# A Numerical Study of an Ellipsoidal Nanoparticles under High Vacuum Using the DSMC Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Approach

#### Drag Force in Slip Regime

## 3. Numerical Methods

#### 3.1. Direct Simulation Monte Calro

_{N}, 10

^{11}–10

^{15}). It approximates the physical phenomenon represented by the Boltzmann equation using a statistical method. In the DSMC method, flow is caused by repeating movement, collision, deposition, and chemical reactions using these representative particles, and various characteristics such as velocity, temperature, and pressure can be expressed through statistical techniques. The collision algorithms are important in DSMC methods and calculate the most complex terms of the Boltzmann or Kac probability equations. The DSMC schemes can be grouped into two groups with respect to collision treatment [32]. The first group is based on the Boltzmann equation and includes TC [33], NTC [34], NC [35], and MFS [36] methods. The second group is based on the Kac stochastic equation, and there are BT [37] and SBT [38,39] methods. In this study, we used the NTC collision scheme, which is the most widely used.

_{N}value according to the temperature and pressure conditions in the initial flow region. It also assigns an initial velocity to every particle for a given initial temperature. At this time, the magnitude of the velocity is determined according to the Maxwell–Boltzmann distribution.

#### 3.2. GPU Computing (CUDA)

## 4. Results and Discussion

#### 4.1. Microchannel Flow

#### 4.2. Nanoparticle Drag Force

_{p}).

#### 4.3. Drag Force of Ellipsoid Particle

_{x}, D

_{y}, and D

_{z}, which are in the x, y, and z-directions, respectively.

_{z}. In addition, to determine the effect of the diameter of the fine particles, the vertical diameter D

