RUI: Hydrodynamic stability and motion of free surface and thin film flows in cylindrical and planar geometries

平面和圆柱几何形状的自由表面和薄膜流动出现在广泛的工业应用中，如喷墨打印、纤维纺丝、农业和工业喷涂、薄膜涂层和窗帘涂层。对于这些流动来说，经历某种形式的不稳定是很常见的，这是一种可能需要也可能不需要的特征。在任何一种情况下，控制不稳定性对于保持应用程序的质量控制都是至关重要的。该项目的成果将有助于从根本上理解流体性质(特别是表面张力、粘度和弹性)、流动条件(斯托克斯和惯性流)以及流动几何形状(平面和圆柱形)对流体动力稳定性的影响。将详细研究三个问题：(1)粘性和粘弹性流体随时间变化的自由表面伸展流动的运动和稳定性；(2)沿涂层金属丝形成的自由表面扰动的动力学；(3)薄膜涂层流动的动力学和稳定性。该项目的范围是跨学科的，包括理论、数值和实验部分。模拟这些流动的方程：Navier-Stokes方程、粘弹性本构模型耦合的柯西方程、一维细长渐近方程和薄膜方程将使用偏微分方程组、线性稳定性分析、渐近分析和数值方法进行分析。实验有两个目的：实验室中的观察将被用来直观地进入数学模型，理论和数值预测将直接与实验数据进行比较，以检查这些结果的有效性。该项目包括对本科生进行数学方法、数值模拟和实验方面的培训，提供跨学科的研究经验。自由表面和薄膜流动出现在广泛的工业应用中，如喷墨打印、纤维纺丝、农业和工业喷涂、薄膜涂层和窗帘涂层。对于这些流动来说，经历某种形式的不稳定是很常见的，这是一种可能需要也可能不需要的特征。在这两种情况下，控制不稳定性对于保持应用程序的质量控制至关重要，无论是产生相同液滴大小的喷雾、减少喷墨打印中的二次液滴形成以防止飞溅在打印页面上，还是在没有滴痕或干斑的均匀涂料涂层上。这个项目的结果将有助于从根本上理解流体性质、流动条件和流动几何对流动稳定性所起的作用。将详细研究三个问题：(1)粘性和粘弹性流体随时间变化的自由表面伸展流动的运动和稳定性；(2)沿涂层金属丝形成的自由表面扰动的动力学；(3)薄膜涂层流动的动力学和稳定性。该项目的范围是跨学科的，包括理论、数值和实验部分。实验有两个目的：实验室中的观察将被用来直观地进入数学模型，理论和数值预测将直接与实验数据进行比较，以检查这些结果的有效性。该项目包括对本科生进行数学方法、数值模拟和实验方面的培训，提供真正的跨学科研究经验。

Free surface and thin film flows in planar and cylindrical geometries appear in a wide variety of industrial applications, such as ink-jet printing, fiber-spinning, agricultural and industrial spraying, film coating and curtain coating. It is common for these flows to undergo some form of instability, a feature that may or may not be desired. In either case, controlling instabilities is paramount to maintaining quality control in applications. The results of this project will contribute to the fundamental understanding of the role that: fluid properties (specifically, surface tension, viscosity and elasticity); flow conditions (Stokes versus inertial flows); and flow geometry (planar versus cylindrical), play on hydrodynamic stability. Three problems will be studied in detail: (1) the motion and stability of time-dependent free surface extensional flows of viscous and viscoelastic fluids; (2) the dynamics of free surface perturbations that form along a coated wire; and (3) the dynamics and stability of thin film coating flows. The project is interdisciplinary in scope with theoretical, numerical and experimental components involved. The equations which model these flows: the Navier-Stokes equations, the Cauchy equations coupled to a viscoelastic constitutive model, 1D slender asymptotic equations and thin film equation will be analyzed using techniques in partial differential equations, linear stability analysis, asymptotic analysis and numerical methods. Experiments serve two purposes: observations in the lab will be used to gain intuition into the mathematical modeling, and theoretical and numerical predictions will be compared directly to experimental data to check the validity of these results. This project includes the training of undergraduate students in mathematical methods, numerical simulation and experiments providing an interdisciplinary research experience.Free surface and thin film flows appear in a wide variety of industrial applications, such as ink-jet printing, fiber-spinning, agricultural and industrial spraying, film coating and curtain coating. It is common for these flows to undergo some form of instability, a feature that may or may not be desired. In either case, controlling instabilities is paramount to maintaining quality control in applications, whether it involves producing a spray of equal drop size, reducing secondary drop formation in ink-jet printing to prevent splatter on the printed page or applying an even coat of paint with no drip marks or dry patches. The results of this project will contribute to the fundamental understanding of the role that fluid properties, flow conditions and flow geometry play on the stability of a flow. Three problems will be studied in detail: (1) the motion and stability of time-dependent free surface extensional flows of viscous and viscoelastic fluids; (2) the dynamics of free surface perturbations that form along a coated wire; and (3) the dynamics and stability of thin film coating flows. The project is interdisciplinary in scope with theoretical, numerical and experimental components involved. Experiments serve two purposes: observations in the lab will be used to gain intuition into the mathematical modeling, and theoretical and numerical predictions will be compared directly to experimental data to check the validity of these results. This project includes the training of undergraduate students in mathematical methods, numerical simulation and experiments providing a truly interdisciplinary research experience.