Draft:Rarefied gas dynamics

The history of rarefied gas dynamics starts with the kinetic theory of gases. In 1860, Maxwell published results on the distribution of molecular velocities for a gas being in equilibrium, which is called the Maxwell-Boltzmann distribution.{{Cite journal |last=Clerk Maxwell |first=James |date=1860 |title=II. Illustrations of the dynamical theory of gases |url=https://www.tandfonline.com/doi/abs/10.1080/14786446008642902 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=20 |issue=130 |pages=21–37 |doi=10.1080/14786446008642902 |issn=1941-5982}}{{Cite journal |last=Clerk Maxwell |first=James |date=1867-01-01 |title=On the Dynamical Theory of Gases |url=https://ui.adsabs.harvard.edu/abs/1867RSPT..157...49C |journal=Philosophical Transactions of the Royal Society of London Series I |volume=157 |pages=49–88|bibcode=1867RSPT..157...49C }} As a result, the assumption that all molecules move with the same speed was abandoned and the random nature of molecular motion was recognized. Then in 1872 , Boltzmann derived an integro-differential equation (Boltzmann equation) to describe the evolution of the velocity distribution function in space and time. {{Cite book |last1=Ferziger |first1=Joel H. |url=https://books.google.com/books?id=mxS2AAAAIAAJ&q=Mathematical+theory+of+transport+processes+in+gases-NH |title=Mathematical Theory of Transport Processes in Gases |last2=Kaper |first2=H. G. |date=1972 |publisher=North-Holland Publishing Company |isbn=978-0-7204-2046-3 |language=en}}Boltzmann, Ludwig. "Further studies on the thermal equilibrium of gas molecules." The kinetic theory of gases: an anthology of classic papers with historical commentary. 2003. 262-349. In the same paper, Boltzmann proved the H-theorem demonstrating that intermolecular collisions always produce the entropy. In other words, the Boltzmann equation is irreversible.

In 1912, Hilbert proved the existence and uniqueness of a solution of the Boltzmann Equation{{Cite book |last=Hilbert |first=D. |title=Grundzüge einer Allgemeinen Theorie der linearen Integralgleichungen |publisher=Chelsea Publishing Co. |year=1953 |isbn=978-1429702362}}. A connection between the kinetic theory and fluid dynamics was done by Chapman and Enskog {{Cite journal |date=1916 |title=VI. On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1916.0006 |journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character |language=en |volume=216 |issue=538–548 |pages=279–348 |doi=10.1098/rsta.1916.0006 |bibcode=1916RSPTA.216..279C |issn=0264-3952 |last1=Chapman |first1=S. }}{{Cite journal |date=1918 |title=V. On the kinetic theory of a gas. Part II.—A composite monatomic gas: diffusion, viscosity, and thermal conduction |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1918.0005 |journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character |language=en |volume=217 |issue=549–560 |pages=115–197 |doi=10.1098/rsta.1918.0005 |bibcode=1918RSPTA.217..115C |issn=0264-3952 |last1=Chapman |first1=S. }} who derived the Euler and Navier-Stokes equations based upon a series expansion of the Boltzmann equation with respect to the Knudsen number. This approach allowed to deduced expressions for the viscosity and thermal conductivity. Furthermore, considering higher order terms of the expansion, Burnett derived more general constitutive equations for the pressure tensor and heat flux.

{{Cite journal | last=Burnett |first=D.|title=The distribution of molecular velocities and the mean motion in non-uniform gas |journal=Proc. London Math. Soc. |date=1935 |volume=40|pages=382–435}} Then, Grad proved the equivalence of the equations of fluid dynamics to an asymptotic form of the Boltzmann equation.{{Cite book |last=Grad |first=Harold |url=https://books.google.com/books?id=qSN4sMWEB-IC&dq=Grad+1965+asymptotic+equivalence+of+the+navier+stokes&pg=PA4 |title=Asymptotic Equivalence of the Navier-Stokes and Nonlinear Boltzmann Equations |date=1964 |publisher=Magneto-Fluid Dynamics Division, Courant Institute of Mathematical Sciences, New York University |language=en}} The scientific branch of rarefied gas dynamics consolidated in the 19th century and became forefront with space exploration. As a result, the first international symposium on rarefied gas dynamics was held in Nice, France in July 1958. Nowadays, this biannual symposium is a international scientific event joining many researchers and engineers from the whole world.

To determine if a gas can be classified as rarefied, the non-dimensional Knudsen number is usually calculated. The Knudsen number \mathrm{Kn} is defined as the ratio of the mean free path $\backslash lambda$, to a characteristic length scale $L$ in the flow, i.e. \mathrm{Kn}=$\backslash lambda/L$. The value of the Knudsen number tells you whether a molecular modeling approach must be used or a macroscopic description is sufficient. High and intermediate Knudsen numbers mean that the gas is rarefied, while small Knudsen numbers imply that the gas is a continuum. Various flow regimes can be distinguished based on the value of the Knudsen number. Namely, a gas flow regime can be characterized as continuum (or viscous), slip, transitional, and free-molecular.

