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

所提出的研究涉及新算法的开发和实施， 液滴剪切破碎的数值研究研究者使用了流体体积（VOF）的方法来模拟牛顿液体，包括惯性的影响，以及低浓度的不溶性表面活性剂。 使用固定的规则网格。的 流体界面的运动通过颜色函数来跟踪，该颜色函数表示网格单元中流体的体积分数。 在数值模拟的每一时间步，颜色函数与流动平流，并利用其梯度重建 界面的位置和计算界面张力。 在先前的工作中， 颜色函数的一阶和二阶导数已经通过有限差分近似。然而，由于颜色函数实际上是一个阶跃函数，因此在非线性组合（如曲率）中使用有限差分导数是有问题的。 传统的补救方法是使用平滑的颜色函数。另一方面，这需要平滑的长度尺度，该尺度相对于物理尺度较小，但相对于网格尺寸仍然较大。在实践中，平滑将数值扩散引入界面张力力。 例如，在液滴破碎模拟中， 网格尺寸的三倍仍然导致卫星液滴体积与空间网格细化的不收敛。 研究者最近发展了PROST方法，它是界面在表面张力作用下的抛物线表示，不需要光滑。该界面是从一个抛物面的最小二乘拟合的颜色函数的值在一个给定的细胞和它的邻居，和曲率和法线检索从这个抛物面。这导致子液滴体积与 网格细化本研究的目的是开发VOF-PROST框架的算法来模拟粘弹性液体与表面活性剂，并实现大规模并行处理系统的代码。 该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。 一个非线性本构模型被用于可溶性表面活性剂，沿界面沿着平流， 悬浮在另一种液体中的液滴的破裂发生在许多化学过程中，例如在回收工业中熔融塑料的商业混合以形成新材料。 在混合时，子液滴的尺寸和分布影响 产品的质量，因此能够控制这些结果非常重要。这是很难预测的商业混合器，因为流体流动的复杂性，导致试错的方法。研究者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。数值算法的设计跟踪液滴变形和破碎的历史，这是一个重要的贡献，涉及混合两种不混溶液体的制造工艺的基础知识库。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， 解决方程的算法，在高性能计算平台上的实现，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，例如生理流体的处理。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重要教育组成部分。

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.