# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 3 Results  
### Simple Shear  
This flow field results from placing the fluid between two parallel plates, with the lower plate stationary and the upper plate moving at a constant velocity \(U_{\text{top}}\). Under viscometric conditions, the equations can be simplified by assuming a fully developed flow so that  
\[\bar{\mathbf{u}}=[\gamma_{0}\bar{y},0]^{\intercal},\hskip 28.452756 pt\frac{ \partial(\cdot)}{\partial x}=0, \tag{30}\]  
where \(\gamma_{0}=U_{\text{top}}/L\) and \(L\) is the plates' separation. Using the same scaling as before simplifies the velocity to \(\mathbf{u}=[y,0]^{\intercal}\), and the velocity gradient tensor reduces to  
\[\boldsymbol{\
abla}\mathbf{u}=\bigg{(} \begin{array}{cc}0&1\\ 0&0\end{array} \bigg{)}. \tag{31}\]  
All initial conditions correspond to equilibrium conditions. For dumbbell models, this means the end-to-end vectors are drawn from a normal distribution. For the closure models, this means the extra stress is zero. The UCM model is an exact closure of the Hookean dumbbell equation, and as expected results from these two models will always agree, this is confirmed in Figure 2, where results from both models in viscometric simple shear flow are compared. Similarly, Figure 3 compares results from the FENE-P model with stochastic simulations of FENE dumbbells. The figure shows that the FENE-P approximation underpredicts the nonlinear behavior of the shear stress, in agreement with previous studies [27]. Figure 3: Extra stress in viscometric simple shear flow for FENE models compared to the FENE-P closure (Equation (24)). All variables are non-dimensionalized using the scaling given in Equation (3) and parameter values given in Table 1.  
Figure 2: Extra stress results in viscometric simple shear flow for Hookean dumbbells (Equation (11)) and UCM model (Equation (17)). The first normal stress difference is calculated as \(N_{1}=\tau_{yy}-\tau_{xx}\). All variables are non-dimensionalized using the scaling given in Equation (3) and parameter values given in Table 1.  
Figure 4 shows that as the nonlinearity, represented by the parameter \(b\), is increased, the discrepancies between the micro and macro representations become more pronounced. For a more in-depth comparison between the FENE model and different FENE closures, the reader is referred to [27]. The advantage of using dumbbell equations, as opposed to closure approximations, is that in addition to the average stress behavior, shown in Figures 2-4, one can also capture the stretching and orientation of dumbbells under flow. In this way, the model predictions can be compared to data obtained from scattering techniques [36]. Figure 5 shows the temporal evolution of the distribution of dumbbell lengths, \(Q_{x}\) vs. \(Q_{y}\), as a function of shear flow both in the linear regime, \(De=10\), and the nonlinear regime, \(De=500\). In the linear regime, dumbbells are stretched and oriented in the direction of the flow, but the distribution of lengths remains Gaussian. In the nonlinear regime, Gaussianity is lost around the same time the overshoot in the shear stress is observed. We can also examine the probability distribution function of the dumbbell orientation as a function of the Deborah number; these results are shown in Figure 6. The figure shows that the nonlinear behavior differs from linear behavior in two main ways. First, the'molecules' are more oriented with respect to the flow, as proven by the fact that the mean orientation angle at steady state for \(De=10\) is not quite zero, while for \(De=500\), the mean angle is zero. Second, narrower distributions for \(De=500\) indicate that more'molecules' are aligned with the flow at steady state compared with molecules in the linear regime. Figure 4: Stochastic simulations for the FENE model with different values of the nonlinear parameter \(b\).