The development and implementation of algorithms to investigate drop fragmentation under shear for viscoelastic liquids with surfactant

本文的研究内容是发展和实现剪切作用下液滴破碎数值研究的新算法。研究人员使用了流体体积(VOF)方法来模拟牛顿液体，包括惰性的影响，以及低浓度下的不溶性表面活性剂。使用固定的规则网格。通过颜色函数跟踪流体界面的运动，该颜色函数表示网格单元中流体的体积分数。在数值模拟的每个时间步，色函数随流动平移，并利用色函数的梯度重建界面位置和计算界面张力。在以前的工作中，颜色函数的一阶和二阶导数都是用有限差分来近似的。然而，由于颜色函数实际上是阶跃函数，在曲率等非线性组合中对导数使用有限差分是有问题的。传统的补救方法是使用平滑的颜色函数。另一方面，这要求平滑的长度比例相对于物理比例很小，但相对于网格大小仍然很大。实际上，平滑在界面张力中引入了数值扩散。例如，在水滴分解模拟中，在两到三倍于网格大小的数量级上进行平滑仍然会导致卫星水滴体积通过空间网格细化而不收敛。研究人员最近发展了PROST方法，它是表面张力中界面的抛物线表示，不需要光滑化。从一个抛物面的最小二乘拟合面重构出给定单元及其邻域中的颜色函数值，并由该抛物面反演曲率和法线。这会导致子滴体积与网格优化的收敛。本研究的目的是开发基于VOF-PROST框架的粘弹性液体表面活性剂模拟算法，并在大规模并行处理系统上实现。数学模型允许液体具有记忆、第一和第二法向应力差以及剪切稀化。可溶表面活性剂采用非线性本构模型，沿界面平流并与体积交换。悬浮在另一液体中的液滴在许多化学过程中发生分解，例如熔融塑料的商业混合，以形成回收工业中的新材料。在混合时，子滴的大小和分布会影响产品的质量，因此能够控制这些结果很重要。对于商用搅拌机来说，这是很难预测的，因为流体流动的复杂性，导致了一种反复试验的方法。研究人员的理论预测的出发点是研究单个液滴在明确定义的流场中的变形。设计了数值算法来跟踪液滴变形和破碎的历史，这是对涉及两种不相容液体混合的制造过程的基础知识库的重要贡献。这项工作的成功取决于液体的物理模型以与现有的实验数据进行比较，求解方程的算法的准确性，在高性能计算平台上的实施，以及与执行受控实验的工程师建立的合作。更广泛的影响是，这些算法适用于更广泛的流体-流体系统，例如生理液体的处理。一名博士后研究助理和一名研究生的参与是该项目的一个基本教育组成部分。

The proposed research concerns the development and implementation of new algorithms for the numerical investigation of drop breakup under shear. The investigator has used thevolume-of-fluid (VOF) approach to simulate Newtonian liquids, including the effect of inertia, and insoluble surfactants at low concentration. A fixed regular grid is used. The motion of fluid interfaces is tracked by means of a color function, which represents the volume fraction for a fluid in a grid cell. At each time step of the numerical simulation, the color function is advected with the flow, and gradients of it are used to reconstruct the position of the interface and to compute the interfacial tension force. In prior work, the first and second derivatives of the color function have been approximated by finite differences. However, since the color function is actually a step function, the use of finitedifferences for derivatives in nonlinear combinations such as curvature is problematic. The conventional remedy is to use a smoothed color function. This, on the other hand, requires a length scale of smoothing which is small relative to physical scales but still large relativeto the mesh size. In practice, smoothing introduces numerical diffusion into the interfacial tension force. In drop breakup simulations, for instance, smoothing on the order of two or three times the mesh size still leads to non-convergence of satellite drop volumes with spatial mesh refinement. The investigator has recently developed the method PROST, which is aparabolic representation of the interface in the surface tension force and which requires nosmoothing. The interface is reconstructed from a least square fit of a paraboloid to the values of the color function in a given cell and its neighbors, and the curvature and normalare retrieved from this paraboloid. This leads to convergence of daughter drop volumes with mesh refinement. The objective of the proposed research is to develop algorithms with the VOF- PROST framework to simulate viscoelastic liquids with surfactants, and to implement the codeon large-scale parallel processing systems. The mathematical model allows the liquids to havememory, first and second normal stress differences, and shear-thinning. A nonlinear constitutive model is used for soluble surfactants, with advection along the interface and exchange with the bulk.The breakup of liquid drops suspended in another liquid occurs in a number of chemical processes, such as the commercial mixing of molten plastics to form new materials in the recycling industry. Upon mixing, the size and distribution of the daughter drops influence the quality of the product, and therefore it is important to be able to control these outcomes. This is difficult to predict for a commercial mixer because of the complexity of thefluid flow, resulting in a trial and error approach. The investigator's starting point for a theoretical prediction is to study the deformation of a single droplet in a well-defined flowfield. Numerical algorithms are devised to track the history of drop deformation and breakup,which is an important contribution to the fundamental knowledge base of manufacturing processes that involve mixing two immiscible liquids. The success of this work depends on thephysical modeling of the liquids to compare with available experimental data, the accuracy ofthe algorithms to solve the equations, the implementation on high-performance computing platforms, and established collaborations with engineers who perform controlled experiments. Abroader impact is that the algorithms apply to a wider class of fluid-fluid systems, such asthe processing of physiological fluids. The participation of a postdoctoral research associate and a graduate student is an essential educational component of the project.