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Topic: Knudsen number

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Laplace number

Power number

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Friction factor

Fluid Mechanics

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Abbe number

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Mean free path

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	Knudsen number - Wikipedia, the free encyclopedia
		The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale.
		The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used.
		If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is no longer a good approximation.
	en.wikipedia.org /wiki/Knudsen_number (188 words)

	Aeronautic and Aerospace Flight Limits
		The Mach number (introduced by J. Ackeret, 1929) defines the appearance of compressibility effects and the changes associated with the shock waves.
		The Reynolds and Mach numbers are independent and are both needed to define the characteristics of speeds in the transonic regime.
		Flows at Knudsen numbers Kn >> 1 are basically collisionless flows (artificial artificial satellites in orbital motion above the Earth); flows at Kn < 1 are in a regime of slight rarefaction, and are called slip flows; flows at intermediate Knudsen numbers are called transitional flows.
	aerodyn.org /Atm-flight/sregime.html (598 words)

	Centre for Microfluidics - Rarefied Gas Dynamics - Flow Around A Micro-Sphere
		The investigation considers the case of low Reynolds number, isothermal, rarefied gas flow past a confined microsphere within a circular pipe and focuses on the estimation of the hydrodynamic drag forces on a stationary (non-rotating) sphere.
		Since the conditions investigated in the study had relatively low Mach numbers compressibility effects were ignored and the flow was assumed to be incompressible and isothermal.
		As the Knudsen number is increased, the velocity distribution becomes more uniform explaining the reduction in the blockage amplification effect.
	www.cse.clrc.ac.uk /ceg/rgdsphere/rgdspherein.shtml (1582 words)

	Unified Flow Model
		It is known from Knudsen's (1909) and Gaede's (1913) experiments in the transition flow regime that there is a minimum in the flowrate in pipe and channel flows at about Kn= 3 and Kn=1, respectively.
		The flowrate data is presented at the average Knudsen number in the channels The Knudsen's minimum is clearly captured by the DSMC results at Kn=1.0.
		The new model predicts Knudsen's minimum for the channel flow, and use of appropriate value of alpha_1 for pipe flows is necessary to predict Knudsen's minimum.
	www.cfm.brown.edu /people/beskok/unifiedmodel.html (1931 words)

	Centre for Microfluidics - Rarefied Gas Dynamics
		In the present study, the Reynolds number was varied from Re=1 to Re=400 whilst the Knudsen number was varied from Kn=0 (continuum flow) to Kn=0.1 (a frequently adopted upper bound for the slip-flow regime).
		The Reynolds and Knudsen numbers were defined using the hydraulic diameter of the cross-section as the characteristic length scale.
		The form of the Knudsen number correction was chosen because the analytical expression for the fully-developed centreline velocity has a similar Knudsen number modification factor.
	www.cse.clrc.ac.uk /ceg/devlen/devlenin.shtml (1659 words)

	Learn more about Fluid mechanics in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		In order to determine whether to use conventional fluid mechanics or statistical mechanics, the Knudsen number is evaluated for the problem.
		The foundational axioms of fluid mechanics are the conservation laws, specifically, conservation of mass, conservation of momentum (also known as Newton's second law), and conservation of energy These are based on classical mechanics and are violated in relativistic mechanics.
		In order to determine whether to use compressible or incompressible fluid mechanics, the Mach number of the problem is evaluated.
	www.onlineencyclopedia.org /f/fl/fluid_mechanics_1.html (961 words)

	[No title]
		Knudsen, was this a totally unique situation, specifically the making of these prints and the number of sets that were made, or was this a kind of a standard procedure for sensitive photo- graphs?
		We have gone over quite a few of your recollec- tions, and we are going to show you, in a second, the color autopsy prints that we have and ask you whether the prints that you are shown are consistent with your recollections of them when you saw them.
		The primary points that we are going to cover are the number and locations of wounds and the other details in the photographs that you described generally, such as the presence of metal probes in the photographs and the presence of rules in the photographs, and what have you.
	mcadams.posc.mu.edu /knudsen_transcript.txt (9119 words)