{{Cite journal |last1=Sun |first1=Quanhua |last2=Boyd |first2=Iain D. |date=March 25, 2002 |title=A Direct Simulation Method for Subsonic, Microscale Gas Flows |url=http://dx.doi.org/10.1006/jcph.2002.7061 |journal=Journal of Computational Physics |volume=179 |issue=2 |pages=400–425 |doi=10.1006/jcph.2002.7061 |bibcode=2002JCoPh.179..400S |issn=0021-9991}}

The following ranges of the Knudsen number for these regimes are usually established:

• \mathrm{Kn} for continuum or viscous regime;
• \mathrm{Kn} for slip flow regime;
• \mathrm{Kn} for transitional regime;
• \mathrm{Kn} $>\; 10$ for free-molecular regime.

This division between the regimes can vary depending on flow type and required accuracy. The mathematical approach and tool to model a gas flow depend on the flow regime. In the continuum regime, the gas is not rarefied and hence can be described by the Euler or Navier-Stokes equations. In the slip flow regime, the Navier-Stokes equations are applied too, but the velocity no-slip and temperature continuity boudary conditions are replaced by the velocity slip and temperature jump conditions. {{Cite journal |last1=Lofthouse |first1=Andrew J. |last2=Scalabrin |first2=Leonardo C. |last3=Boyd |first3=Iain D. |date=23 May 2012 |title=Velocity Slip and Temperature Jump in Hypersonic Aerothermodynamics |url=https://arc.aiaa.org/doi/10.2514/1.31280 |journal=Journal of Thermophysics and Heat Transfer |language=en |volume=22 |issue=1 |pages=38–49 |doi=10.2514/1.31280 |hdl=2027.42/76244 |issn=0887-8722 |via=AIAA}} {{cite journal |last= Sharipov |first=Felix |date=2011|title= Data on the velocity slip and temperature jump on a gas-solid interface |journal= J. Phys. Chem. Ref. Data. |volume=40 |issue=2 |pages=023101 |doi=10.1063/1.3580290|bibcode=2011JPCRD..40b3101S }} All approaches to model gas flows in the transitional regime are based on the Boltzmann transport equation. Since the intermolecular collisions are neglected in the free-molecular regime, the Boltzmann equation is applied in its simplified form.

When the continuum assumption is fulfilled, the molecular nature of the gas is neglected and the spatial and temporal variations of the flow are described in terms of its macroscopic properties such as velocity, density, pressure, and temperature. In this case, the Navier-Stokes equations provides the appropriate mathematical model of fluid flows. However, the continuum assumption is broken in a rarefied gas flow, therefore a modeling of such a flow is based on the Boltzmann transport equation. In this case, the molecular nature of the gas must be considered {{Cite journal |last=Muntz |first=E. |date=1989-01-01 |title=Rarefied Gas Dynamics |url=http://dx.doi.org/10.1146/annurev.fluid.21.1.387 |journal=Annual Review of Fluid Mechanics |volume=21 |issue=1 |pages=387–417 |doi=10.1146/annurev.fluid.21.1.387 |issn=0066-4189}} by means of the velocity distribution function $f(t,\backslash mathbf\{r\},\backslash mathbf\{v\})$ where $t$ is the time, $\backslash mathbf\{r\}$ is a vector of spatial coordinates and $\backslash mathbf\{v\}$ is a velocity of molecules. This function provides a statistical description of the gas on the molecular level. Taking moments of the velocity distribution function leads to macroscopic properties of a gas flow, thus providing the connection between microscopic and macroscopic levels. The evolution of the velocity distribution function is described by the Boltzmann equation, which is valid for a dilute gas. In such a gas, the mean distance between molecules is significantly larger than the molecular size, meaning that the intermolecular collisions are predominantly binary.{{Cite book |last1=Boyd |first1=Iain D. |url=https://www.cambridge.org/core/books/nonequilibrium-gas-dynamics-and-molecular-simulation/E317E76AA03C6C2C0A78A67EAB122C30 |title=Nonequilibrium Gas Dynamics and Molecular Simulation |last2=Schwartzentruber |first2=Thomas E. |date=2017 |publisher=Cambridge University Press |isbn=978-1-107-07344-9 |series=Cambridge Aerospace Series |location=Cambridge}} Another important assumption in the derivation of the Boltzmann transport equation is molecular chaos, which means that the velocities of particles during their free motion between collisions are uncorrelated, and independent of their positions.

When a rarefied gas is restricted by a solid surface, the interaction of the gas molecules with this surfaces should be considered. Such an interaction leads to the momentum and heat transfer between the gas and the solid surface. This implies that the Boltzmann equation must be accompanied by boundary conditions determined by the gas-surface interaction.

The Boltzmann equation is valid for a dilute gas flow in the whole range of the Knudsen number. However, a general solution of this equation requires significant computational effort, therefore several mathematical approaches to model rarefied gases were developed depending on the flow regime.

In the slip flow regime, the gas rarefaction is moderate allowing to use a Navier-Stokes solver subject to the velocity slip and temperature jump boundary conditions on a solid surface. {{Cite journal |last1=Aoki |first1=Kazuo |last2=Baranger |first2=Céline |last3=Hattori |first3=Masanari |last4=Kosuge |first4=Shingo |last5=Martalò |first5=Giorgio |last6=Mathiaud |first6=Julien |last7=Mieussens |first7=Luc |date=2017-10-07 |title=Slip Boundary Conditions for the Compressible Navier–Stokes Equations |url=http://dx.doi.org/10.1007/s10955-017-1886-8 |journal=Journal of Statistical Physics |volume=169 |issue=4 |pages=744–781 |doi=10.1007/s10955-017-1886-8 |bibcode=2017JSP...169..744A |issn=0022-4715}} The slip and jump coefficients calculated on the molecular level are universal and can be applied to any geometrical configuration of the gas flow.

