# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Method

- (a)
- Calculate the Lagrangian node trajectory by solving (8) backwards in time, starting at the predefined location $\mathbf{x}\left({t}_{n+1}\right)$.
- (b)
- Interpolate the stress $\mathit{\tau}({t}_{n},\mathbf{x}\left({t}_{n}\right))$ to the Lagrangian node from the known node values at time ${t}_{n}$.
- (c)

^{®}[37], an in-house CFD code that was developed at the Fraunhofer–Chalmers Research Institute for Industrial Mathematics in Gothenburg, Sweden. In addition to viscoelastic flow, the solver has previously been employed to simulate conjugated heat transfer [38,39,40], and fluid-structure interaction [41], as well as free surface flow of shear-thinning fluids with applications for seam sealing [42,43], adhesive application [44], and 3D-bioprinting [45].

## 4. Results

#### 4.1. Planar Poiseuille Flow

#### 4.2. Die Swell

#### 4.3. Jet Buckling

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Oldroyd-B Fluid Fully Developed Channel Flow

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**Figure 1.**Two-dimensional grid with one refinement level, showing cell centers (×) and grid nodes (•).

**Figure 3.**Basis for interpolating properties stored at the fluid grid to Lagrangian nodes in areas with uniform grid spacing (

**a**) and near refinements (

**b**).

**Figure 7.**Computed errors with respect to the analytic solution for velocity (

**a**) and viscoelastic normal stress (

**b**).

**Figure 9.**Grid M1 used in the grid dependence study, as defined in Table 1.

**Figure 10.**Simulated profiles for ${S}_{R}=2.5$ across the channel at $x=5h$, obtained with grid M1, compared to the analytic solution.

**Figure 11.**Free surface position in the expansion zone of the die swell geometry, simulated for grids M1, M2, and M3 for ${S}_{R}=2.5$.

**Figure 12.**Snapshots of die swell simulation with ${S}_{R}=1.0$, interface between viscoelastic phase (green) and Newtonian phase (white) visualized by $\alpha =0.5$.

**Figure 13.**Snapshots of die swell simulation with ${S}_{R}=2.5$, interface between the viscoelastic phase (green) and Newtonian phase (white) visualized by $\alpha =0.5$.

**Figure 16.**Isosurface $\alpha =0.5$ in viscoelastic jet buckling simulation with $\lambda =0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ and $\mathrm{Wi}=10$ obtained with the finest cell sizes $\Delta {x}_{min}=D/8,D/16,D/32$.

**Figure 17.**First normal stress difference ${N}_{1}={\tau}_{yy}-{\tau}_{xx}$ along the line $x/D=5$ in viscoelastic jet buckling simulation with $\lambda =0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ and $\mathrm{Wi}=10$ obtained with the finest cell sizes $\Delta {x}_{min}=D/8,D/16,D/32$.

**Figure 19.**Viscoelastic jet buckling simulation with $\lambda =0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

Grid | $\mathit{h}/\Delta {\mathit{x}}_{\mathbf{base}}$ | $\mathit{h}/\Delta {\mathit{x}}_{\mathbf{channel}}$ | $\mathit{h}/\Delta {\mathit{x}}_{\mathbf{corner}}$ | Num. Cells Total. |
---|---|---|---|---|

M1 | 5 | 10 | 20 | 4829 |

M2 | 10 | 20 | 40 | 19,685 |

M3 | 20 | 40 | 80 | 78,740 |

Work | Method | Re |
---|---|---|

Current work | VOF, Lagrangian-Eulerian | $0.5$ |

Crochet & Keunings [11] | Mixed FEM | 0 |

Tomé et al. [14] | GENSMAC | $0.5$ |

Habla et al. [22] | pseudo-VOF | $0.5$ |

Comminal et al. [24] (CCU) | VOF, Geometric scheme | 0 |

Comminal et al. [24] (HRIC) | VOF, Algebraic scheme | 0 |

Comminal et al. [24] (RheoTool) | VOF, Algebraic scheme (MULES) | $0.01$ |

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**MDPI and ACS Style**

Ingelsten, S.; Mark, A.; Kádár, R.; Edelvik, F.
A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows. *Appl. Sci.* **2021**, *11*, 439.
https://doi.org/10.3390/app11010439

**AMA Style**

Ingelsten S, Mark A, Kádár R, Edelvik F.
A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows. *Applied Sciences*. 2021; 11(1):439.
https://doi.org/10.3390/app11010439

**Chicago/Turabian Style**

Ingelsten, Simon, Andreas Mark, Roland Kádár, and Fredrik Edelvik.
2021. "A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows" *Applied Sciences* 11, no. 1: 439.
https://doi.org/10.3390/app11010439