# Condition for the Incipient Motion of Non-Cohesive Particles Due to Laminar Flows of Power-Law Fluids in Closed Conduits

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction and Objective

^{2}section and 6 m long), found that the incipient motion condition can be expressed in terms of a particle Reynolds number based on the mean flow velocity adapted to consider the duct geometry, and a modified Archimedes number, defined as $A{r}^{*}=fAr$, where $f$ is the static friction coefficient between the particle and the bottom and $Ar=\rho \left({\rho}_{S}-\rho \right)g{d}^{3}/{\mu}^{2}$, with $\mu $ as the fluid dynamic viscosity. $Ar$ is also known as the Galileo (or Galilei) number, $Ga$. Hong et al. [21], from experiments in the facility used by Lobkovsky et al. [19], reported the critical condition in a Shields fashion graph. In their paper, both the Shields stress and particle Reynolds number are defined in terms of the shear rate. Other authors [22,23,24] have approached the problem as one of instability of a deformable bed, predicting the threshold condition in terms of the Shields stress, Galileo number, and $d/{h}_{f}$ [22,23]. Ouriemi et al. [24], present Charru and co-workers’ results as a relation between the pipe Reynolds number and $Ga{\left({h}_{f}/d\right)}^{2}$.

## 2. Experimental Setup and Materials

^{2}, and the pipe is 10 m long. The inclination angle of the pipe with respect to a horizontal datum can change between −17.5 and 17.5°. The circuit is closed by means of two hoses that join the square pipe to a cylindrical one, connected to a stainless steel centrifugal pump. A heat exchanger installed in the system allows the control of temperature during the experiments. Air and sediment traps are located in the downstream end of the square pipe. Two reservoirs are located at the upstream end of the pipe. One of them contains the sediments used to generate the granular bed in the pipe, as described later. The second reservoir serves two purposes: (i) as a mixing tank to prepare the CMC aqueous solutions to be tested, and (ii) as a reservoir that contains the non-Newtonian mixtures that will fill the closed circuit before the experiment starts. A built-in mixer in the container permits achieving the first purpose. The square pipe was built using a transparent perspex, 10 mm thick. The discharge was measured with a magnetic flowmeter Siemens model SITRANS F M MAG 3100 (Siemens, Lille, France). Eight pressure transducers made by OMEGA model PX409-2.5DWUV (Measurements Specialties, Inc., Hampton, VA, USA) were installed along the pipe and connected to a data acquisition system. Particle motions were recorded by a camera NIKON D800 (Nikon, Tokyo, Japan) installed normal to the top wall, and by a camera GoPro Hero 3 (GoPro, San Mateo, CA, USA) placed normal to one of the vertical walls.

^{n}. Density was practically equal to the water in which the CMC was mixed. The solutions resulting from the CMC–water mixture are transparent, allowing visualization of the bed particles.

^{3}/h. Entrance length was estimated from the Kim et al. relationship [27], which indicates that for the experimental conditions tested, the length to have a fully developed flow was around 10 cm or less.

## 3. Dimensional analysis

## 4. Experimental Results

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Rheogram corresponding to the water- sodium carboxymethyl cellulose (CMC) solution used in Experiment 13. The solid line is the best fitting curve to the measurements, given by $\tau =0.194{\dot{\gamma}}^{0.678}$, with a correlation coefficient ${R}^{2}=0.99834$.

**Figure 3.**Relation between the modified particle Reynolds number and the modified Galileo number. Error bars are also included.

**Figure 4.**Condition of incipient motion in terms of the modified particle Reynolds number and the modified Galileo number. Continuous line corresponds to the best fit of the data. Segmented line is the relationship $R{e}_{*p}=0.434\sqrt{G{a}_{K}}$.

**Figure 5.**Condition of incipient motion presented in the standard Shields diagram with data corresponding to laminar flows of Newtonian and power-law fluids. The grey line is the Rouse–Mantz fitting to the movement threshold condition. The experimental points corresponding to the incipient condition in a pipe flow (this research, black filled circles) show the value of the flow index. The black line is the threshold condition obtained from Equation (14).

Sand | Size Range (mm) | $\mathit{d}$ (mm) | $\mathit{\alpha}$ (°) | ${\mathit{\rho}}_{\mathit{S}}$ (kg/m^{3}) |
---|---|---|---|---|

A | 1.0–1.3 | 1.15 | 29 | 2700 |

B | 0.4–0.6 | 0.50 | 31 | 2650 |

C | 1.6–2.36 | 1.98 | 29 | 2700 |

D | 2.36–3.35 | 2.86 | 29 | 2700 |

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**MDPI and ACS Style**

Tamburrino, A.; Traslaviña, C.
Condition for the Incipient Motion of Non-Cohesive Particles Due to Laminar Flows of Power-Law Fluids in Closed Conduits. *Water* **2020**, *12*, 1295.
https://doi.org/10.3390/w12051295

**AMA Style**

Tamburrino A, Traslaviña C.
Condition for the Incipient Motion of Non-Cohesive Particles Due to Laminar Flows of Power-Law Fluids in Closed Conduits. *Water*. 2020; 12(5):1295.
https://doi.org/10.3390/w12051295

**Chicago/Turabian Style**

Tamburrino, Aldo, and Cristóbal Traslaviña.
2020. "Condition for the Incipient Motion of Non-Cohesive Particles Due to Laminar Flows of Power-Law Fluids in Closed Conduits" *Water* 12, no. 5: 1295.
https://doi.org/10.3390/w12051295