_{x}was analyzed at 50 nm and 100 nm.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Brenner, H. The Slow Motion of a Sphere through a Viscous Fluid towards a Plane Surface. Chem. Eng. Sci.
**1961**, 16, 242–251. [Google Scholar] [CrossRef] - Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1983; Volume 1, ISBN 9024728770. [Google Scholar]
- Loussaief, H.; Pasol, L.; Feuillebois, F. Motion of a Spherical Particle in a Viscous Fluid along a Slip Wall. Q. J. Mech. Appl. Math.
**2015**, 68, 115–144. [Google Scholar] [CrossRef] [Green Version] - Maude, A.D. End Effects in a Falling-Sphere Viscometer. Br. J. Appl. Phys.
**1961**, 12, 293. [Google Scholar] [CrossRef] - O’Neill, M.E. A Slow Motion of Viscous Liquid Caused by a Slowly Moving Solid Sphere. Mathematika
**1964**, 11, 67–74. [Google Scholar] [CrossRef] - Goren, S.L. The Hydrodynamic Force Resisting the Approach of a Sphere to a Plane Wall in Slip Flow. J. Colloid Interface Sci.
**1973**, 44, 356–360. [Google Scholar] [CrossRef] - Luo, H.; Pozrikidis, C. Effect of Surface Slip on Stokes Flow Past a Spherical Particle in Infinite Fluid and near a Plane Wall. J. Eng. Math.
**2008**, 62, 1–21. [Google Scholar] [CrossRef] - Goswami, P.; Baier, T.; Tiwari, S.; Lv, C.; Hardt, S.; Klar, A. Drag Force on Spherical Particle Moving near a Plane Wall in Highly Rarefied Gas. J. Fluid Mech.
**2020**, 883, A47. [Google Scholar] [CrossRef] - Bird, G.A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows; Clarendon Press: Oxford, UK, 1994. [Google Scholar]
- Chapman, S.; Cowling, T.G. The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases; Cambridge University Press: Cambridge, UK, 1990; ISBN 052140844X. [Google Scholar]
- Bird, G.A. Direct Simulation and the Boltzmann Equation. Phys. Fluids
**1970**, 13, 2676–2681. [Google Scholar] [CrossRef] - Russo, G.; Pareschi, L.; Trazzi, S.; Shevyrin, A.A.; Bondar, Y.A.; Ivanov, M.S. Plane Couette Flow Computations by TRMC and MFS Methods. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2005; Volume 762, pp. 577–582. [Google Scholar]
- Muntz, E.P.; Weaver, D.P.; Campbell, D.H. Rarefied Gas Dynamics: Theoretical and Computational Techniques; International Symposium, 16th, Pasadena, CA, July 10–16, 1988, Technical Papers; American Institute of Aeronautics and Astronautics, Inc.: Washington, DC, USA, 1989. [Google Scholar]
- Ozawa, T.; Levin, D.A.; Nompelis, I.; Barnhardt, M.; Candler, G.V. Particle and Continuum Method Comparison of a High-Altitude, Extreme-Mach-Number Reentry Flow. J. Thermophys. Heat Trans.
**2010**, 24, 225–240. [Google Scholar] [CrossRef] - Sohn, I.; Li, Z.; Levin, D.A.; Modest, M.F. Coupled DSMC-PMC Radiation Simulations of a Hypersonic Reentry. J. Thermophys. Heat Trans.
**2012**, 26, 22–35. [Google Scholar] [CrossRef] - Bird, G.A. The Velocity Distribution Function within a Shock Wave. J. Fluid Mech.
**1967**, 30, 479–487. [Google Scholar] [CrossRef] - Bird, G.A. The Structure of Normal Shock Waves in a Binary Gas Mixture. J. Fluid Mech.
**1968**, 31, 657–668. [Google Scholar] [CrossRef] - Bird, G.A. Aspects of the Structure of Strong Shock Waves. Phys. Fluids
**1970**, 13, 1172–1177. [Google Scholar] [CrossRef] - Swaminathan-Gopalan, K.; Stephani, K.A. Recommended Direct Simulation Monte Carlo Collision Model Parameters for Modeling Ionized Air Transport Processes. Phys. Fluids
**2016**, 28, 027101. [Google Scholar] [CrossRef] - Stefanov, S.; Roohi, E.; Shoja-Sani, A. A Novel Transient-Adaptive Subcell Algorithm with a Hybrid Application of Different Collision Techniques in Direct Simulation Monte Carlo (DSMC). Phys. Fluids
**2022**, 34, 092003. [Google Scholar] [CrossRef] - Bird, G.A. Direct Molecular Simulation of a Dissociating Diatomic Gas. J. Comput. Phys.
**1977**, 25, 353–365. [Google Scholar] [CrossRef] - Ozawa, T.; Zhong, J.; Levin, D.A. Development of Kinetic-Based Energy Exchange Models for Noncontinuum, Ionized Hypersonic Flows. Phys. Fluids
**2008**, 20, 046102. [Google Scholar] [CrossRef] - Li, Z.; Ozawa, T.; Sohn, I.; Levin, D.A. Modeling of Electronic Excitation and Radiation in Non-Continuum Hypersonic Reentry Flows. Phys. Fluids
**2011**, 23, 066102. [Google Scholar] [CrossRef] - Shariati, V.; Roohi, E.; Ebrahimi, A. Numerical Study of Gas Flow in Super Nanoporous Materials Using the Direct Simulation Monte-Carlo Method. Micromachines
**2023**, 14, 139. [Google Scholar] [CrossRef] - Schwartzentruber, T.E.; Boyd, I.D. A Hybrid Particle-Continuum Method Applied to Shock Waves. J. Comput. Phys.
**2006**, 215, 402–416. [Google Scholar] [CrossRef] - Cunningham, E. On the Velocity of Steady Fall of Spherical Particles through Fluid Medium. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character
**1910**, 83, 357–365. [Google Scholar] - Millikan, R.A. The Isolation of an Ion, a Precision Measurement of Its Charge, and the Correction of Stokes’s Law. Science
**1910**, 32, 436–448. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Knudsen, M.; Weber, S. Luftwiderstand Gegen Die Langsame Bewegung Kleiner Kugeln. Ann. Phys.
**1911**, 341, 981–994. [Google Scholar] [CrossRef] [Green Version] - Millikan, R.A. The General Law of Fall of a Small Spherical Body through a Gas, and Its Bearing upon the Nature of Molecular Reflection from Surfaces. Phys. Rev.
**1923**, 22, 1. [Google Scholar] [CrossRef] [Green Version] - Birge, R.T. The 1944 Values of Certain Atomic Constants with Particular Reference to the Electronic Charge. Am. J. Phys.
**1945**, 13, 63–73. [Google Scholar] [CrossRef] - Kim, J.H.; Mulholland, G.W.; Kukuck, S.R.; Pui, D.Y.H. Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer (Nano-DMA) for Knudsen Number from 0.5 to 83. J. Res. Natl. Inst. Stand. Technol.
**2005**, 110, 31. [Google Scholar] [CrossRef] - Roohi, E.; Stefanov, S. Collision Partner Selection Schemes in DSMC: From Micro/Nano Flows to Hypersonic Flows. Phys. Rep.
**2016**, 656, 1–38. [Google Scholar] [CrossRef] - Bird, G.A. Shock-Wave Structure in a Rigid Sphere Gas. In Proceedings of the 4th International Symposium on Rarefied Gas Dynamics, Toronto, ON, Canada, 14–17 July 1964; Academic Press: New York, MY, USA, 1965; Volume 2, pp. 216–222. [Google Scholar]
- Bird, G.A. Perception of Numerical Methods in Rarefied Gasdynamics. Prog. Astronaut. Aeronaut.
**1989**, 117, 211–226. [Google Scholar] - Koura, K. Null-collision Technique in the Direct-simulation Monte Carlo Method. Phys. Fluids
**1986**, 29, 3509–3511. [Google Scholar] [CrossRef] - Ivanov, M.S.; Rogasinskii, S.V. Theoretical Analysis of Traditional and Modern Schemes of the DSMC Method. In Rarefied Gas Dynamics; VCH Verlagsgesellschaft mbH: Weinheim, Germany, 1991; pp. 629–642. [Google Scholar]
- Yanitskiy, V. Operator Approach to Direct Simulation Monte Carlo Theory in Rarefied Gas Dynamics. In Proceedings of the 17th Symposium on Rarefied Gas Dynamics, Aachen, Germany, 8–14 July 1990; VCH New York: New York, NY, USA, 1990; pp. 770–777. [Google Scholar]
- Stefanov, S.K. Particle Monte Carlo Algorithms with Small Number of Particles in Grid Cells. In Proceedings of the Numerical Methods and Applications: 7th International Conference, NMA 2010, Borovets, Bulgaria, 20–24 August 2010; Revised Papers 7. Springer: Berlin/Heidelberg, Germany, 2011; pp. 110–117. [Google Scholar]
- Stefanov, S.K. On DSMC Calculations of Rarefied Gas Flows with Small Number of Particles in Cells. SIAM J. Sci. Comput.
**2011**, 33, 677–702. [Google Scholar] [CrossRef] - Piekos, E.S.; Breuer, K.S. Numerical Modeling of Micromechanical Devices Using the Direct Simulation Monte Carlo Method. J. Fluids Eng. Sep.
**1996**, 118, 464–469. [Google Scholar] [CrossRef] - Shu, C.; Mao, X.H.; Chew, Y.T. Particle Number per Cell and Scaling Factor Effect on Accuracy of DSMC Simulation of Micro Flows. Int. J. Numer. Methods Heat Fluid Flow
**2005**, 15, 827–841. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Velocity profiles for microchannel flow. (

