# Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

**I**is the identity tensor, ${\mathsf{\tau}}_{0}$ is the yield stress, k is the consistency index, n is the flow behavior index and ${\stackrel{{\scriptstyle \phantom{\rule{-0.11108pt}{0ex}}\square}}{\mathsf{\tau}}}_{P}$ is the Gordon-Schowalter derivative given by

## 3. Numerical Method

- Start by computing the raw volume fraction values for each cell in the computational domain using the VOF method.
- Use the raw volume fraction data to generate an initial estimate of the interface position and orientation using a simple threshold operation. Cells with volume fraction values above a certain threshold (e.g., 0.5) are labeled as Fluid A, while cells with volume fraction values below the threshold are labeled as Fluid B.
- Generate an initial estimate of the distance function from the interface [44]. This distance function is used to define an initial estimate of the interface normal and curvature at each cell.
- Use the initial estimate of the distance function to calculate an RDF that better estimates the local interface position and orientation. Here, the gradient of the RDF is defined as the difference between the interface normal estimated from the distance function and the normal estimated from the initial threshold operation.
- Update the interface position and orientation at each cell using the RDF, and repeat the previous step until the RDF converges to a desired tolerance.
- Use the updated interface position and orientation to generate a new estimate of the distance function and repeat the previous steps until a desired level of accuracy is achieved.
- Finally, use the updated interface position and orientation to calculate the interface curvature and normal at each cell, which can be used in subsequent calculations, such as interface advection or pressure-velocity coupling.

Algorithm1 Multiphase viscoelastic PLIC-RDF isoAdvector (MVP-RIA) algorithm |

Require: Mesh, physical properties, boundary conditions, initial conditionsEnsure: Velocity, pressure, viscoelastic stress tensor and interface geometry fields1: Initialize fields 2: Set time step, $\Delta t$, or Courant number, $Co$ 3: Start time loop and set end time for simulation 4: Set the number of outer correctors, $nOuterCorrectors$, and pressure correctors, $nCorrectors$ 5: Set current iteration count $n=1$ and pressure correctors count $m=1$ 6: while not converged or $n<nOuterCorrectors$ (PIMPLE corrector loop) do7: Compute face fluxes 8: Update interface geometry using PLIC-RDF isoAdvector algorithm 9: Compute viscoelastic stress tensor (Equations (4) or (5)) 10: Compute linear momentum equation (Equation (2)) 11: while $m<nCorrectors$ (PISO corrector loop) do12: Solve the pressure equation and momentum corrector 13: Increment iteration count m 14: end while15: Increment iteration count n 16: end while17: Output results |

## 4. Validation Case Studies

#### 4.1. Buoyancy-Driven Rise of a Bubble in a Newtonian Fluid

#### 4.2. Buoyancy-Driven Rise of a Bubble through a Viscoelastic Shear-Thinning Fluid

#### 4.3. Buoyancy-Driven Rise of a Bubble through an Elastoviscoplastic Fluid

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ADI | Alternating-Direction Implicit |

