# Macro–Micro-Coupled Simulations of Dilute Viscoelastic Fluids

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## Abstract

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## 1. Introduction

## 2. Governing Equations

#### 2.1. Macroscopic Flow Equations

- The Reynolds number, $Re=\frac{\rho UL}{\eta}$, which is the ratio of inertial to viscous forces.
- The ratio of the solvent to total zero-shear viscosity: $\beta =\frac{{\eta}_{s}}{{\eta}_{s}+{\eta}_{p}}$.
- The Deborah number, $De=\lambda \phantom{\rule{0.166667em}{0ex}}\frac{U}{L}$, which is the ratio of the fluid relaxation time to a characteristic time for the macroscopic flow. Note that in the case of viscometric flow, in which $U/L={\dot{\gamma}}_{0}$, this is also referred to as the Weissenberg number.

#### 2.2. Constitutive Equations for Extra Stress

#### 2.2.1. Microscopic Description—Dumbbell Models

**Hookean dumbbells:**For Hookean dumbbells, a linear form is assumed for the spring force, $F\left(\mathbf{Q}\right)=H\mathbf{Q}$, where H is the spring constant. With this, Equations (5) and (6) become$$\begin{array}{ccc}\hfill d{\tilde{\mathbf{Q}}}_{t}& =& {\left(\tilde{{\displaystyle \nabla}}\tilde{\mathbf{u}}\right)}^{\u22ba}\xb7{\tilde{\mathbf{Q}}}_{t}-\frac{2H}{\zeta}{\tilde{\mathbf{Q}}}_{t}+\sqrt{\frac{4{k}_{B}T}{\zeta}}\phantom{\rule{0.166667em}{0ex}}d{\mathbf{W}}_{t},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \tilde{\mathit{\tau}}& =& nH\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a-n{k}_{B}T\phantom{\rule{0.166667em}{0ex}}\mathbf{I}.\hfill \end{array}$$These equations are scaled using Equation (3) and the characteristic microscopic length,$$\begin{array}{ccc}\hfill x& =& \sqrt{\frac{{k}_{B}T}{H}}.\hfill \end{array}$$With these,$$\begin{array}{c}\hfill {\eta}_{p}=n{k}_{B}T\lambda ,\phantom{\rule{28.45274pt}{0ex}}\mathit{\tau}=\tilde{\mathit{\tau}}\xb7\left(\frac{\lambda}{{\eta}_{p}}\right)=\tilde{\mathit{\tau}}\xb7\left(\frac{1}{n{k}_{B}T}\right),\phantom{\rule{28.45274pt}{0ex}}\mathbf{Q}=\tilde{\mathbf{Q}}\xb7\left(\sqrt{\frac{H}{{k}_{B}T}}\right).\end{array}$$In addition, the longest relaxation time is defined as$$\begin{array}{ccc}\hfill \lambda & =& \frac{\zeta}{4H}.\hfill \end{array}$$The resulting non-dimensional equations are then$$\begin{array}{ccc}\hfill d{\mathbf{Q}}_{t}& =& {\left({\displaystyle \nabla}\mathbf{u}\right)}^{\u22ba}\xb7{\mathbf{Q}}_{t}-\frac{1}{2\phantom{\rule{0.166667em}{0ex}}De}{\mathbf{Q}}_{t}+\frac{1}{\sqrt{De}}d{\mathbf{W}}_{t},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \mathit{\tau}& =& \u2329\mathbf{Q}\mathbf{Q}\u232a-\mathbf{I},\hfill \end{array}$$**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 Warner’s finitely extensible nonlinear connector force law [15],$$\begin{array}{c}\hfill F\left(\tilde{\mathbf{Q}}\right)=\frac{H\tilde{\mathbf{Q}}}{1-{\left(\tilde{Q}/{Q}_{\mathrm{max}}\right)}^{2}},\end{array}$$Introducing this spring law into Equation (5) and non-dimensionalizing as before gives$$\begin{array}{ccc}\hfill d{\mathbf{Q}}_{t}& =& {\left({\displaystyle \nabla}\mathbf{u}\right)}^{\u22ba}\xb7{\mathbf{Q}}_{t}-\frac{1}{2De}\left(\frac{{\mathbf{Q}}_{t}}{1-{Q}^{2}/b}\right)+\frac{1}{\sqrt{De}}d{\mathbf{W}}_{t},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \mathit{\tau}& =& \u2329\frac{\mathbf{Q}\mathbf{Q}}{1-{Q}^{2}/b}\u232a-\mathbf{I},\hfill \end{array}$$