In the free molecular regime, the intermolecular collisions are neglected so that each particle moves independently on each other. In this case, the Boltzmann equation is significantly simplified. All macroscopic properties of a gas flow around a convex body are obtained in the form of integral, since a gas molecule undergoes only one interaction with the body surface. Internal flows and external flows around a concave body involve multiple gas-surface interactions. In this case, the macroscopic characteristics are calculated from an integral equation. An alternative approach is the test particle Monte Carlo method consisting in tracking of a large number of model particles. Since the intermolecular collisions are neglected, the molecular trajectories are independent of each other and can be modeled serially. All these approaches are described in details in the book by Bird.

The transitional regime is hardest for mathematical modeling, because the continuum assumption is broken and the intermolecular collisions cannot be neglected. Therefore, the Boltzmann equation should be solved for any specific gas flow. The discrete velocity method and fast spectral method are typically used for this task. However, an application of this method to the Boltzmann equation in its exact form is costly very much so that, till now, only few very simple problems have been solved by this way. {{Cite journal |last1=Sharipov |first1=Felix |last2=Bertoldo |first2=Guilherme |date=2009-06-18 |title=Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number |url=https://pubs.aip.org/aip/pof/article-abstract/21/6/067101/257147/Poiseuille-flow-and-thermal-creep-based-on-the?redirectedFrom=fulltext |journal=Physics of Fluids |volume=21 |issue=6 |pages=067101–067101–8 |doi=10.1063/1.3156011 |bibcode=2009PhFl...21f7101S |issn=1070-6631}} {{Cite journal |last1=Dodulad |first1=O. I. |last2=Tcheremissine |first2=F. G. |date=June 2013 |title=Computation of a shock wave structure in monatomic gas with accuracy control |url=http://link.springer.com/10.1134/S0965542513060055 |journal=Computational Mathematics and Mathematical Physics |language=en |volume=53 |issue=6 |pages=827–844 |doi=10.1134/S0965542513060055 |bibcode=2013CMMPh..53..827D |issn=0965-5425}}{{Cite journal |last1=Jaiswal |first1=Shashank |last2=Alexeenko |first2=Alina A. |last3=Hu |first3=Jingwei |date=February 2019 |title=A discontinuous Galerkin fast spectral method for the full Boltzmann equation with general collision kernels |url=https://linkinghub.elsevier.com/retrieve/pii/S0021999118307198 |journal=Journal of Computational Physics |language=en |volume=378 |pages=178–208 |arxiv=1809.10186 |bibcode=2019JCoPh.378..178J |doi=10.1016/j.jcp.2018.11.001}} To reduce the computational cost, the Boltzmann equation is replaced by its simplified form called a model kinetic equation. Three commonly used model equations are Bhatnagar, Gross, Krook (BGK) {{Cite journal |last=Bhatnagar |first=P. L. |date=1954 |title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.511 |journal=Physical Review |volume=94 |issue=3 |pages=511–525 |doi=10.1103/PhysRev.94.511|bibcode=1954PhRv...94..511B }} , Ellipsoidal Statistical (ES-model),{{Cite journal |last=Holway |first=Lowell H. |date=1966-09-01 |title=New Statistical Models for Kinetic Theory: Methods of Construction |url=https://pubs.aip.org/pfl/article/9/9/1658/945826/New-Statistical-Models-for-Kinetic-Theory-Methods |journal=The Physics of Fluids |language=en |volume=9 |issue=9 |pages=1658–1673 |doi=10.1063/1.1761920 |bibcode=1966PhFl....9.1658H |issn=0031-9171}} and Shakhov (S-model) {{Cite journal |last=Shakhov |first=E. M. |date=1972 |title=Generalization of the Krook kinetic relaxation equation |url=http://link.springer.com/10.1007/BF01029546 |journal=Fluid Dynamics |language=en |volume=3 |issue=5 |pages=95–96 |doi=10.1007/BF01029546 |issn=0015-4628}}. Computational effort to solve these model equations is essentially smaller than that to solve the Boltzmann equation itself and, at the same time, they are appropriate to solve practical problems with a reasonable accuracy. Some examples of numerical solvers are based on the model equations can be found here.{{Cite journal |last1=Xu |first1=Kun |last2=Huang |first2=Juan-Chen |date=October 2010 |title=A unified gas-kinetic scheme for continuum and rarefied flows |url=https://linkinghub.elsevier.com/retrieve/pii/S0021999110003475 |journal=Journal of Computational Physics |language=en |volume=229 |issue=20 |pages=7747–7764 |bibcode=2010JCoPh.229.7747X |doi=10.1016/j.jcp.2010.06.032}}{{Cite journal |last1=Tumuklu |first1=Ozgur |last2=Li |first2=Zheng |last3=Levin |first3=Deborah A. |date=December 2016 |title=Particle Ellipsoidal Statistical Bhatnagar–Gross–Krook Approach for Simulation of Hypersonic Shocks |url=https://arc.aiaa.org/doi/10.2514/1.J054837 |journal=AIAA Journal |language=en |volume=54 |issue=12 |pages=3701–3716 |bibcode=2016AIAAJ..54.3701T |doi=10.2514/1.J054837 |issn=0001-1452}}{{Cite journal |last1=Alexeenko |first1=Alina |last2=Galitzine |first2=Cyril |last3=Alekseenko |first3=Alexander |date=2008-06-23 |title=High-Order Discontinuous Galerkin Method for Boltzmann Model Equations |url=https://arc.aiaa.org/doi/10.2514/6.2008-4256 |journal=AIAA 40th Thermophysics Conference |series=Fluid Dynamics and Co-located Conferences |language=en |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2008-4256 |isbn=978-1-60086-991-4}}

