# Dynamic Behaviours of a Filament in a Viscoelastic Uniform Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Numerical Method

**C**) being the trace of the tensor $\mathit{C}$. The conformation of tensor $\mathit{C}$ is determined by the following transport equations [18,31,32]:

**X**is the position vector of a point on the filament, s is the Lagrangian coordinate along the filament, $T={E}_{s}\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">\frac{\partial \mathbf{X}}{\partial {s}_{0}}$ is the tensile stress, ${E}_{s}$ is the stretching coefficient, ${s}_{0}$ is the Lagrangian coordinate of the undeformed filament, ${E}_{b}$ is the bending rigidity, $\mathit{g}$ is the gravitational acceleration, and ${\mathit{F}}_{H}$ is the hydrodynamic force exerted on the filament by the ambient fluid.

## 3. Validation

#### 3.1. A 2D Filament Flapping in a Newtonian Uniform Flow

#### 3.2. Grid Convergence Study of the Filament Flapping Motion in a Viscoelastic Uniform Flow

#### 3.3. Determination of the Prandlt Number $Pr$

## 4. Results and Discussion

#### 4.1. The Filament Behaviours in a Giesekus Uniform Flow

#### 4.1.1. The Effects of Wi and Re on the Flapping Motion of the Filament

#### 4.1.2. The Effects of Wi and Re on the Drag and Lift Coefficients of the Filament

#### 4.2. The Filament Motion in a Uniform FENE-CR Flow.

#### 4.2.1. The Effects of Wi and Re on the Flapping Motion of the Filament

#### 4.2.2. The Effects of Wi and Re on the Drag and Lift Coefficients of the Filament

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MRT | multi relaxation time |

SRT | single relaxation time |

IB-LBM | immersed boundary-lattice Boltzmann method |

FSI | fluid–structure interaction |

ALE-FEM | arbitrary Lagrangian Eulerian finite element method |

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**Figure 2.**A 2D filament flapping in a Newtonian uniform flow: the time history of the transverse displacement of the trailing edge node.

**Figure 3.**A 2D filament flapping in a Newtonian uniform flow: the time histories of (

**a**) the drag coefficient and (

**b**) the lift coefficient of the flapping filament at $Re=200$.

**Figure 4.**A 2D filament flapping in a Newtonian uniform flow: The time history of the transverse displacement of the trailing edge node at $Re=200$ in domains with different sizes ($8{L}_{f}\times 8{L}_{f}$, $12{L}_{f}\times 12{L}_{f}$ and $16{L}_{f}\times 16{L}_{f}$).

**Figure 5.**The time histories of the free-end position of the filament at $Re=200$ and $Wi=1.0$ in the (

**a**) FENE-CR and (

**b**) Giesekus fluids at different values of $Pr$.

**Figure 6.**The time histories of the free-end position of the filament at $Re=200$ and $Wi=1.0$ in the (

**a**) FENE-CR and (

**b**) Giesekus fluids at different values of $Pr$.

**Figure 7.**The time histories of the free-end position of the filament in Newtonian and Giesekus uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

**Figure 8.**The phase diagram of a filament motion as a function of the Reynolds number and Weissenberg number. □—stable oscillations; $\Delta $—damped oscillations.

**Figure 9.**The time histories of the drag coefficient on the filament in Newtonian and Giesekus uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

**Figure 10.**The non-dimensional shear rate ${\dot{\gamma}}^{*}=\frac{\partial {u}_{x}}{\partial y}/\frac{{U}_{c}}{{L}_{f}}$ along the filament at $t=50$, $Re=10$ and $Wi=0$, 0.2, 0.6 and 1.0.

**Figure 11.**Contours of the pressure coefficient at $Re=100$, and $Wi=$ (

**a**) 0 (at $t=$ 30.935), (

**b**) 0.2 (at $t=$ 30.825), (

**c**) 0.6 (at $t=$ 30.605) and (

**d**) 1.0 (at $t=$ 30.375).

**Figure 12.**The time histories of the lift coefficient on the filament in Newtonian and Giesekus uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

**Figure 13.**The time histories of the free-end position of the filament in Newtonian and FENE-CR uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

**Figure 14.**The phase diagram of a filament motion as a function of the Reynolds number and Weissenberg number. □, stable oscillations; $\Delta $ damped oscillations.

**Figure 15.**The time histories of the drag coefficient on the filament in Newtonian and FENE-CR uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

**Figure 16.**The non-dimensional shear rate ${\dot{\gamma}}^{*}=\frac{\partial {u}_{x}}{\partial y}/\frac{{U}_{c}}{{L}_{f}}$ along the filament at $t=50$, $Re=10$ and $Wi=0$, 0.2, 0.6 and 1.0.

**Figure 17.**Contours of the pressure coefficient at $Re=100$, and $Wi=$ (

**a**) 0 (at $t=$ 30.935), (

**b**) 0.2 (at $t=$ 30.87), (

**c**) 0.6 (at $t=$ 30.825) and (

**d**) 1.0 (at $t=$ 30.795).

**Figure 18.**The time histories of the lift coefficient on the filament in Newtonian and FENE-CR uniform flows at $Re=$ (

**a**) 10, (

**b**) 25, (

**c**) 50, (

**d**) 100 and (

**e**) 200.

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**MDPI and ACS Style**

Ma, J.; Tian, F.-B.; Young, J.; Lai, J.C.S.
Dynamic Behaviours of a Filament in a Viscoelastic Uniform Flow. *Fluids* **2021**, *6*, 90.
https://doi.org/10.3390/fluids6020090

**AMA Style**

Ma J, Tian F-B, Young J, Lai JCS.
Dynamic Behaviours of a Filament in a Viscoelastic Uniform Flow. *Fluids*. 2021; 6(2):90.
https://doi.org/10.3390/fluids6020090

**Chicago/Turabian Style**

Ma, Jingtao, Fang-Bao Tian, John Young, and Joseph C. S. Lai.
2021. "Dynamic Behaviours of a Filament in a Viscoelastic Uniform Flow" *Fluids* 6, no. 2: 90.
https://doi.org/10.3390/fluids6020090