# Flows of Dense Suspensions of Polymer Particles through Oblique Bifurcating Channels: Two Continua Approach

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Algorithm

## 4. Results

**Remark**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\varphi}_{f}$ | 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, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right]$ |

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

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

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

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

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

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

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

p | pressure, [Pa] |

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

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

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

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

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

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

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

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

${D}_{d}$ | deviatoric part of D, $\left[{\mathrm{s}}^{-1}\right]$ |

I | identity matrix, dimensionless |

$\mathbf{j}$ | total momentum, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{2}\xb7\mathrm{s}}\right]$ |

$\mathbf{l}$ | flux of mass concentration of particles, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{2}\xb7\mathrm{s}}\right]$ |

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

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

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

$\tau $ | yield stress of the solid phase, [cp/s] |

k | interphase friction, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{3}\xb7\mathrm{s}}\right]$ |

B | gravitation mobility, [s] |

$\frac{{d}_{s}}{dt}$ | 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 |

$\langle f\rangle $ | time-average value of quantity f |

${b}_{N}$ | reduced Bingham number, dimensionless |

l | dimensionless length of channel |

w | dimensionless width of channel |

$|{\nabla}_{br}\phantom{\rule{0.166667em}{0ex}}p|$ | dimensionless pressure drop along channel branch |

$Re$ | Reynolds number, dimensionless |

$Fr$ | Froude number, dimensionless |

${M}^{p}$ | dimensionless mass of particles that passed through the branch |

${M}^{f}$ | dimensionless mass of fluid that passed through the branch |

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**Figure 1.**Scheme of flow domain. $AB$ is the inlet boundary, $CD$ and $EF$ are the outlet boundaries. $ABLK$ is the inlet branch, $MNDC$ is the main branch, $LFEN$ is the side branch. $\angle END$ is the bifurcation angle $\alpha $. The boundaries $AC$, $BL$, $LF$, $ND$, $NE$ are impenetrable.

**Figure 2.**The pulsatile mode of injection. Calculated snapshots of particle mass concentration for the bifurcation angle $\alpha ={30}^{\circ}$ and the data (43) at the dimensionless times $t=20,50,60,100,200,300$ from left to right and from top down.

**Figure 3.**Inlet (red), outlet (blue) and side (green) particle mass flow rates ${Q}_{in}^{p}$, ${Q}_{out}^{p}$, and ${Q}_{br}^{p}$ versus time for the data (43) and the bifurcation angle $\alpha ={30}^{\circ}$.

**Figure 4.**Inlet (red), outlet (blue) and side (green) values of particle masses ${M}_{in}^{p}$, ${M}_{out}^{p}$, and ${M}_{br}^{p}$ for the data (43) and for the bifurcation angle at ${30}^{\circ}$.

**Figure 5.**Cross-section average concentration $\overline{c}$ versus time for the bifurcation angle ${30}^{\circ}$ and data (43). The red, blue and green lines are for the inlet cross-section $KL$, for the outlet cross-section $MN$, and for the branch cross-section $EF$, respectively, (see Figure 1).

**Figure 6.**Particle-loss curve on the plane ($\alpha ,y$). The vertical coordinate $y={M}_{br}^{p}/{M}_{out}^{p}$ means the relative mass loss of particles into the side branch.

**Figure 7.**Calculated correlation between $f={M}_{out}^{f}/{M}_{in}^{f}$ and $p={M}_{out}^{p}/{M}_{in}^{p}$ for the bifurcation angle $\alpha ={90}^{\circ}$.

**Figure 8.**Pressure p versus s, where s is the distance from the bifurcation point, with $\alpha ={90}^{\circ}$. Green lines are for the side branch and blue lines are for the main branch. The left column corresponds to ${w}_{br}=0.5$ and the right column is for ${w}_{br}=0.2$. The pictures from top down correspond to $t=100,200,300$, respectively.

**Figure 9.**Concentration c versus s where s is the distance from the bifurcation point along the midline of the side branch with the bifurcation angle $\alpha ={90}^{\circ}$. The left and right pictures correspond to ${w}_{br}=0.5$ and ${w}_{br}=0.2$, respectively. The pictures from top down correspond to $t=100,200,300$, respectively.

