# Recursive Settling of Particles in Shear Thinning Polymer Solutions: Two Velocity Mathematical Model

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Incompressible Isothermal Flows

## 4. Gravitation Mobility of Particles in a Solution of Polymers

## 5. Settling in 2D Tube

## 6. Kinematic Sedimentation Equation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

V | arbitrary volume of suspension, $\left[{m}^{3}\right]$ |

${V}_{f}$ | volume of fluid part in arbitrary volume V, $\left[{m}^{3}\right]$ |

${V}_{p}$ | volume of dry particles in arbitrary volume V, $\left[{m}^{3}\right]$ |

${V}_{M}$ | volume of mud in arbitrary volume V, $\left[{m}^{3}\right]$ |

${V}_{s}$ | $={V}_{p}+{V}_{M}$ volume of solid phase in arbitrary volume V, $\left[{m}^{3}\right]$ |

${m}_{f}$ | mass of fluid in arbitrary volume V, $\left[kg\right]$ |

${m}_{p}$ | mass of dry particles in arbitrary volume V, $\left[kg\right]$ |

${m}_{M}$ | mass of mud in arbitrary volume V, $\left[kg\right]$ |

${m}_{s}$ | $={m}_{p}+{m}_{M}$ mass of solid phase in arbitrary volume V, $\left[kg\right]$ |

m | $={m}_{f}+{m}_{p}+{m}_{M}$ mass of arbitrary suspension volume V,$\left[kg\right]$ |

c | $=\frac{{m}_{p}}{m}$ mass concentration of particles, dimensionless |

${\overline{\rho}}_{f}$ | fluid density, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\overline{\rho}}_{p}$ | density of dry particles, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\overline{\rho}}_{M}$ | mud density, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\varphi}_{f}$ | volume fraction of fluid, dimensionless |

${\varphi}_{p}$ | volume fraction of dry particles, dimensionless |

${\varphi}_{M}$ | volume fraction of mud, dimensionless |

${\varphi}_{s}$ | $={\varphi}_{p}+{\varphi}_{M}$ volume fraction of solid phase, dimensionless |

${\rho}_{f}$ | =${\varphi}_{f}{\overline{\rho}}_{f}$ partial fluid density, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\rho}_{p}$ | $={\varphi}_{p}{\overline{\rho}}_{p}$ partial particle density, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\rho}_{M}$ | $={\varphi}_{M}{\overline{\rho}}_{M}$ partial mud density, $\left[\frac{kg}{{m}^{3}}\right]$ |

${\rho}_{s}$ | $={\rho}_{p}+{\rho}_{M}$ partial density of solid phase, $\left[\frac{kg}{{m}^{3}}\right]$ |

$\rho $ | $={\rho}_{f}+{\rho}_{p}+{\rho}_{M}$ total density $\left[\frac{kg}{{m}^{3}}\right]$ |

${\overline{\rho}}_{s}$ | $=\frac{{\rho}_{s}}{{\varphi}_{s}}$, density of the solid phase, $\left[\frac{kg}{{m}^{3}}\right]$ |

${R}_{0}$ | $=\frac{{\overline{\rho}}_{s}}{{\overline{\rho}}_{f}}$, ratio of densities, dimensionless |

g | gravitational acceleration, $980\left[\frac{cm}{{s}^{2}}\right]$ |

p | pressure, $\left[Pa\right]$ |

${\mathbf{v}}_{f}$ | velocity of fluid phase, $\left[\frac{m}{s}\right]$ |

${\mathbf{v}}_{s}$ | velocity of solid phase, $\left[\frac{m}{s}\right]$ |

$\mathbf{u}$ | $={\mathbf{v}}_{s}-{\mathbf{v}}_{f}$ difference of velocities, $\left[\frac{m}{s}\right]$ |

$\mathbf{v}$ | $={\varphi}_{f}{\mathbf{v}}_{f}+{\varphi}_{s}{\mathbf{v}}_{s}$ mean volume velocity, $\left[\frac{m}{s}\right]$ |

$\tilde{\mathbf{v}}$ | $=(1-c){\mathbf{v}}_{f}+c{\mathbf{v}}_{s}$ mean mass velocity, $\left[\frac{m}{s}\right]$ |

${T}_{f}$ | viscous part of stress tensor of fluid phase, $\left[Pa\right]$ |

${T}_{s}$ | viscous part of stress tensor of solid phase, $\left[Pa\right]$ |

D | rate of strain tensor, $\left[{s}^{-1}\right]$ |

I | identity matrix, dimensionless |

$\mathbf{j}$ | total momentum, $\left[\frac{kg}{{m}^{2}\xb7s}\right]$ |

$\mathbf{l}$ | flux of mass concentration of particles, $\left[\frac{kg}{{m}^{2}\xb7s}\right]$ |

${\eta}_{f}$ | dynamic viscosity of fluid phase, $\left[cp\right]$ |

${\eta}_{s}$ | dynamic viscosity of solid phase, $\left[cp\right]$ |

${\eta}_{s}^{0}$ | consistency of solid phase, $\left[cp\right]$ |

$\tau $ | shear stress, $[cp/s]$ |

$\dot{\gamma}$ | dimensionless shear strain |

k | interphase friction, $\left[\frac{kg}{{m}^{3}\xb7s}\right]$ |

B | gravitation mobility, $\left[s\right]$ |

${V}_{St}$ | Stokes settling velocity, $\left[\frac{m}{s}\right]$ |

$H\left(c\right)$ | hindered settling function, dimensionless |

$\frac{{d}_{s}}{dt}$ | differential operator of material derivative related to velocity ${\mathbf{v}}_{s}$ |

