# Particle Orientation in a shear flow for power law and viscoelastic Fluids

Patrice Laure, Julien Férec, Luisa Silva

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Patrice Laure, Julien Férec, Luisa Silva. Particle Orientation in a shear flow for power law and viscoelastic Fluids. 56ème congrès du GFR, Oct 2022, Rennes, France. . [hal-03862195](https://hal.science/hal-03862195)

HAL Id: hal-03862195

https://hal.science/hal-03862195

Submitted on 21 Nov 2022

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
## Objective

Influence on the particle shape and fluid behavior on the validity of Jeffery' equation

$\dot{\mathbf{p}} = \mathbf{W} \cdot \mathbf{p} + \xi (\mathbf{D} \cdot \mathbf{p} - [\mathbf{D}:\mathbf{p}] \mathbf{p})$ 
W vorticity tensor; $\mathbf{D}$ deformation rate tensor

$$ \xi = \frac{\beta^{2}-1}{\beta^{2}+1} \text{ Shape factor} $$

$$ \beta = \frac{L}{D} \text{ Aspect ratio} $$

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

Prolate and oblate ellipsoids; cylinders

## Direct numerical simulations

### Fictitious domain method:

* particle defined by a level set

* Lagrange multiplier for rigidity condition

* Velocity imposed on upper and lower walls

* Finite element solution
  Navier-Stokes + extra-stress

→ velocity and pressure on the whole domain

<figure><img src="image_7.png" /></figure>

### Particle position update:

* Velocity of center of mass: $\mathbf{u}(X_c)$

* Rotation vector: $\vec{\omega}' = \frac{1}{2} \nabla \times \mathbf{u}(X_c)$

* Adams-Bashfort scheme

$$ \frac{dX_c}{dt} = \mathbf{u}(X_c); \quad \frac{d\mathbf{p}}{dt} = \vec{\omega}' \times \mathbf{p} $$

## Power Law fluid

$$ \eta = K \dot{\gamma}^{n-1} $$

### Jeffery's orbit: $n = 0.1$

<figure><img src="image_8.png" /></figure>

$$ T = \frac{2\pi(\beta^2 + 1)}{\dot{\gamma}\beta} \rightarrow \beta_{\text{equivalent}} $$

– prolate

$T_{\text{Newt}} < T_{\text{Pow}} \rightarrow \beta_{\text{equivalent}} \nearrow$

– oblate

$T_{\text{Newt}} > T_{\text{Pow}} \rightarrow \beta_{\text{equivalent}} \downarrow$

<table><thead><tr><th rowspan="2">n</th><th colspan="3">β = 3</th><th colspan="3">l = 1/3</th></tr><tr><th>T<sub>Newt</sub></th><th>T<sub>Pow</sub></th><th>ζ<sub>Newt</sub></th><th>⟨|Δp|⟩</th><th>T<sub>Newt</sub></th><th>T<sub>Pow</sub></th><th>ξ<sub>Newt</sub></th><th>⟨|Δp|⟩</th></tr></thead><tbody><tr><td>0.1</td><td>21.08</td><td>3.02</td><td>0.8129</td><td>0.007</td><td>21.91</td><td>0.320</td><td>0.8139</td><td>0.004</td></tr><tr><td>0.5</td><td>21.85</td><td>3.16</td><td>0.8181</td><td>0.02</td><td>21.34</td><td>0.326</td><td>0.8082</td><td>0.04</td></tr><tr><td>0.2</td><td>22.76</td><td>3.32</td><td>0.8338</td><td>0.04</td><td>20.92</td><td>0.334</td><td>0.7995</td><td>0.08</td></tr><tr><td>0.1</td><td>23.83</td><td>3.51</td><td>0.8497</td><td>0.05</td><td>20.83</td><td>0.336</td><td>0.7975</td><td>0.11</td></tr></tbody></table>

$$ \beta_{\text{equivalent}} = \text{Cte} ? \qquad \|\xi(\mathbf{p})\| = \frac{\sqrt{\omega_x^2 + \omega_y^2 + (\omega_x + \frac{\dot{\gamma}}{2})^2}}{\frac{\dot{\gamma}}{2}\sqrt{(p_x p_z)^2 + (p_y p_z)^2 + (1 - p_y^2 - p_z^2)^2}} $$

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

## Oldroyd-B fluid

$$ \sigma_f = -p\mathbf{I} + 2\eta_f\varepsilon(\mathbf{u}) + \tau ; \quad \tau = \frac{\eta_e}{\lambda}(3\mathbf{s} - \mathbf{I}) $$

$$ \frac{\partial \mathbf{s}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{s} - (\nabla \mathbf{u} \cdot \mathbf{s} + \mathbf{s} \cdot \nabla \mathbf{u}^t) + \frac{1}{\lambda}(3\mathbf{s} - \mathbf{I}) = 0 $$

$\lambda$ relaxation time from 0.1 to 5 → Main axis // vorticity axis

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Prolate ellipsoid</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>oblate ellipsoid</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Rotation around vorticity axis $e_r$ or $e_z$</figcaption></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

### References:

- T. Laurencin, P. Laure, L. Orgeas, P.J.J. Dumont, L. Silva, S. Rolland du Roscoat., J. Non-Newt. Fluid Mech. 273 (2019) 104187

- J. Férec, E. Bertévas, B.C. Khoo, G. Ausias, N. Phan-Thien, J. Fluid Mech. 798 (2016) 350-370

- J. Férec, E. Bertévas, B. C. Khoo, G. Ausias, N. Phan-Thien, Physics of Fluids 33, 103320 (2021)