**Figure 10.**Partitioning of particles for the bifurcation angle $\alpha ={90}^{\circ}$. The mass of particles ${M}^{p}$ that passed through the branch during the period of pulsation. The left and right pictures correspond to ${w}_{br}=0.5$ and ${w}_{br}=0.2$, respectively.

**Figure 11.**Solid phase velocity profiles in a branch for the bifurcation angle $\alpha ={90}^{\circ}$ at the terminal time instant. Projection ${v}_{s}$ of the solid phase velocity on the midline of the branch versus x, where x is a transversal variable in the branch. The side branch and the main branch are from top down, the values ${w}_{br}=0.5$ and ${w}_{br}=0.2$ are from left to right. The solid, dashed and dotted lines correspond to the branch locations $0.25\phantom{\rule{0.166667em}{0ex}}l,0.5\phantom{\rule{0.166667em}{0ex}}l$ and $0.75\phantom{\rule{0.166667em}{0ex}}l$ reckoned from the bifurcation point.

**Figure 12.**Fluid velocity profiles in a branch for the bifurcation angle $\alpha ={90}^{\circ}$ at the terminal time instant. Projection ${v}_{f}$ of the fluid velocity on the midline of the branch versus x, where x is a transversal variable in the branch. The side branch and the main branch are from top down, the values ${w}_{br}=0.5$ and ${w}_{br}=0.2$ are from left to right. The solid, dashed and dotted lines correspond to the branch locations $0.25\phantom{\rule{0.166667em}{0ex}}l,0.5\phantom{\rule{0.166667em}{0ex}}l$ and $0.75\phantom{\rule{0.166667em}{0ex}}l$ reckoned from the bifurcation point.

**Figure 13.**Fluid velocity profiles in a branch for the bifurcation angle $\alpha ={90}^{\circ}$ in the case of the zero Bingham number ${b}_{N}$. Projection ${v}_{f}$ of the fluid velocity on the midline of the branch versus x, where x is a transversal variable in the branch. The side branch and the main branch are from top down, the values ${w}_{br}=0.5$ and ${w}_{br}=0.2$ are from left to right. The solid, dashed and dotted lines correspond to the branch locations $0.25\phantom{\rule{0.166667em}{0ex}}l,0.5\phantom{\rule{0.166667em}{0ex}}l$ and $0.75\phantom{\rule{0.166667em}{0ex}}l$ reckoned from the bifurcation point.

**Figure 14.**Solid phase velocity profiles in a branch for the bifurcation angle $\alpha ={90}^{\circ}$ in the case of the zero Bingham number ${b}_{N}$. Projection ${v}_{s}$ of the solid phase velocity on the midline of the branch versus x, where x is a transversal variable in the branch. The side branch and the main branch are from top down, the values ${w}_{br}=0.5$ and ${w}_{br}=0.2$ are from left to right. The solid, dashed and dotted lines correspond to the branch locations $0.25\phantom{\rule{0.166667em}{0ex}}l,0.5\phantom{\rule{0.166667em}{0ex}}l$ and $0.75\phantom{\rule{0.166667em}{0ex}}l$ reckoned from the bifurcation point.

${w}_{br}$ | 1 | 0.5 | 0.2 |

$\langle {b}_{N}^{in}\rangle $ | 3.953 | 4.053 | 4.062 |

$\langle {b}_{N}^{out}\rangle $ | 7.068 | 4.719 | 4.387 |

$\langle {b}_{N}^{br}\rangle $ | 8.062 | 10.006 | 22.324 |

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**MDPI and ACS Style**

Shelukhin, V.; Antonov, A.
Flows of Dense Suspensions of Polymer Particles through Oblique Bifurcating Channels: Two Continua Approach. *Polymers* **2022**, *14*, 3880.
https://doi.org/10.3390/polym14183880

**AMA Style**

Shelukhin V, Antonov A.
Flows of Dense Suspensions of Polymer Particles through Oblique Bifurcating Channels: Two Continua Approach. *Polymers*. 2022; 14(18):3880.
https://doi.org/10.3390/polym14183880

**Chicago/Turabian Style**

Shelukhin, Vladimir, and Andrey Antonov.
2022. "Flows of Dense Suspensions of Polymer Particles through Oblique Bifurcating Channels: Two Continua Approach" *Polymers* 14, no. 18: 3880.
https://doi.org/10.3390/polym14183880