# Stability of the Plane Bingham–Poiseuille Flow in an Inclined Channel

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## Abstract

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## 1. Introduction

## 2. Laminar Open Flow of a Bingham Fluid Down an Inclined Channel

## 3. Linear Stability Analysis

## 4. Linear-Energy Stability of Perturbations

#### 4.1. Stability of Streamwise Perturbations

#### 4.2. Stability of Spanwise Perturbations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Inclined layer of an angle $\beta $. The light grey layer is the shear layer which is moving with a parabolic law. The wall corresponding to $z=0$ is the rigid terrain. The superior layer, the plug region ${D}_{S}\le z\le D$, is rigidly moving at a constant velocity $\mathbf{U}\left({D}_{S}\right)$. D is the depth of the layer, ${D}_{S}$ is the depth of the shear layer and $D-{D}_{S}$ is the depth of the plug region. $\mathbf{g}$ is the gravity.

**Figure 2.**Real part of the time decay coefficient, $\mathrm{Re}\left(C\right)$, when a runs from 0 to $2.6$, and the Reynolds number $\mathrm{Re}$ is in the interval $[1000,11000]$. The meshed plane corresponds to $\mathrm{Re}\left(C\right)=0$, the surfaces corresponds to the set of points $(a,\mathrm{Re},\mathrm{Re}(C\left)\right)$, for sample Bingham values $B=0$ (purple), $B=2$ (green) and, $B=5$ (cyan).

**Figure 3.**Critical Orr–Reynolds numbers versus Bingham number. The plot shows the stabilizing effect of the Bingham number.

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**MDPI and ACS Style**

Falsaperla, P.; Giacobbe, A.; Mulone, G.
Stability of the Plane Bingham–Poiseuille Flow in an Inclined Channel. *Fluids* **2020**, *5*, 141.
https://doi.org/10.3390/fluids5030141

**AMA Style**

Falsaperla P, Giacobbe A, Mulone G.
Stability of the Plane Bingham–Poiseuille Flow in an Inclined Channel. *Fluids*. 2020; 5(3):141.
https://doi.org/10.3390/fluids5030141

**Chicago/Turabian Style**

Falsaperla, Paolo, Andrea Giacobbe, and Giuseppe Mulone.
2020. "Stability of the Plane Bingham–Poiseuille Flow in an Inclined Channel" *Fluids* 5, no. 3: 141.
https://doi.org/10.3390/fluids5030141