title: Simulation of Viscoelastic Fluids using Brownian Configuration Fields
Creators: name: Meier, Andreas
Publisher: name: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Year: 2024

description: Non-Newtonian viscoelastic fluids are widely present in various industrial and medical applications. For instance, solutions containing DNA molecules or blood can be classified as viscoelastic fluids. An even larger field are polymeric fluids, where the presence of macro molecules (polymer chains) leads to significant quantitative rheological distinctions when compared to Newtonian fluids. Notable flow phenomena of polymeric fluids include the rod climbing effect, drag reduction, and extrudate swell. The drag reduction effect is widely exploited in industry, e.g., transport of oil through pipelines or to increase the effectiveness of fire fighting equipment. There are multiple approaches to model viscoelastic fluids. For instance, the well-known Oldroyd-B model belongs to the class of constitutive models. Alternatively, multiscale models, using kinetic theory to describe the dynamics on the micro scale, provide more accurate models. These models include bead-spring-rod models, which represent polymer chains as beads interconnected by springs or rods. In the pure bead-rod case, the chain length remains constant w.r.t. possible deformations, whereas the chain length in the bead-spring case is allowed to vary. Concerning the constant length of a polymer chain, the bead-rod chain model can be viewed as more accurate model for approximating polymer chains. Brownian dynamic simulations of bead-rod-spring models w.r.t. simple homogeneous flows (shear flow, extensional flow) were already analyzed multiple times. However, assuming a prescribed constant flow field is a strong simplification and prevents the simulation of instationary flows. The coupling of the bead-rod-spring micro-scale models to the Navier-Stokes equations enables the consideration of more realistic scenarios in which polymer chains can actually affect the underlying flow field. The numerical simulation of such fluids in complex domains is very challenging since high-stress profiles near the boundary require fine discretization meshes. Simulation of coupled models is computationally expensive and was done for low dimensional bead-spring chain models or constitutive models. Coupling the higher-dimensional bead-rod chain model to Navier-Stokes is even more computationally expensive since bead-spring models are simplified coarse-grained versions of bead-rod models. To the best of our knowledge, no such simulations have been published to date in the literature. In this thesis, we simulate non-Newtonian polymeric fluids described by the microscopic bead-spring dumbbell and bead-rod chain model, also known as Kramers chain, coupled with the Navier-Stokes equations and close the above-mentioned research gap. We consider advanced scenarios with flows fully resolving the underlying geometry such as the start-up shear flow, flow around a cylinder and flow through a planar contraction. In these scenarios, we analyze how the bead-rod chain length affects the macroscopic velocity and drag coefficient in multiple studies by varying the number of beads and thus the chain lengths. Additionally, quantitative comparison to bead-spring dumbbell simulations are made. In principle, there are two options for the simulations of multiscale models: Monte-Carlo methods that compute many realizations of a small problem on the micro scale or alternatively solving a corresponding Fokker-Planck equation. For Kramers chains like treated in this thesis, a Monte-Carlo method seems to be the only feasible approach, since otherwise one had to solve a prohibitively high-dimensional Fokker-Planck equation, where the dimension scales with the number of beads (curse of dimensionality). We combine finite elements with the Brownian configuration field (BCF) approach. Due to the constant rod length constraints appearing in the bead-rod chain model, additional numerical challenges need to be carefully treated. This type of explicit constant length constraint does not arise in bead-spring type models either. As computational resources are now readily available, we are able to begin with the study of bead-rod models coupled to Navier-Stokes equations.