# Fractal Analysis of a Non-Newtonian Fluid Flow in a Rough-Walled Pipe

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## Abstract

**:**

## 1. Introduction

## 2. Laminar Flow of a Non-Newtonian Fluid through a Pipe with a Smooth Surface

## 3. Laminar Flow of a Non-Newtonian Fluid through a Rough-Walled Pipe

#### 3.1. Velocity Profile

#### 3.2. Darcy Friction Factor

## 4. Results and Discussion

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Two examples of fractal rough surfaces generated numerically. The root-mean-square surface height is the same for both surfaces, but the surface fractal dimensions are different.

**Figure 4.**Plot of velocity profile defined by Equation (15) for different values of n, with ${r}^{*}=r/R$, ${v}_{x}^{*}={v}_{x}/{v}_{max}\left(n\right)$.

**Figure 8.**Influence of the surface fractal dimension ${D}_{s3}$ on the velocity profile ${v}_{x}$ (m/s) defined by Equation (27) for different values of n, with ${r}^{*}=r/{R}_{e}$, ${R}_{e}=5$ mm, $\Delta p=5$ Pa, ${K}_{0}={10}^{-3}$ Pa·s

^{n}, and $c=100$ m

^{$1-\Gamma $}.

**Figure 9.**Plot of the friction factor defined by Equation (33) with respect to ${D}_{s3}$ for different values of n with ${R}_{e}=5$ mm, $\Delta p=[5,10,20,30,40,50]$ Pa, $\rho =1000$ kg/m

^{3}, ${K}_{0}={10}^{-3}$ Pa.s

^{n}and $c=100$ m

^{$1-\mathsf{\Gamma}$}.

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**MDPI and ACS Style**

Bouchendouka, A.; Fellah, Z.E.A.; Larbi, Z.; Louna, Z.; Ogam, E.; Fellah, M.; Depollier, C.
Fractal Analysis of a Non-Newtonian Fluid Flow in a Rough-Walled Pipe. *Materials* **2022**, *15*, 3700.
https://doi.org/10.3390/ma15103700

**AMA Style**

Bouchendouka A, Fellah ZEA, Larbi Z, Louna Z, Ogam E, Fellah M, Depollier C.
Fractal Analysis of a Non-Newtonian Fluid Flow in a Rough-Walled Pipe. *Materials*. 2022; 15(10):3700.
https://doi.org/10.3390/ma15103700

**Chicago/Turabian Style**

Bouchendouka, Abdellah, Zine El Abiddine Fellah, Zakaria Larbi, Zineeddine Louna, Erick Ogam, Mohamed Fellah, and Claude Depollier.
2022. "Fractal Analysis of a Non-Newtonian Fluid Flow in a Rough-Walled Pipe" *Materials* 15, no. 10: 3700.
https://doi.org/10.3390/ma15103700