# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 2 Governing Equations
### Constitutive Equations for Extra Stress
#### 2.2.1 Microscopic Description--Dumbbell Models  
The solvent is assumed to be an incompressible Newtonian fluid of viscosity \(\eta_{s}\). The configuration of the dumbbell, shown in Figure 1, is described by the end-to-end connector vector \(\mathbf{Q}=\mathbf{r}_{2}-\mathbf{r}_{1}\) and the center-of-mass vector \(\mathbf{r}_{c}=\frac{1}{2}(\mathbf{r}_{1}+\mathbf{r}_{2})\). Here, we assume a homogeneous distribution, so that the only variable of =interest is \(\mathbf{Q}\).  
The stochastic differential equation describing the evolution of the end-to-end vector is given by [15; 29],  
\[d\tilde{\mathbf{Q}}_{t}=\big{(}\boldsymbol{\
abla}\tilde{\mathbf{u}}\big{)}^{ \intercal}\cdot\tilde{\mathbf{Q}}_{t}-\frac{2}{\xi}F\big{(}\tilde{\mathbf{Q}}_{ t}\big{)}+\sqrt{\frac{4k_{B}T}{\xi}}\,d\mathbf{W}_{t}, \tag{5}\]  
where \(F\left(\cdot\right)\) denotes the functional form of the spring force, \(\mathbf{W}_{t}\) is a two-dimensional Wiener process, \(\zeta\) is the drag coefficient acting on each bead, and \(k_{B}T\) is the thermal energy.  
The viscoelastic contribution to the stress is given by [PERSON]'s relation [15; 29],  
\[\tilde{\boldsymbol{\tau}}=n\big{\langle}F(\tilde{\mathbf{Q}})\,\tilde{\mathbf{ Q}}\big{\rangle}-nk_{B}T\,\mathbf{I}, \tag{6}\]  
where \(n\) is the dumbbells' number density, \(n\langle F(\mathbf{Q})\,\mathbf{Q}\rangle\) is the contribution of the tension on the spring with the spring law \(F(\mathbf{Q})\), and \(nk_{B}T\) captures effects due to Brownian motion.  
* **Hookean dumbbells:** For Hookean dumbbells, a linear form is assumed for the spring force, \(F(\mathbf{Q})=H\mathbf{Q}\), where \(H\) is the spring constant. With this, Equations (5) and (6) become \[d\tilde{\mathbf{Q}}_{t} = \big{(}\boldsymbol{\
abla}\tilde{\mathbf{u}}\big{)}^{\intercal} \cdot\tilde{\mathbf{Q}}_{t}-\frac{2H}{\xi}\tilde{\mathbf{Q}}_{t}+\sqrt{\frac{4 k_{B}T}{\xi}}\,d\mathbf{W}_{t},\] (7a) \[\tilde{\boldsymbol{\tau}} = nH\big{\langle}\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\big{\rangle} -nk_{B}T\,\mathbf{I}.\] (7b) These equations are scaled using Equation ( 3 ) and the characteristic microscopic length, \[x = \sqrt{\frac{k_{B}T}{H}}.\] (8) With these, \[\eta_{p}=nk_{B}T\lambda,\qquad\quad\boldsymbol{\tau}=\tilde{\boldsymbol{\tau}} \cdot\left(\frac{\lambda}{\eta_{p}}\right)=\tilde{\boldsymbol{\tau}}\cdot \left(\frac{1}{nk_{B}T}\right),\qquad\quad\mathbf{Q}=\tilde{\mathbf{Q}}\cdot \left(\sqrt{\frac{H}{k_{B}T}}\right).\] (9) In addition, the longest relaxation time is defined as \[\lambda = \frac{\xi}{4H}.\] (10)  
Figure 1: Dumbbell configuration given by the coordinates of each dumbbell, \(\vec{r}_{1}\) and \(\vec{r}_{2}\), or its center of mass, \(\vec{r}_{c}\) and end-to-end vector, \(\vec{Q}\).  
The resulting non-dimensional equations are then \[d\mathbf{Q}_{t} = (\mathbf{\
abla}\mathbf{u})^{\mathsf{T}}\cdot\mathbf{Q}_{t}-\frac{1}{2 \,De}\mathbf{Q}_{t}+\frac{1}{\sqrt{De}}d\mathbf{W}_{t},\] (11a) \[\mathbf{\tau} = \langle\mathbf{Q}\mathbf{Q}\rangle-\mathbf{I},\] (11b) where the Deborah number is defined as before, \(De=\lambda U/L\).
* **FENE Dumbbells** If instead of a linear spring connector a finitely extensible spring is considered, we obtain FENE-type models. These models are derived by introducing [PERSON]'s finitely extensible nonlinear connector force law [15], \[F(\tilde{\mathbf{Q}})=\frac{H\tilde{\mathbf{Q}}}{1-\big{(}\tilde{Q}/Q_{\text{ max}}\big{)}^{2}},\] (12) where \(Q^{2}=\mathbf{Q}\cdot\mathbf{Q}\) and \(Q_{\text{max}}\) is the maximum allowed extension of the dumbbell. Introducing this spring law into Equation (5) and non-dimensionalizing as before gives \[d\mathbf{Q}_{t} = (\mathbf{\
abla}\mathbf{u})^{\mathsf{T}}\cdot\mathbf{Q}_{t}-\frac{1} {2 De}\bigg{(}\frac{\mathbf{Q}_{t}}{1-Q^{2}/b}\bigg{)}+\frac{1}{\sqrt{De}}d \mathbf{W}_{t},\] (13a) \[\mathbf{\tau} = \bigg{\langle}\frac{\mathbf{Q}\mathbf{Q}}{1-Q^{2}/b}\bigg{\rangle} -\mathbf{I},\] (13b) where the nonlinear FENE parameters are \[b=H\,Q_{\text{max}}^{2}/(k_{B}T)\].