ALE | Arbitrary Lagrangian Eulerian |

BiCGStab | Bi-Conjugate Gradient-Stable Algorithm |

CSF | Continuum Surface Force |

CUBISTA | Convergent and Universally Bounded Interpolation Scheme |

for the Treatment of Advection | |

FTM | Front-Tracking Method |

GAMG | Geometric-Algebraic Multi-Grid |

IBM | Immersed Boundary Method |

IFEM | Immersed-Finite-Element Method |

ILU | Incomplete Lower-Upper |

LSM | Level-Set Method |

MAC | Marker-And-Cell |

MULES | Multidimensional Universal Limiter with Explicit Solution |

MVP-RIA | Multiphase Viscoelastic PLIC-RDF isoAdvector |

OpenFOAM | Open Source Field Operation and Manipulation |

PFM | Phase Field Method |

PIMPLE | Mixture of PISO and SIMPLE |

PISO | Pressure Implicit with Splitting of Operator |

PLIC | Piecewise Linear Interface Construction |

RDF | Reconstructed Distance Function |

SIMPLE | Semi-Implicit Method for Pressure Linked Equations |

VOF | Volume-Of-Fluid |

Nomenclature | |

Physical and mathematical quantities | |

u | Velocity vector |

$\rho $ | Density |

p | Pressure |

g | Gravity acceleration |

${\mathit{f}}_{\mathit{s}}$ | Surface tension force |

$\mathsf{\tau}$ | Stress tensor |

${\mathsf{\tau}}_{S}$ | Newtonian (Solvent) stress tensor |

${\mathsf{\tau}}_{P}$ | Polymeric extra-stress tensor |

${\eta}_{S}$ | Solvent dynamic viscosity |

${\eta}_{P}$ | Polymeric dynamic viscosity |

$\alpha $ | Mobility parameter |

$\lambda $ | Relaxation factor, relaxation time |

$\epsilon $ | Extensibility parameter |

${\sigma}_{D}$ | Deviatoric part of stress tensor |

$\overline{\sigma}$ | Second invariant of the deviatoric stress tensor |

$\mathit{I}$ | Identity tensor |

${\mathsf{\tau}}_{0}$ | Yield stress |

${\stackrel{{\scriptstyle \phantom{\rule{-0.11108pt}{0ex}}\square}}{\mathsf{\tau}}}_{P}$ | Gordon-Schowalter derivative |

$\zeta $ | Non-affine deformation parameter |

${\stackrel{\u25bf}{\mathsf{\tau}}}_{P}$ | Upper-convective time derivative of the polymeric extra-stress tensor |

$\mathit{A}$ | Conformation tensor |

$\gamma $ | Volume fraction |

u${}_{r}$ | Relative velocity vector of two fluids |

$\sigma $ | Surface tension coefficient |

${U}^{*}$ | Calculated rise velocity |

$\mathit{\theta}$ | Natural logarithm of the conformation tensor |

k | Consistency index |

n | Shear-thinning exponent |

G | Elastic modulus of the material |

Geometrical parameters | |

W | Domain width |

H | Domain height |

d | Initial bubble diameter |

R | Initial bubble radius |

${V}_{b}$ | Initial bubble volume |

${R}_{eff}$ | Effective bubble radii |

Non-dimensional numbers | |

$Ar$ | Archimedes number |

$Bo$ | Bond (Eötvös) number |

$Mo$ | Morton number |

$Re$ | Reynolds number |

$Wi$ | Weissenberg number |

$Bn$ | Bingham number |

$Eg$ | Elastogravity number |

Operators | |

∇ | Gradient |

$\nabla .$ | Divergence |

$\frac{D(.)}{Dt}$ | Total time derivative |

$\mathrm{tr}(.)$ | Trace operator |

$\mathrm{max}(.,.)$ | Maximum operator |

${(.)}^{T}$ | Transpose operator |

: | Double dot product |

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**Figure 1.**The geometry and boundary conditions for the buoyancy-driven rise of a bubble in a Newtonian fluid.

**Figure 2.**Comparison of our predicted bubble rise velocity with experimental data reported by Maxworthy et al. [6] and numerical simulation results by Tsamopoulos et al. [7]. The comparison is performed for two selected values of $Mo$ number, representing different flow conditions in a Newtonian liquid.

**Figure 3.**Contour plots of the fluid’s volume fraction for $Mo=2.174\times {10}^{-7}$ (

**left**) and $Mo=3.769\times {10}^{-4}$ (

**right**) with $d=0.7\phantom{\rule{3.33333pt}{0ex}}$mm (

**top**) and $d=6\phantom{\rule{3.33333pt}{0ex}}$mm (

**bottom**).

**Figure 4.**The geometry and boundary conditions for the buoyancy-driven rise of a bubble through a viscoelastic shear-thinning fluid.

**Figure 5.**Comparison of our predicted bubble rise velocity with experimental data reported by Pilz and Brenn [74] and numerical simulation results by Ji et al. [9]. The comparison is conducted for various initial bubble volumes as they ascend through a shear-thinning viscoelastic fluid described by the Giesekus constitutive model.

**Figure 6.**Contour plots of the fluid’s volume fraction for initial bubble volumes of ${V}_{b}=40,\phantom{\rule{3.33333pt}{0ex}}50,\phantom{\rule{3.33333pt}{0ex}}100$ and $400\phantom{\rule{3.33333pt}{0ex}}$mm${}^{3}$ as they ascend through a shear-thinning viscoelastic fluid described by the Giesekus constitutive model.

**Figure 7.**Contour plots illustrating the magnitude of the natural logarithm $\mathit{\theta}$ of the conformation tensor are presented for various initial bubble volumes, including (

**a**) ${V}_{b}=40$, (

**b**) ${V}_{b}=50$, (

**c**) ${V}_{b}=100$, and (

**d**) ${V}_{b}=400$ mm${}^{3}$, when they ascend through a shear-thinning viscoelastic fluid described by the Giesekus constitutive model.

