Positive definiteness preserving approaches for viscoelastic flow of Oldroyd-B and FENE-CR types: Applications to particulate flow

颗粒在流体中的运动不仅具有基本的理论意义，而且在涉及颗粒负载材料的工业过程中的许多应用中也具有重要意义。虽然模拟牛顿流体中粒子运动的数值方法已经非常成功，但在粘弹性流体中数值模拟粒子运动是非常复杂和具有挑战性的。通过本项目探索的计算方法，将进行有效的模拟，以研究和理解粘弹性流体中复杂的流体-颗粒和颗粒-颗粒动力学，特别是颗粒在三维通道中的沉降、迁移、上升和再悬浮。该项目开发的模拟工具将具有重要的工程和生物医学应用，例如油气井水力压裂作业中的支撑剂输送，以及用于开发具有成本效益的芯片实验室（如细胞计数设备）的弹性惯性粒子聚焦。粘弹性流体中粒子运动的数值模拟方法非常复杂和具有挑战性。粘弹性流动模拟的难点之一是数值方法的失效。人们普遍认为，在整个时间积分过程中，构象张量在离散水平上缺乏正确定性保持特性是导致分解的原因之一。为了保持构象张量的正定性，可以将本构方程重新表述为构象张量的矩阵对数方程，以保持构象张量的正定性，如Fattal和Kupferman所做的。另一种方法是对构象张量进行因式分解，然后在离散水平上写出新的关联方程，这样就像Lozinski和Owens所做的那样，迫使构象张量具有正确定性。在这个项目中，我们的目标是扩展和结合基于虚拟域的分布式拉格朗日乘子方法，用于模拟流体中粒子的运动，或者是Lozinski和Owens的分解方法，或者是通过算子分裂技术的对数构象张量方法，以保持模拟Oldroyd-B和FENE-P型粘弹性流体中粒子运动的构象张量的正确定性。通过本项目提出的计算方法，将进行有效的模拟，以研究和理解粘弹性流体中复杂的流体-颗粒和颗粒-颗粒动力学，特别是颗粒在二维和三维通道中的沉降、迁移、上升和再悬浮。

The motion of particles in fluids is not only of fundamental theoretical interest, but is also of importance in many applications to industrial processes involving particle-laden materials. While numerical methods for simulating particle motion in Newtonian fluids have been very successful, numerically simulating particle motion in viscoelastic fluids is quite complicated and challenging. Through the computational methodologies explored in this project, efficient simulations will be performed to investigate and understand the complex fluid-particle and particle-particle dynamics in viscoelastic fluids, especially the sedimentation, migration, lift-off and resuspension of particles in three dimensional channels. The simulation tools developed in this project will have significant engineering and biomedical applications, such as proppant transport in hydraulic fracturing operations used in oil and gas wells and the elasto-inertial particle focusing for developing cost-effective labs-on-a-chip such as cell counting devices.Numerical methods for simulating particle motion in viscoelastic fluids is quite complicated and challenging. One of the difficulties for simulating viscoelastic flows is the breakdown of the numerical methods. It has been widely believed that the lack of positive definiteness preserving property of the conformation tensor at the discrete level during the entire time integration is one of the reasons for the breakdown. To preserve the positive definiteness property of the conformation tensor, the constitutive equation can be reformulated as equations for the matrix logarithm of the conformation tensor to preserve the property of the positive definiteness as Fattal and Kupferman did. Another approach is to factorize the conformation tensor and then to write down the new associated equations at the discrete level, and hence the positive definiteness of the conformation tensor is forced as Lozinski and Owens did. In this project, we aim to extend and combine fictitious domain based distributed Lagrange multiplier methods, which is for simulating particle motion in fluid, with either the Lozinski and Owens' factorization approach or the log-conformation tensor approach via an operator splitting technique to preserve the positive definiteness property of the conformation tensor for simulating particle motion in viscoelastic fluids of Oldroyd-B and FENE-P types. Through the computational methodologies proposed in this project, efficient simulations will be performed to investigate and understand the complex fluid-particle and particle-particle dynamics in viscoelastic fluids, especially the sedimentation, migration, lift-off and resuspension of particles in two and three dimensional channels.