# Diffusion-Slip Boundary Conditions for Isothermal Flows in Micro- and Nano-Channels

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and the New Models

#### 2.1. Governing Equations

#### 2.2. Slip boundary Conditions

#### 2.3. Effects of Rarefaction

#### 2.4. Analytical Solution

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of pressure driven flow of a rarefied gas through a micro-channel of length L, height h and width w ($L,w\gg h$).

**Figure 4.**Variation in non-linearity of pressure profiles along the micro-channel for a pressure ratio $\mathcal{P}=4$, plotted for various outlet pressure conditions.

**Figure 5.**(Color Online) Comparison of normalised stream-wise velocity plotted as a function of normalised height, $y/H$, at location X = 0.9 along the micro-channel, for various Knudsen numbers as predicted by current model (in black) and N-S with no-slip condition (in red). The velocity profiles are normalised with the corresponding maximum value for each case which corresponds to the exit velocity along the center-line of the pipe.

**Figure 6.**(Color Online) Stream-wise velocity contours predicted by the new model for outlet pressure conditions corresponding to Knudsen numbers (

**a**) 10, (

**b**) 1, (

**c**) 0.1 and (

**d**) 0.01. The velocity contour lines are plotted over the stream-wise velocity contour field for a typical parabolic velocity profile predicted by N-S equations with no slip boundary condition.

Experimental Parameters | Value |
---|---|

Gas used | Helium |

Length, L | 9.39 ± 0.1 mm |

Height, h | 9.38 ± 0.2 $\mathsf{\mu}$m |

Width, w | 492 ± 1 $\mathsf{\mu}$m |

Avg. Temperature, T | 296 K |

Viscosity, $\mu $ | $1.967\times {10}^{-5}$ Pa s |

Gas Constant, $\mathcal{R}$ | 2078.5 J/(kg K) |

Inlet Pressure range | 60.4–109,825 Pa |

Outlet Pressure range | 12.2–22,633 Pa |

Average Kn range | 0.027–50.2 |

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**MDPI and ACS Style**

Tomy, A.M.; Dadzie, S.K.
Diffusion-Slip Boundary Conditions for Isothermal Flows in Micro- and Nano-Channels. *Micromachines* **2022**, *13*, 1425.
https://doi.org/10.3390/mi13091425

**AMA Style**

Tomy AM, Dadzie SK.
Diffusion-Slip Boundary Conditions for Isothermal Flows in Micro- and Nano-Channels. *Micromachines*. 2022; 13(9):1425.
https://doi.org/10.3390/mi13091425

**Chicago/Turabian Style**

Tomy, Alwin Michael, and S. Kokou Dadzie.
2022. "Diffusion-Slip Boundary Conditions for Isothermal Flows in Micro- and Nano-Channels" *Micromachines* 13, no. 9: 1425.
https://doi.org/10.3390/mi13091425