The Direct Simulation Monte Carlo (DSMC) is a probabilistic method proposed by G.A. Bird.{{Cite book |last=Bird |first=G. A. |title=The DSMC method |date=2013 |publisher=G. A. Bird |isbn=978-1-4921-1290-7 |edition=Version 1.2 |location=S.l.}} It consists of tracking of a huge number of model particles simultaneously decoupling the processes of their free motion and collisions between them. After an alternating these two processes many times, the macroscopic characteristics are calculated based on the information about positions and velocities of the model particles. A result obtained by the DSMC method is equivalent to that obtained from the Boltzmann equation.The DSMC method is easily applied to complicated geometrical configurations and to flows with chemical reactions. That is why it became a widely used method to solve scientific and engineering problems related to highly non-equilibrium gas flows.{{Cite journal |last1=Alexander |first1=Francis J. |last2=Garcia |first2=Alejandro L. |date=1997-11-01 |title=The Direct Simulation Monte Carlo Method |url=https://pubs.aip.org/cip/article/11/6/588/136775/The-Direct-Simulation-Monte-Carlo-Method |journal=Computers in Physics |language=en |volume=11 |issue=6 |pages=588–593 |bibcode=1997ComPh..11..588A |doi=10.1063/1.168619 |issn=0894-1866}}{{Cite journal |last1=Schwartzentruber |first1=Thomas E. |last2=Grover |first2=Maninder S. |last3=Valentini |first3=Paolo |date=October 2018 |title=Direct Molecular Simulation of Nonequilibrium Dilute Gases |url=http://dx.doi.org/10.2514/1.t5188 |journal=Journal of Thermophysics and Heat Transfer |volume=32 |issue=4 |pages=892–903 |doi=10.2514/1.t5188 |issn=0887-8722}} For weakly non-equilibrium flows, the discrete velocity method applied to the Boltzmann equation or to its model is more efficient than DSMC method.{{Cite journal |last1=Clarke |first1=Peter |last2=Varghese |first2=Philip |last3=Goldstein |first3=David |date=January 2018 |title=A low noise discrete velocity method for the Boltzmann equation with quantized rotational and vibrational energy |url=https://linkinghub.elsevier.com/retrieve/pii/S0021999117306460 |journal=Journal of Computational Physics |language=en |volume=352 |pages=326–340 |doi=10.1016/j.jcp.2017.08.065|bibcode=2018JCoPh.352..326C }}{{Cite journal |last1=Oblapenko |first1=Georgii |last2=Goldstein |first2=David |last3=Varghese |first3=Philip |last4=Moore |first4=Christopher |date=2021 |title=Velocity-Space Hybridization of Direct Simulation Monte Carlo and a Quasi-Particle Boltzmann Solver |url=https://arc.aiaa.org/doi/abs/10.2514/1.T6074 |journal=Journal of Thermophysics and Heat Transfer |volume=35 |issue=4 |pages=788–799 |doi=10.2514/1.T6074 |osti=1765750 |issn=0887-8722}}

hypersonic flight, atmospheric entry: Satellites in orbit, spacecraft or rockets re-entering Earth atmosphere, or entering atmosphere of another planet, or hypersonic cruise vehicles. These application areas often need RGD methods{{Cite book |last1=Nagnibeda |first1=Ekaterina |url=https://link.springer.com/book/10.1007/978-3-642-01390-4 |title=Non-Equilibrium Reacting Gas Flows |last2=Kustova |first2=Elena |date=2009 |isbn=978-3-642-01389-8 |series=Heat and Mass Transfer |language=en |doi=10.1007/978-3-642-01390-4}}{{Cite journal |last1=Ivanov |first1=M. S. |last2=Gimelshein |first2=S. F. |date=January 1998 |title=Computational Hypersonic Rarefied Flows |url=https://www.annualreviews.org/doi/10.1146/annurev.fluid.30.1.469 |journal=Annual Review of Fluid Mechanics |language=en |volume=30 |issue=1 |pages=469–505 |doi=10.1146/annurev.fluid.30.1.469 |bibcode=1998AnRFM..30..469I |issn=0066-4189}}{{Cite journal |last1=Theofilis |first1=Vassilis |last2=Pirozzoli |first2=Sergio |last3=Martin |first3=Pino |date=February 2022 |title=Special issue on the fluid mechanics of hypersonic flight |url=https://link.springer.com/10.1007/s00162-022-00605-2 |journal=Theoretical and Computational Fluid Dynamics |language=en |volume=36 |issue=1 |pages=1–8 |doi=10.1007/s00162-022-00605-2 |bibcode=2022ThCFD..36....1T |hdl=11573/1629119 |issn=0935-4964}}{{Cite journal |last=Sharipov |first=Felix |date=2003-06-01 |title=Hypersonic flow of rarefied gas near the Brazilian satellite during its reentry into atmosphere |url=https://www.scielo.br/j/bjp/a/GzMQYNKqMJkBVHddPsp5LFk/?lang=en |journal=Brazilian Journal of Physics |language=en |volume=33 |issue=2 |pages=398–405 |doi=10.1590/S0103-97332003000200044 |bibcode=2003BrJPh..33..398S |issn=0103-9733}} In addition, even within a mostly continuum flow around a high-speed vehicle, there may be local regions of rarefied flow as in the wake of the vehicle or near sharp-leading edges. These rarefied flow regions must be modelled appropriately in order to determine, e.g., the drag and heating to the vehicle and the radiation emitted and plasma blackout. It also includes understanding the damage impact of micrometer aerosol atmospheric particles on the material erosion and heating at the surface.{{Cite journal |last1=Habeck |first1=Joseph B. |last2=Kroells |first2=Michael D. |last3=Schwartzentruber |first3=Thomas E. |last4=Candler |first4=Graham V. |date=2023-11-29 |title=Characterization of Particle-Surface Impacts on a Sphere-Cone at Hypersonic Flight Conditions |url=https://arc.aiaa.org/doi/10.2514/1.J062945 |journal=AIAA Journal |volume=62 |issue=2 |language=en |pages=460–475 |doi=10.2514/1.J062945 |issn=0001-1452}}

