# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 3 Results
### Capillary Thinning  
Transient elongational rheometry is used to obtain comparative information about the elongational behavior of polymer solutions. The investigation of the elongational behavior of complex fluids is relevant to applications defined by spraying and jetting, such as microfluidic devices, printing, fiber spinning, and understanding how droplet emissions from respiratory events affect the transmission of infectious diseases [1; 2; 3; 4; 5; 6]. [PERSON] et al. [34] showed that three conditions should be satisfied for an elongational experiment to be successful: an elastocapillary balance must be established, the range of diameters of the liquid jets should be experimentally accessible, and the formation of secondary droplets along the thread must be suppressed. Clearly, understanding the underlying microstructural dynamics that govern one or more of these conditions is important to evaluate the optimal performance both under experimental and application settings. While most of the theoretical analysis is carried out using homogenized views of the microstructure in the form of macroscopic constitutive equations, using methods that allow for the straightforward explicit integration of microstructural dynamics is appealing. Here, our goal is to illustrate how coarse-grained mesoscopic simulations can also be used to gain insight into the development of droplets in a viscoelastic fluid jet. We note that the objective of this publication is to demonstrate the feasibility of the methods, and we take advantage of symmetry and one-dimensional flow fields to simplify many of the computations. More in-depth studies will be left for future investigations. In capillary thinning experiments, a piece of fluid is stretched between two separating surfaces, with surface tension being the main driver of the fluid motion. When a Newtonian fluid is stretched, its radius decreases due to surface tension until it breaks into many droplets. For viscoelastic materials, the underlying microstructure can give rise to different configurations. In polymeric materials, these dynamics involve nonlinear effects derived from the finite extensibility of polymer chains within the fluid. As the fluid jet starts to form droplets, the polymer inside the drops remains relaxed, while polymer chains in the filaments between drops become strongly oriented and stretched, essentially leading to the formation of bridges with high tensile strains. This instability, characteristic of fluids that have some elasticity, has become known as a \"beads-on-a-string\" structure [38; 39; 40; 41]. Moreover, under certain flow regimes, when there is a balance of capillary, viscous, and inertial effects, additional structures can appear. For a more in-depth review of the formation of bead-on-a-string structures, see [40]. To solve this flow, a thin film formulation is used where only the leading-order approximation in an expansion on the jet radius, \(R(z,t)\), is considered [42]. This essentially assumes that the flow inside the fluid jet is one-dimensional so that the only relevant  
Figure 9: Comparison of FENE-P model (Equation (24)) with stochastic simulations (Equation (13)) under viscometric elongational flow. variables are the radius and the axial velocity, \(v(z,t)\). In this case, the non-dimensional conservation of mass and of momentum equations are simplified as [43],  
\[\frac{\partial R^{2}}{\partial t}+\frac{\partial}{\partial z}(vR^{2})=0 \tag{36a}\] \[\left(\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial z}\right)=- \frac{\partial\kappa}{\partial z}+3\,Oh\frac{\beta}{R^{2}}\frac{\partial}{ \partial z}\left(R^{2}\frac{\partial v}{\partial z}\right)+\frac{Oh}{De}\frac {(1-\beta)}{R^{2}}\frac{\partial}{\partial z}\Big{(}R^{2}(\tau_{zz}-\tau_{rr} )\Big{)}, \tag{36b}\]  
where  
\[\kappa=\frac{1}{R(1+R_{z}^{2})^{1/2}}-\frac{R_{zz}}{(1+R_{z}^{2})^{3/2}} \tag{37}\]  
is the curvature and \(\mathbf{\tau}\) is the polymer contribution to the stress.