**Figure 8.**Bubble steady-state velocity ${U}^{*}$ as a function of the bubble initial radius ${R}_{eff}$. The dot-dashed line represent the results obtained with our MVP-RIA algorithm, the dashed line represent the results obtained with the ALE algorithm from Moschopoulos et al. [79] while the symbols represent the experimental data from Lopez et al. [80].

**Figure 9.**(

**a**) Experimental bubble shape by Lopez et al. [80] (black and white), and the numerical simulation of Moschopoulos et al. [79] (red line), (

**b**) Contour plots of the natural logarithm $\mathit{\theta}$ of the conformation tensor for ${R}_{eff}=0.004\phantom{\rule{3.33333pt}{0ex}}$m obtained with the newly developed MVP-RIA algorithm.

**Figure 10.**(

**a**) Experimental bubble shape by Lopez et al. [80] (black and white), and the numerical simulation of Moschopoulos et al. [79] (red line), (

**b**) Contour plots of the natural logarithm $\mathit{\theta}$ of the conformation tensor for ${R}_{eff}=0.0107\phantom{\rule{3.33333pt}{0ex}}$m obtained with the newly developed MVP-RIA algorithm.

**Figure 11.**(

**a**) Experimental bubble shape by Lopez et al. [80] (black and white), and the numerical simulation of Moschopoulos et al. [79] (red line), (

**b**) Contour plots of the natural logarithm $\mathit{\theta}$ of the conformation tensor for ${R}_{eff}=0.0163\phantom{\rule{3.33333pt}{0ex}}$m obtained with the newly developed MVP-RIA algorithm.

**Table 1.**Fluid properties for simulation of the buoyancy-driven rise of a bubble in a Newtonian fluid.

Fluid | Dynamic Viscosity | Density | Surface Tension | Morton Number |
---|---|---|---|---|

${\mathit{\eta}}_{\mathit{S}}$ (Ns/m${}^{2}$) | $\mathit{\rho}$ (kg/m${}^{3}$) | $\mathit{\sigma}$ (N/m) | $\mathbf{Mo}$$=\mathit{g}{\mathit{\eta}}_{\mathit{S}}^{4}/\mathit{\rho}{\mathit{\sigma}}^{3}$ | |

B-1 | $9.45\times {10}^{-3}$ | 1153.8 | 0.06782 | $2.174\times {10}^{-7}$ |

B-2 | $6.01\times {10}^{-2}$ | 1208.5 | 0.06550 | $3.769\times {10}^{-4}$ |

A | $1.48\times {10}^{-5}$ | 1 |

**Table 2.**Fluid properties for simulation of the buoyancy-driven rise of a bubble in a viscoelastic shear-thinning fluid.

Fluid | Solvent Viscosity | Polymer Viscosity | Density | Surface Tension | Relaxation Time | Mobility Factor |
---|---|---|---|---|---|---|

${\mathit{\eta}}_{\mathit{s}}$ (Ns/m${}^{2}$) | ${\mathit{\eta}}_{\mathit{p}}$ (Ns/m${}^{2}$) | $\mathit{\rho}$ (kg/m${}^{3}$) | $\mathit{\sigma}$ (N/m) | $\mathit{\lambda}$ (s) | $\mathit{\alpha}$ | |

B | $1.0\times {10}^{-3}$ | $1.511$ | 1000.90 | 0.076 | 0.207 | 0.6 |

A | $1.7\times {10}^{-5}$ | 1.25 |

**Table 3.**Dimensionless numbers employed for the simulation of the buoyancy-driven rise of a bubble through an elastoviscoplastic fluid.

${\mathit{R}}_{\mathbf{eff}}$ [m] | $\mathit{Ar}$ | $\mathit{Bn}$ | $\mathit{Bo}$ | $\mathit{Eg}$ |
---|---|---|---|---|

0.004 | 3.610 | 0.119 | 2.150 | 0.971 |

0.0083 | 8.929 | 0.057 | 9.347 | 2.024 |

0.0107 | 12.090 | 0.044 | 15.385 | 2.597 |

0.0163 | 20.410 | 0.029 | 35.704 | 3.956 |

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**MDPI and ACS Style**

Fakhari, A.; Fernandes, C.
Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm. *Polymers* **2023**, *15*, 3437.
https://doi.org/10.3390/polym15163437

**AMA Style**

Fakhari A, Fernandes C.
Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm. *Polymers*. 2023; 15(16):3437.
https://doi.org/10.3390/polym15163437

**Chicago/Turabian Style**

Fakhari, Ahmad, and Célio Fernandes.
2023. "Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm" *Polymers* 15, no. 16: 3437.
https://doi.org/10.3390/polym15163437