# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 1 Introduction
It is not surprising, then, that because of the lack of a single constitutive equation and due to their ubiquity in many industrial and biological processes, numerical methods for solving the flow of these viscoelastic materials under flows fields of different levels of complexity have been a very active research field. Some extensive reviews in this subject can be found in [9; 16; 17; 18; 19; 20]. Broadly speaking, the link between polymer fluids and external fields is relatively well-understood: _how a polymeric fluid behaves under a given field is directly governed by the dynamics of their corresponding underlying microstructure_. For polymeric fluids, this microstructure comprises polymer chains. The underlying dynamics include coiling and uncoiling processes, hindered motion due to physical entanglements or hydrodynamic effects caused by the presence of other polymer molecules in their vicinity, and in some cases, physical cross-linking between chains. In this paper, we focus on a subclass of the available computational techniques: the so-called micro-macro approaches [18]. These approaches combine the solution of the conservation equations at the macroscale with models derived from kinetic theory at the microscale. The advantage of using kinetic models is that instead of a homogenized view of the rheology of the material through a macroscopic constitutive equation, the viscoelastic contribution to the stress is obtained using coarse-grained representations of the microstructural dynamics. For more comprehensive reviews on these methods, the reader is referred to [18; 21; 22; 23; 24; 25]. Micro-macro methods are especially appealing when modifications to the microstructural dynamics need to be introduced without incurring large numerical changes or extreme computational costs. This is because the evolution of the microscopic description is somehow decoupled from the macroscopic flow solvers. This has the potential of making the model highly versatile and applicable to a range of deformation conditions. In addition, micro-macro methods facilitate the investigation of the effects of various parameters to gain insights into the underlying physics of a given phenomenon and lead to an optimization of a range of industrial processes. On the other hand, micro-macro methods are more computationally demanding than their corresponding macroscopic counterparts. In practice, one must choose between models that are computationally tractable but perhaps do not offer a direct link between the flow-induced development of the micro-structure and the applied flow field or be limited to simple geometries and viscometric flows when using macro-micro methods. Within the context of microscopic models, different models have been developed to describe the coupling between macroscopic response and microstructure dynamics in polymeric fluids. One family of models are the so-called bead-spring models. These models use molecular coarse-graining to describe the behavior of polymer chains represented as beads connected by massless springs. The simplest of these models considers only two beads, and it is known as the elastic dumbbell model. Although it is widely recognized that a single dumbbell is too simple to be able to describe any complicated dynamics in polymeric systems, it is also known that stretching and orientation alone suffice to give a qualitative description of steady-state rheological properties and flows with slow characteristic time scales [15]. Accordingly, these models have been extensively used to develop \"an elementary but broad understanding of the relation between macro-molecular motions and rheological phenomena\" [15]. One of the most common dumbbell models is the Hookean dumbbell model, where the spring connecting the two beads is assumed to follow a linear elastic law ( Hooke's law). This model, although physically unrealistic, has the advantage of being the only model in its class with an exact closure [15,26]; this closure leads to the well-known macroscopic constitutive equation called the Oldroyd-B model.