An example of rarefied gas dynamics that combines both low density and small length scales involve thrusters on spacecraft used for maneuvering in outer space as well as rocket plumes impinging on surfaces.{{Cite journal |last=Cai |first=Chunpei |date=December 2014 |title=Rocket Plume Modeling |url=https://arc.aiaa.org/doi/10.2514/1.J053351 |journal=AIAA Journal |language=en |volume=52 |issue=12 |pages=2907–2910 |doi=10.2514/1.J053351 |bibcode=2014AIAAJ..52.2907C |issn=0001-1452}}{{Cite journal |last1=Karpuzcu |first1=Irmak T. |last2=Levin |first2=Deborah A. |date=November 2023 |title=Study of Side-Jet Interactions over a Hypersonic Cone Flow Using Kinetic Methods |url=https://arc.aiaa.org/doi/10.2514/1.J063020 |journal=AIAA Journal |language=en |volume=61 |issue=11 |pages=4741–4751 |doi=10.2514/1.J063020 |bibcode=2023AIAAJ..61.4741K |issn=0001-1452}}{{Cite journal |last1=Morris |first1=Aaron B. |last2=Goldstein |first2=David B. |last3=Varghese |first3=Philip L. |last4=Trafton |first4=Laurence M. |date=March 2015 |title=Approach for Modeling Rocket Plume Impingement and Dust Dispersal on the Moon |url=https://arc.aiaa.org/doi/10.2514/1.A33058 |journal=Journal of Spacecraft and Rockets |language=en |volume=52 |issue=2 |pages=362–374 |doi=10.2514/1.A33058 |bibcode=2015JSpRo..52..362M |issn=0022-4650}}{{Cite journal |last1=Gimelshein |first1=Natalia E. |last2=Lyons |first2=Robert B. |last3=Reuster |first3=James G. |last4=Gimelshein |first4=Sergey F. |date=October 2009 |title=Analysis of Ultraviolet Radiation Predictions from High Altitude Two-Phase Plumes |url=https://arc.aiaa.org/doi/10.2514/1.44067 |journal=AIAA Journal |language=en |volume=47 |issue=10 |pages=2486–2492 |doi=10.2514/1.44067 |bibcode=2009AIAAJ..47.2486G |issn=0001-1452}}

Modeling gas flow around porous carbon-fiber materials at the microscale when designing heat shields for spacecraft{{Cite journal |last1=Kroells |first1=Michael |last2=Ramjatan |first2=Sahadeo |last3=Schwartzentruber |first3=Thomas E. |date=2024-07-27 |title=Aerothermal Loads on a Representative Porous Mesostructure at Flight Conditions Using Direct Simulation Monte Carlo |url=https://arc.aiaa.org/doi/10.2514/1.T6894 |journal=Journal of Thermophysics and Heat Transfer |language=en |volume=38 |issue=4 |pages=491–507 |doi=10.2514/1.T6894 |issn=0887-8722}}{{Cite journal |last1=Fu |first1=Rui |last2=Martin |first2=Alexandre |last3=Ramjatan |first3=Sahadeo |last4=Kroells |first4=Michael |last5=Schwartzentruber |first5=Thomas E. |date=2024-05-17 |title=Microscale Thermal–Structural Modeling for Carbon Fibers Subjected to a Hypersonic Boundary Layer |url=https://arc.aiaa.org/doi/10.2514/1.A35916 |journal=Journal of Spacecraft and Rockets |volume=62 |language=en |pages=7–17 |doi=10.2514/1.A35916 |issn=0022-4650}}{{Cite journal |last1=Jambunathan |first1=Revathi |last2=Levin |first2=Deborah A. |last3=Borner |first3=Arnaud |last4=Ferguson |first4=Joseph C. |last5=Panerai |first5=Francesco |date=March 2019 |title=Prediction of gas transport properties through fibrous carbon preform microstructures using Direct Simulation Monte Carlo |url=https://linkinghub.elsevier.com/retrieve/pii/S0017931018326619 |journal=International Journal of Heat and Mass Transfer |language=en |volume=130 |pages=923–937 |doi=10.1016/j.ijheatmasstransfer.2018.11.006|bibcode=2019IJHMT.130..923J }}{{Cite journal |last1=Poovathingal |first1=Savio J. |last2=Soto |first2=Brendan M. |last3=Brewer |first3=Cameron |date=March 2022 |title=Effective Permeability of Carbon Composites Under Reentry Conditions |url=https://arc.aiaa.org/doi/10.2514/1.J060630 |journal=AIAA Journal |language=en |volume=60 |issue=3 |pages=1293–1302 |doi=10.2514/1.J060630 |bibcode=2022AIAAJ..60.1293P |issn=0001-1452}}{{Cite journal |last1=Poovathingal |first1=Savio |last2=Stern |first2=Eric C. |last3=Nompelis |first3=Ioannis |last4=Schwartzentruber |first4=Thomas E. |last5=Candler |first5=Graham V. |date=March 2019 |title=Nonequilibrium flow through porous thermal protection materials, Part II: Oxidation and pyrolysis |url=https://linkinghub.elsevier.com/retrieve/pii/S0021999118301268 |journal=Journal of Computational Physics |language=en |volume=380 |pages=427–441 |doi=10.1016/j.jcp.2018.02.043|bibcode=2019JCoPh.380..427P }} with the objective of characterizing the aerothermal loading (e.g. heat flux, shear stress) and concentrations of atomic species inside the material microstructure. Here the mean free path approaches the length scale of the carbon-fibers of $10\; \backslash mu\; m$.

