并将继续使用数字 模拟指导控制方程的分析， 为了确定时间尺度， 振幅和预测的其他特征 模型的。