**b**) Pressure profile ($Kn$ = 0.0439) of microchannel flow. (

**c**) Deviation in the pressure ($Kn$= 0.0439) of microchannel flow.

**Figure 3.**(

**a**) Schematic diagram of the calculation domain for the drag force. (

**b**) Force-per-area contour on the particle surface. (

**c**) Force per area on particle surface for various angles.

**Figure 5.**(

**a**) The shape of the ellipsoid with aspect ratio. (

**b**) Force per area on ellipsoidal particle surface for various angles.

**Figure 6.**(

**a**) Drag force for various aspect ratios (${D}_{x}=50\mathrm{nm}$). (

**b**) Drag force for various aspect ratios (${D}_{x}=100\mathrm{nm}$).

**Figure 7.**(

**a**) Normalized drag force according to various aspect ratios (${D}_{x}=50\mathrm{nm}$). (

**b**) Normalized drag force according to various aspect ratios (${D}_{x}=100\mathrm{nm}$).

Case | Inlet Pressure [Pa] | Outlet Pressure [Pa] | $\mathit{K}\mathit{n}$ |
---|---|---|---|

1 | $1.4\times {10}^{5}$ | $1.0\times {10}^{5}$ | 0.0402 |

2 | $1.2\times {10}^{5}$ | $1.0\times {10}^{5}$ | 0.0439 |

Knudsen Number | Pressure [Pa] |
---|---|

10 | 13,851 |

20 | 6925 |

50 | 2770 |

100 | 1385 |

500 | 277 |

1000 | 139 |

5000 | 28 |

10,000 | 14 |

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**MDPI and ACS Style**

Jang, J.; Son, Y.; Lee, S.
A Numerical Study of an Ellipsoidal Nanoparticles under High Vacuum Using the DSMC Method. *Micromachines* **2023**, *14*, 778.
https://doi.org/10.3390/mi14040778

**AMA Style**

Jang J, Son Y, Lee S.
A Numerical Study of an Ellipsoidal Nanoparticles under High Vacuum Using the DSMC Method. *Micromachines*. 2023; 14(4):778.
https://doi.org/10.3390/mi14040778

**Chicago/Turabian Style**

Jang, Jinwoo, Youngwoo Son, and Sanghwan Lee.
2023. "A Numerical Study of an Ellipsoidal Nanoparticles under High Vacuum Using the DSMC Method" *Micromachines* 14, no. 4: 778.
https://doi.org/10.3390/mi14040778