Very small length scales can also result in rarefied gas phenomena. Micro-Electro-Mechanical systems (MEMS) and Nanoelectromechanical systems (NEMS), which involve the fabrication and operation of microscopic devices, involve the motion of gases at very small length scales including $1\; \backslash mu\; m$ or $10^\{-6\}$ m resulting in rarefied gas dynamics{{Citation |last1=Aluru |first1=Narayan R. |title=Numerical Simulation Of Microflows And Nanoflows |date=2010-03-25 |work=Micro/Nano Technology Systems for Biomedical Applications |pages=53–120 |url=http://dx.doi.org/10.1093/acprof:oso/9780199219698.003.0003 |access-date=2024-11-22 |publisher=Oxford University Press |last2=Karniadakis |first2=George Em|doi=10.1093/acprof:oso/9780199219698.003.0003 |isbn=978-0-19-921969-8 }}{{Citation |last=Li |first=Jun |title=Simulation Methods for Rarefied Gas Flows |date=2020 |work=Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation |pages=67–117 |url=http://link.springer.com/10.1007/978-3-030-26466-6_3 |access-date=2024-11-22 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-26466-6_3 |isbn=978-3-030-26465-9}}{{Cite journal |last1=Patronis |first1=Alexander |last2=Lockerby |first2=Duncan A. |last3=Borg |first3=Matthew K. |last4=Reese |first4=Jason M. |date=2013-12-15 |title=Hybrid continuum–molecular modelling of multiscale internal gas flows |url=https://www.sciencedirect.com/science/article/pii/S0021999113005718 |journal=Journal of Computational Physics |volume=255 |pages=558–571 |doi=10.1016/j.jcp.2013.08.033 |bibcode=2013JCoPh.255..558P |issn=0021-9991}}{{Cite journal |last1=Varoutis |first1=S. |last2=Valougeorgis |first2=D. |last3=Sharipov |first3=F. |date=2009-11-01 |title=Simulation of gas flow through tubes of finite length over the whole range of rarefaction for various pressure drop ratios |url=https://pubs.aip.org/jva/article/27/6/1377/243946/Simulation-of-gas-flow-through-tubes-of-finite |journal=Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films |language=en |volume=27 |issue=6 |pages=1377–1391 |doi=10.1116/1.3248273 |bibcode=2009JVSTA..27.1377V |issn=0734-2101}}

Solid or liquid particulates or droplets suspended in a gas flow often have size scales of a micro-m or less, leading to Kn near the particle of the order 1 and the need to include rarefaction effects. Computation of important quantities such as particle drag or heat transfer between the particle and the gas may need to use rarefied gas dynamic methods.{{Cite journal |last1=Clercx |first1=H.J.H. |last2=Livi |first2=C. |last3=Di Staso |first3=G. |last4=Toschi |first4=F. |date=November 2024 |title=Modeling drag coefficients of spheroidal particles in rarefied flow conditions |url=https://linkinghub.elsevier.com/retrieve/pii/S0997754624000967 |journal=European Journal of Mechanics - B/Fluids |language=en |volume=108 |pages=90–103 |doi=10.1016/j.euromechflu.2024.07.008|bibcode=2024EuJMB.108...90C }}{{Cite journal |last1=Cheremisin |first1=A. A. |last2=Kushnarenko |first2=A. V. |date=2013-08-01 |title=Photophoretic interaction of aerosol particles and its effect on coagulation in rarefied gas medium |url=https://www.sciencedirect.com/science/article/abs/pii/S0021850213000724 |journal=Journal of Aerosol Science |volume=62 |pages=26–39 |doi=10.1016/j.jaerosci.2013.03.011 |bibcode=2013JAerS..62...26C |issn=0021-8502}}{{Cite journal |last1=Filippov |first1=A. V. |last2=Rosner |first2=D. E. |date=2000-01-01 |title=Energy transfer between an aerosol particle and gas at high temperature ratios in the Knudsen transition regime |url=https://www.sciencedirect.com/science/article/pii/S0017931099001131 |journal=International Journal of Heat and Mass Transfer |volume=43 |issue=1 |pages=127–138 |doi=10.1016/S0017-9310(99)00113-1 |bibcode=2000IJHMT..43..127F |issn=0017-9310}}

