# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 3 Results
### Capillary Thinning
The equations have been non-dimensionalized by taking the initial radius of the jet, \(R_{0}\), as the characteristic macroscopic length scale and the Rayleigh time, \(t_{R}=\sqrt{\rho R_{0}^{3}/\gamma}\), as the macroscopic characteristic time scale; this gives the characteristic velocity as \(\sqrt{\gamma/(\rho R_{0})}\). Finally, the Ohnesorge number is defined as \(Oh=\eta_{0}/\sqrt{\rho\gamma R_{0}}\), and all other dimensionless groups are defined as before. With this scaling, the dynamics of the axisymmetric slender jet are governed by three non-dimensional groups: the Ohnesorge number, \(Oh\), which is the ratio of surface tension to inertial forces; the parameter \(\beta\), which is the ratio between the solvent and fluid viscosities; and the Deborah number, \(De\), which measures the ratio of the material's characteristic time to the time scale of the flow. An additional parameter captures the nonlinear viscoelastic behavior of the polymer chains, which, in the case of FENE models, is the finite extensibility parameter, \(b\), defined above. The nonlinear system of partial differential equations (PDEs), Equation (36), is solved numerically via finite differencing [28]. All spatial terms, except convection, are discretized using second-order central differencing. For the convective terms, a velocity-weighted upwind discretization scheme is used. In particular, the derivative is interpolated between pure central differencing and pure upwind. The result is that in regions of slow flow, a central differencing is used; however, as the velocity increases, the weighting moves closer to a pure upwind differencing. For the convective terms where the derivative depends on the velocity, the first derivative is calculated as follows:  
\[\left(\frac{\partial f}{\partial z}\right)_{j}=\frac{(1-g_{j})(f_{j+1}-f_{j}) }{2\Delta z}+\frac{(1+g_{j})(f_{j}-f_{j-1})}{2\Delta z}, \tag{38}\]  
where \(g_{j}=v_{j}/v_{max}\). We use the implicit Euler method for time discretization combined with time filters, which increase the order of accuracy from first to second order [44]. Additionally, we have implemented variable step sizes, which greatly enhances the speed and efficiency of the solver. The numerical solver has been benchmarked against existing simulations of the Oldroyd-B model [34]. In the spatial domain, periodic boundary conditions are used for the velocity, the radius, and the polymer configurations. The initial condition considers a homogeneous base state corresponding to a uniform cylindrical shape and molecules at equilibrium conditions. To this base state a small amplitude perturbation is added. We note that the eigenvalues of the linearized system resulting from expanding the governing equations to first order in the amplitude of these perturbations determines how fast the perturbations grow towards a necked state or whether they decay to leave a uniform filament. Following [34], we set the initial shape of the jet as  
\[R=R_{0}(1+0.01\cos(kz)), \tag{39}\]where \(k\) is the wavenumber. At each time step, the system of PDEs is solved to find the velocity field. The field is then used to find the velocity gradient, \(\mathbf{\
abla}\mathbf{u}\), at each grid point. Using these values, the dumbbell configurations are updated using Equation (13a) and the extra stress calculated with Equation (13b). The resulting values for the extra stress at each grid point are introduced in the macroscopic simulation through Equation (36b). Below, we show the results obtained when the Deborah number, the wavelength values, and the maximum extensibility of the FENE dumbbells are varied. As a metric for microstructural organization, we use the expansion ratio defined in Equation (32).