$\frac{{d}_{f}}{dt}$ | differential operator of material derivative related to velocity ${\mathbf{v}}_{f}$ |

$\frac{d}{dt}$ | differential operator of material derivative related to mean mass velocity $\tilde{\mathbf{v}}$ |

${r}_{f}$ | $={\rho}_{f}/{\overline{\rho}}_{f}$, dimensionless density of fluid phase |

${r}_{s}$ | $={\rho}_{s}/{\overline{\rho}}_{f}$, dimensionless density of solid phase |

l | dimensionless length of channel |

$Re$ | Reynolds number, dimensionless |

$Fr$ | Froude number, dimensionless |

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**Figure 1.**Calculated profiles of average concentration over the cross section of the 2D tube for different time instances in a vertical vessel in the case of mobility ${B}_{rz}$.

**Figure 2.**Calculated profiles of average concentration over the cross section of the 2D tube for different time instances in a vertical vessel in the case of the shear thinning mobility ${B}_{sh}$. Effect of two-fold particle sedimentation.

**Figure 3.**Calculated profiles of the concentration wave at different time instances for the shear thinning flux ${F}_{sh}\left(c\right)$ in the case when all the dissipation effects are negligible. The initial and boundary data are ${c}_{0}=0.09$, ${c}_{bot}=1$, ${c}_{top}=0$.

**Figure 4.**Experimental data for mean concentration versus time at different heights in the case of a Newtonian fluid [4].

**Figure 5.**Calculated values of mean concentration versus time at different vertical locations for the flux ${F}_{rz}$, with $m=1$.

**Figure 6.**Experimental values of average concentration versus time at different height levels in a polymer solution [4].

**Figure 7.**Calculated values of the average concentration versus time at different vertical locations for the shear thinning flux ${F}_{sh}\left(c\right)$.

**Figure 8.**Calculated snapshots of concentration in the case of $n=1$ for the flux ${B}_{rz}$ for both the vertical and tilted vessels. The inclination angle is 30°.

**Figure 9.**Calculated snapshots of concentration in the case of $n=0.34$ for the flux ${B}_{sh}$ for both the vertical and tilted vessels. The inclination angle is 30°.

**Figure 10.**Reduced volumetric rate $V\left(t\right)$ of the clear fluid zone $c=0$ versus dimensionless time for the inclination angles $\alpha ={30}^{\xb0}$ from the bottom upwards: Newtonian fluid.

**Figure 11.**Reduced volumetric rate $V\left(t\right)$ of the clear fluid zone $c=0$ versus dimensionless time for the inclination angles $\alpha ={30}^{\xb0}$ from the bottom upwards: non-Newtonian fluid.

**Figure 12.**Streamlines of the mean volume velocity $\mathbf{v}$ for the vertical and inclined cells in the case of non-Newtonian fluid.

**Figure 13.**Calculated profiles of the concentration wave at different time instances for the flux ${F}_{rz}\left(c\right)$ with $m=4.65$. The initial and boundary data are ${c}_{0}=0.09$, ${c}_{bot}=1$, ${c}_{top}=0$. The rising discontinuity wave is followed by a centered rarefaction wave.

**Figure 14.**Calculated profiles of the concentration wave at different time instances for the flux ${F}_{rz}\left(c\right)$ with $m=4.65$. The initial and boundary data are ${c}_{0}=0.09$, ${c}_{bot}=0.45$, ${c}_{top}=0$. The rising discontinuity wave is not followed by a centered rarefaction wave.

**Figure 15.**Calculated values of concentration versus time at different vertical locations for the flux ${F}_{rz}$. There is monotonicity with the initial and boundary data ${c}_{0}=0.09$, ${c}_{bot}=1$, ${c}_{top}=0$.

**Figure 16.**Calculated values of concentration versus time at different vertical locations for the flux ${F}_{rz}$. Loss of monotonicity with the initial and boundary data ${c}_{0}=0.19$, ${c}_{bot}=1$, ${c}_{top}=0$.

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**MDPI and ACS Style**

Neverov, V.; Shelukhin, V.
Recursive Settling of Particles in Shear Thinning Polymer Solutions: Two Velocity Mathematical Model. *Polymers* **2022**, *14*, 4241.
https://doi.org/10.3390/polym14194241

**AMA Style**

Neverov V, Shelukhin V.
Recursive Settling of Particles in Shear Thinning Polymer Solutions: Two Velocity Mathematical Model. *Polymers*. 2022; 14(19):4241.
https://doi.org/10.3390/polym14194241

**Chicago/Turabian Style**

Neverov, Vladimir, and Vladimir Shelukhin.
2022. "Recursive Settling of Particles in Shear Thinning Polymer Solutions: Two Velocity Mathematical Model" *Polymers* 14, no. 19: 4241.
https://doi.org/10.3390/polym14194241