A variety of interesting phenomena and micro-device concepts have been investigated that rely on the effects of temperature differences acting in the rarefied or transitional gas-flow regime.{{Cite journal |last1=Alexeenko |first1=Alina A. |last2=Strongrich |first2=A. D. |last3=Cofer |first3=A. G. |last4=Pikus |first4=A. |last5=Sebastiao |first5=I. B. |last6=Tholeti |first6=S. S. |last7=Shivkumar |first7=G. |date=2016-11-15 |title=Microdevices enabled by rarefied flow phenomena |url=https://pubs.aip.org/aip/acp/article-abstract/1786/1/080001/886077/Microdevices-enabled-by-rarefied-flow-phenomena |journal=AIP Conference Proceedings |volume=1786 |issue=1 |pages=080001 |bibcode=2016AIPC.1786h0001A |doi=10.1063/1.4967594 |issn=0094-243X}}{{Cite journal |last1=Ketsdever |first1=Andrew |last2=Gimelshein |first2=Natalia |last3=Gimelshein |first3=Sergey |last4=Selden |first4=Nathaniel |date=May 2012 |title=Radiometric phenomena: From the 19th to the 21st century |url=https://linkinghub.elsevier.com/retrieve/pii/S0042207X12000723 |journal=Vacuum |language=en |volume=86 |issue=11 |pages=1644–1662 |bibcode=2012Vacuu..86.1644K |doi=10.1016/j.vacuum.2012.02.006}}{{Cite journal |last1=Selden |first1=N. |last2=Ngalande |first2=C. |last3=Gimelshein |first3=S. |last4=Muntz |first4=E. P. |last5=Alexeenko |first5=A. |last6=Ketsdever |first6=A. |date=2009-04-06 |title=Area and edge effects in radiometric forces |url=https://link.aps.org/doi/10.1103/PhysRevE.79.041201 |journal=Physical Review E |language=en |volume=79 |issue=4 |page=041201 |bibcode=2009PhRvE..79d1201S |doi=10.1103/PhysRevE.79.041201 |issn=1539-3755 |pmid=19518216}} Notable examples include the Crookes radiometer, Knudsen compressors,{{Cite journal |last1=Akhlaghi |first1=Hassan |last2=Roohi |first2=Ehsan |last3=Stefanov |first3=Stefan |date=January 2023 |title=A comprehensive review on micro- and nano-scale gas flow effects: Slip-jump phenomena, Knudsen paradox, thermally-driven flows, and Knudsen pumps |url=https://linkinghub.elsevier.com/retrieve/pii/S0370157322003787 |journal=Physics Reports |language=en |volume=997 |pages=1–60 |bibcode=2023PhR...997....1A |doi=10.1016/j.physrep.2022.10.004}}{{Cite journal |last1=Han |first1=Y.-L. |last2=Muntz |first2=E. P. |date=2007-04-12 |title=Experimental investigation of micro-mesoscale Knudsen compressor performance at low pressures |url=https://pubs.aip.org/avs/jvb/article-abstract/25/3/703/468574/Experimental-investigation-of-micro-mesoscale?redirectedFrom=fulltext |journal=Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena |volume=25 |issue=3 |pages=703–714 |bibcode=2007JVSTB..25..703H |doi=10.1116/1.2723755 |issn=1071-1023}} radiometric force actuators and photophoretic phenomena.{{Cite journal |last1=Sharipov |first1=Felix |last2=Schafer |first2=Benjamin C. |last3=Keith |first3=David W. |date=2024-08-23 |title=Modeling the photophoretic force on a perforated membrane |url=https://pubs.aip.org/aip/pof/article-abstract/36/8/081707/3309568/Modeling-the-photophoretic-force-on-a-perforated?redirectedFrom=fulltext |journal=Physics of Fluids |volume=36 |issue=8 |pages=081707 |bibcode=2024PhFl...36h1707S |doi=10.1063/5.0223475 |issn=1070-6631}}{{Cite journal |last1=Cheremisin |first1=A.A. |last2=Kushnarenko |first2=A.V. |date=August 2013 |title=Photophoretic interaction of aerosol particles and its effect on coagulation in rarefied gas medium |url=https://linkinghub.elsevier.com/retrieve/pii/S0021850213000724 |journal=Journal of Aerosol Science |language=en |volume=62 |pages=26–39 |bibcode=2013JAerS..62...26C |doi=10.1016/j.jaerosci.2013.03.011}}