#### 2.2.2. Macro-Constitutive Equations—Closure Models

**Upper convected Maxwell model:**If the spring is linear (Hookean), where $F\left(\mathbf{Q}\right)=H\mathbf{Q}$, the resulting constitute equation obtained from Equation (14) is known as the upper convected Maxwell (UCM) model$$\begin{array}{cc}& \tilde{\mathit{\tau}}=nH\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a-n{k}_{B}T\mathbf{I},\end{array}$$$$\begin{array}{cc}& {\displaystyle {\left(\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a\right)}_{\left(1\right)}+\frac{4H}{\zeta}\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a=\frac{4{k}_{B}T}{\zeta}\mathbf{I}.}\end{array}$$Using the same scaling as before leads to the following non-dimensional constitutive equations:$$\begin{array}{cc}& \mathit{\tau}=\u2329\mathbf{Q}\mathbf{Q}\u232a-\mathbf{I},\end{array}$$$$\begin{array}{cc}& De{\left(\u2329\mathbf{Q}\mathbf{Q}\u232a\right)}_{\left(1\right)}+\u2329\mathbf{Q}\mathbf{Q}\u232a=\mathbf{I}.\end{array}$$**FENE-P model**In a FENE model, with the spring law given by Equation (12), the evolution equation, Equation (14), becomes$$\begin{array}{c}\hfill {\left(\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a\right)}_{\left(1\right)}+\frac{4H}{\zeta}\u2329\frac{H\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}}{1-{(\tilde{Q}/{Q}_{\mathrm{max}})}^{2}}\u232a=\frac{4{k}_{B}T}{\zeta}\mathbf{I}.\end{array}$$Different FENE models can be formulated based on the approximation used to calculate the second term in Equation (18). Here, we focus on the FENE-P model, which is obtained by the approximation$$\begin{array}{c}\hfill \u2329\frac{H\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}}{1-{(\tilde{Q}/{Q}_{\mathrm{max}})}^{2}}\u232a=\frac{\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a}{1-\u2329{\tilde{Q}}^{2}\u232a/{Q}_{\mathrm{max}}^{2}}.\end{array}$$With this approximation, the constitutive equations become$$\begin{array}{cc}& {\displaystyle \tilde{\mathit{\tau}}=\frac{nH}{1-\u2329{\tilde{Q}}^{2}\u232a/{\tilde{Q}}_{\mathrm{max}}^{2}}\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a-n{k}_{B}T\mathbf{I},}\\ \hfill & & \end{array}$$$$\begin{array}{cc}& {\displaystyle {\left(\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a\right)}_{\left(\tilde{1}\right)}+\frac{4H}{\zeta}\frac{\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a}{1-\u2329{\tilde{Q}}^{2}\u232a/{\tilde{Q}}_{\mathrm{max}}^{2}}=\frac{4{k}_{B}T}{\zeta}\mathbf{I}.}\end{array}$$Scaling as before, we obtain$$\begin{array}{cc}& \mathit{\tau}=\frac{\u2329\mathbf{Q}\mathbf{Q}\u232a}{{\displaystyle 1-\u2329{Q}^{2}\u232a/b}}-\mathbf{I},\end{array}$$$$\begin{array}{cc}& {\displaystyle De{\left(\u2329\mathbf{Q}\mathbf{Q}\u232a\right)}_{\left(1\right)}+\frac{\u2329\mathbf{Q}\mathbf{Q}\u232a}{1-\u2329{Q}^{2}\u232a/b}=\mathbf{I}.}\end{array}$$For convenience, one can define the non-dimensional configuration tensor, $\mathbf{A}$, as$$\begin{array}{c}\hfill \mathbf{A}\equiv d\phantom{\rule{0.166667em}{0ex}}\frac{\u2329\tilde{\mathbf{Q}}\tilde{\mathbf{Q}}\u232a}{{\u2329{\tilde{Q}}^{2}\u232a}_{0}},\end{array}$$Note that if we scale the end-to-end vector as before, we have$$\begin{array}{c}\hfill \mathbf{A}\equiv d\phantom{\rule{0.166667em}{0ex}}\frac{\frac{{k}_{B}T}{H}\phantom{\rule{0.166667em}{0ex}}\u2329\mathbf{Q}\mathbf{Q}\u232a}{{\u2329{\tilde{Q}}^{2}\u232a}_{0}}=d\u2329\mathbf{Q}\mathbf{Q}\u232a\left(\frac{{k}_{B}T}{H\phantom{\rule{0.166667em}{0ex}}{\u2329{\tilde{Q}}^{2}\u232a}^{2}}\right)=d\u2329\mathbf{Q}\mathbf{Q}\u232a\left(\frac{1}{d}\right)=\u2329\mathbf{Q}\mathbf{Q}\u232a.\end{array}$$In addition, trace$\left(\mathbf{A}\right)=d\u2329{Q}^{2}\u232a.$The constitutive equation for $\mathbf{A}$ in two dimensions is$$\begin{array}{cc}& {\displaystyle \mathit{\tau}=\frac{\mathbf{A}}{1-\mathrm{trace}\left(\mathbf{A}\right)/\left(2b\right)}-\mathbf{I}.}\\ \hfill & & \end{array}$$$$\begin{array}{cc}& {\displaystyle De{\mathbf{A}}_{\left(1\right)}+\frac{\mathbf{A}}{1-\mathrm{trace}\left(\mathbf{A}\right)/\left(2b\right)}=\mathbf{I},}\\ \hfill \end{array}$$