text:{{Cite journal |last1=Nanbu |first1=K. |last2=Morimoto |first2=T. |last3=Suetani |first3=M. |date=Oct./1999 |title=Direct simulation Monte Carlo analysis of flows and etch rate in an inductively coupled plasma reactor |url=https://ieeexplore.ieee.org/document/799816 |journal=IEEE Transactions on Plasma Science |volume=27 |issue=5 |pages=1379–1388 |doi=10.1109/27.799816|bibcode=1999ITPS...27.1379N }}{{Cite journal |last1=Bruno |first1=D |last2=Capitelli |first2=M |last3=Longo |first3=S |last4=Minelli |first4=P |last5=Taccogna |first5=F |date=2003-11-01 |title=Particle kinetic modelling of rarefied gases and plasmas |url=https://iopscience.iop.org/article/10.1088/0963-0252/12/4/024 |journal=Plasma Sources Science and Technology |volume=12 |issue=4 |pages=S89–S97 |doi=10.1088/0963-0252/12/4/024 |bibcode=2003PSST...12S..89B |issn=0963-0252}}{{Cite journal |last=Boyd |first=Iain D. |date=2001 |title=Review of Hall Thruster Plume Modeling |url=https://arc.aiaa.org/doi/abs/10.2514/2.3695 |journal=Journal of Spacecraft and Rockets |volume=38 |issue=3 |pages=381–387 |doi=10.2514/2.3695 |bibcode=2001JSpRo..38..381B |hdl=2027.42/76574 |issn=0022-4650}}{{Cite journal |last1=Nishii |first1=Keita |last2=Levin |first2=Deborah A |date=2023-11-01 |title=Kinetic simulation of ion thruster plume neutralization in a vacuum chamber |url=https://iopscience.iop.org/article/10.1088/1361-6595/ad0836 |journal=Plasma Sources Science and Technology |volume=32 |issue=11 |pages=115009 |doi=10.1088/1361-6595/ad0836 |arxiv=2306.16693 |bibcode=2023PSST...32k5009N |issn=0963-0252}}{{Cite journal |last1=Rebrov |first1=A.K. |last2=Maltsev |first2=R.V. |last3=Safonov |first3=A.I. |last4=Timoshenko |first4=N.I. |date=May 2011 |title=Activated gas jet deposition |url=https://linkinghub.elsevier.com/retrieve/pii/S0040609011003555 |journal=Thin Solid Films |language=en |volume=519 |issue=14 |pages=4542–4544 |doi=10.1016/j.tsf.2011.01.290|bibcode=2011TSF...519.4542R }}{{Cite journal |last1=Tumuklu |first1=Ozgur |last2=Levin |first2=Deborah A. |date=September 2018 |title=Particle Simulations of the Effects of Atomic Oxygen on Ion Thruster Plumes |url=http://dx.doi.org/10.2514/1.a34053 |journal=Journal of Spacecraft and Rockets |volume=55 |issue=5 |pages=1154–1165 |doi=10.2514/1.a34053 |bibcode=2018JSpRo..55.1154T |issn=0022-4650}}

Some of the many Application areas for RGD methods and research are given here with example references. Note that some of these flows have vastly different density/Kn in some spatial or temporal regions where they transition between continuum and free-molecular conditions (plumes; hypersonic flows around complex shapes) and thus may require hybrid solution methods.{{Cite journal |last1=Hash |first1=D. B. |last2=Hassan |first2=H. A. |date=1996 |title=Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods |url=https://arc.aiaa.org/doi/abs/10.2514/3.781 |journal=Journal of Thermophysics and Heat Transfer |volume=10 |issue=2 |pages=242–249 |doi=10.2514/3.781 |issn=0887-8722}}{{Cite journal |last1=Schwartzentruber |first1=Thomas E. |last2=Scalabrin |first2=Leonardo C. |last3=Boyd |first3=Iain D. |date=January 2008 |title=Hybrid Particle-Continuum Simulations of Nonequilibrium Hypersonic Blunt-Body Flowfields |url=https://arc.aiaa.org/doi/10.2514/1.30216 |journal=Journal of Thermophysics and Heat Transfer |language=en |volume=22 |issue=1 |pages=29–37 |doi=10.2514/1.30216 |hdl=2027.42/77319 |issn=0887-8722}}{{Cite journal |last1=Fei |first1=Fei |last2=Jenny |first2=Patrick |date=January 2021 |title=A hybrid particle approach based on the unified stochastic particle Bhatnagar-Gross-Krook and DSMC methods |url=https://linkinghub.elsevier.com/retrieve/pii/S002199912030632X |journal=Journal of Computational Physics |language=en |volume=424 |pages=109858 |doi=10.1016/j.jcp.2020.109858|bibcode=2021JCoPh.42409858F }}

"The International Symposium on Rarefied Gas Dynamics is a biennial academic conference. The symposia are a forum for the presentation of recent advances in the field of rarefied gas dynamics. Research presented encompasses applications of space, materials, and propulsion, as well as the basic physics of molecular interactions, gas surface interactions, kinetic theory, astronomical observations, gas transport, multi-phase flows, combustion, non-equilibrium hypersonic gas dynamics, and plasma processing. The first symposium was held in 1958 in Nice, France. Since that time, the symposia have been organized in various countries in North America, Europe, Asia, and Australia."{{Cite web |title=About – Rarefied Gas Dynamics |url=https://www.rarefiedgasdynamics.org/about/ |access-date=2025-01-07 |language=en-US}}

Held every two years in Santa Fe, New Mexico, the DSMC conference "brings together outstanding DSMC researchers from around the world." The goal of this meeting is to bring together developers and practitioners of the Direct Simulation Monte Carlo (DSMC) method.{{Cite web |title=DSMC 2023 Conference |url=https://www.sandia.gov/dsmc/ |access-date=2024-06-18 |website=Direct Simulation Monte Carlo DSMC |language=en-US}}

• https://www.rarefiedgasdynamics.org/
• https://www.sandia.gov/dsmc/
• [https://www.linkedin.com/company/rgd-next-gen/posts/?feedView=all https://www.linkedin.com/company/rgd-next-gen/]

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