#### 2.3. Numerical Considerations

**Parameter values:**Unless otherwise noted, the parameter values used in the simulations are given in Table 1.

Parameter | Symbol—Eqn | Units | Value | Ref. |
---|---|---|---|---|

Density | $\rho $ | kg/m${}^{3}$ | 1000 | |

Solvent viscosity | ${\eta}_{s}$ | Pa s | 0.001 | |

Zero shear rate viscosity | ${\eta}_{0}={\eta}_{s}+{\eta}_{p}$ | Pa s | 0.0037 | [33] |

Characteristic relaxation time | $\lambda $ | s | 0.010 | [34] |

Characteristic velocity | $U=\sqrt{\gamma /\left(\rho {R}_{0}\right)}$ | m/s | 0.260 | |

Characteristic length | L | m | 0.003 | |

Surface tension | $\gamma $ | N/m | 0.070 | [34] |

Initial jet radius | ${R}_{0}$ | m | 0.001 | [34] |

Reynolds number | $Re=\rho UL/{\eta}_{0}$ | Dimensionless | 78 | |

Ratio of viscosities | $\beta ={\eta}_{s}/{\eta}_{0}$ | Dimensionless | 0.270 | |

Deborah number | $De=\lambda U/L$ | Dimensionless | 0.800 | |

Ohnesorge number | $Oh={\eta}_{0}/\sqrt{\rho \gamma {R}_{0}}$ | Dimensionless | 0.038 | |

FENE parameter | b | Dimensionless | ${10}^{5}$ | |

Wavenumber | k | Dimensionless | 0.8 |

#### 2.3.1. Hookean Dumbbells

#### 2.3.2. FENE Dumbbells

## 3. Results

#### 3.1. Simple Shear

#### 3.2. Elongational Flow

#### 3.3. Capillary Thinning

#### 3.3.1. Deborah Number

#### 3.3.2. Wavelength

#### 3.3.3. Nonlinear FENE Parameter

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Flow Diagram

#### Appendix A.2. Computational Resources

**Figure A2.**Wall clock time comparison for viscometric computations in simple shear flow for Hookean dumbbells. Speedup using GPU increases with the number of dumbbells considered.

#### Appendix A.3. Capillary Thinning Results Using FENE-P Model

**Figure A3.**Principal stress difference obtained from simulations of the FENE-P model. These results are comparable to those shown in Figure 10. The differences arise because the FENE-P model is not an exact closure for the FENE dumbbell model and some information is lost by the approximation used in the FENE-P model.

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**Figure 1.**Dumbbell configuration given by the coordinates of each dumbbell, ${\overrightarrow{r}}_{1}$ and ${\overrightarrow{r}}_{2}$, or its center of mass, ${\overrightarrow{r}}_{c}$ and end-to-end vector, $\overrightarrow{Q}$.

**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 5.**Scatter plots of ${Q}_{x}$ and ${Q}_{y}$ in simple shear flow for two values of the Deborah number $De$. Note: the scale of the scatter plots is not the same across the different plots.

**Figure 6.**Probability distribution function (PDF) of dumbbells’ orientation as a function of time and Deborah number ($De$).

**Figure 7.**Expansion ratio as a function of time and Deborah number. The ratio is calculated using Equation (32).

**Figure 8.**Axes of extension as a function of time and $De=500$. Distribution plots at the bottom have been normalized by their corresponding maximum values. Between 0 and 100, the system is in the linear regime, and the stretching in the x- and y-directions are proportional to each other. In the nonlinear regime, $t>100$, the molecules stretch more in the direction of the flow field, i.e., the x-direction. As the stretching approaches the maximum dumbbell length ($b=300$), the stretching is ’accumulated’ at the end of the spectrum.

**Figure 9.**Comparison of FENE-P model (Equation (24)) with stochastic simulations (Equation (13)) under viscometric elongational flow.

**Figure 10.**Filament stretching evolution coupled with microstructure evolution. In this model, the microstructure is represented by non-interacting dumbbells, and their conformation is given by the expansion ratio in Equation (32). Other parameters are $\beta =0.27$, $Oh=0.04$, $k=0.8$, and $b={10}^{5}$.

**Figure 11.**Filament and microstructure evolution for different wavelength numbers. Other parameters are $\beta =0.27$, $Oh=0.04$, $De=0.8$, and $b={10}^{5}$.

**Figure 12.**Effects of different FENE parameter. Other parameters are $\beta =0.27$, $Oh=0.04$, $k=0.8$, and $De=0.8$.

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**MDPI and ACS Style**

Cromer, M.; Vasquez, P.A.
Macro–Micro-Coupled Simulations of Dilute Viscoelastic Fluids. *Appl. Sci.* **2023**, *13*, 12265.
https://doi.org/10.3390/app132212265

**AMA Style**

Cromer M, Vasquez PA.
Macro–Micro-Coupled Simulations of Dilute Viscoelastic Fluids. *Applied Sciences*. 2023; 13(22):12265.
https://doi.org/10.3390/app132212265

**Chicago/Turabian Style**

Cromer, Michael, and Paula A. Vasquez.
2023. "Macro–Micro-Coupled Simulations of Dilute Viscoelastic Fluids" *Applied Sciences* 13, no. 22: 12265.
https://doi.org/10.3390/app132212265