	Dictionary of Technical Terms for Aerospace Use - K
		Various types of Knudsen gage are distinguished by the names of the inventors and differ mainly in the shape and method of suspension of the movable element.
		A number used to describe the flow of a low density gas, equal to the ratio λ/l where λ is the mean free path of the gas molecule and l is a characteristic length, such as boundary layer thickness, or apparatus dimension.
		The Knudsen number is used most commonly to define the extend to which the gas behaves like a collection of independent particles (free-molecule regime, Knudsen number much larger than unity), or like a viscous fluid (continuous regime, Knudsen number much smaller than unity).
	roland.lerc.nasa.gov /~dglover/dictionary/k.html (813 words)

	Abstract (Site not responding. Last check: 2007-10-21)
		Owing to the later the unified description of the kinetic, hydrodynamic and diffusion processes for all values of the Knudsen number becomes possible.
		The existence of second term allows to give the general definition of the heat flux for any values of the Knudsen number which is proportional to the gradient of entropy.
		This general definition for small Knudsen number and a constant pressure leads to the Fourier law.
	www.icmp.lviv.ua /journal/zbirnyk.04/003/abstract.html (427 words)

	KTH in your Pocket: Non-linear wave propagation in fluids
		Nonlinear acoustics usually includes terms to second order in the Mach number as well as the product of the Mach number and the Knudsen number (the Knudsen number is the mean free path divided by the wavelength).
		But when a shock is present, kinetic effects (second order in the Knudsen number) are expected to be as important as terms to third order in the Mach number.
		To meet this, a set of equations has been derived, which is correct to second order in the Knudsen number.
	www.kth.se /forskning/pocket/project.asp?id=3370 (177 words)

	Henning Struchtrup (Site not responding. Last check: 2007-10-21)
		The other main method for developing continuum models for larger Knudsen numbers is Grad's moment method, which extends the number of continuum variables, and derives balance equations for extended sets of variables from the Boltzmann equation..
		This method faces several problems, in particular a large number of variables is required to describe certain flows, and the equations introduce unphysical subshocks into shock structures.
		The main result is on the number of moments that one will need to describe some basic processes in non-equilibrium radiative transfer in order to match solutions of the radiative transfer equation.
	www.engr.uvic.ca /~struchtr/research2004.html (1737 words)

	3 Comments on the electron heat flux
		This conclusion is misleading, since, despite the smallness of the Knudsen number, the heat flux is strongly non classical.
		The simulations suggest that this is not easily achievable because the driving of the wind to supersonic velocities does precisely requires the plasma to be sufficiently collisional at the sonic point.
		Even though the qualitative behavior is similar it appears that a high mass ratio is in favor of a stronger acceleration of the wind.
	www.arcetri.astro.it /~slandi/wind/node7.html (1041 words)

	Research
		The degree of rarefaction of a gas is expressed through the Knudsen number, which is the ratio of the mean free path to the characteristic dimension.
		When the local Knudsen number exceeds 0.1, the traditional Navier-Stokes equations are not valid.
		The nuclei number density contours in the plume domain (2 x 5 m) are shown now.
	www.personal.psu.edu /users/j/x/jxz194/resch.html (534 words)

	Mazur Group Publication
		Instead of treating bulk and boundary-layer effects separately, both molecular and surface scattering are included from the beginning in the collisional part of the Boltzmann equation, and the surface is treated as one component of a multicomponent mixture.
		A simple first-order solution of this problem is sufficient to account for the dependence of the transport coefficients on the Knudsen number in the presence of a field.
		Detailed results for the longitudinal and transverse viscomagnetic effects in a single gas are presented, and shown to be in good agreement with experimental data for CO and N2.
	mazur-www.harvard.edu /publications.php?function=display&rowid=93 (142 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The non-zero Knudsen number flows are nonequilibrium in sense of Boltzmann.
		The deviation from the local Maxwellian equilibrium state is of the order of the Knudsen number Kn, and exerts an appreciable influence on the physical state of the system through inter-molecular collisions.
		Thus, to numerically solve the Boltzmann equation for non-zero Knudsen number flows, one must ensure theoretically that the numerical error is much smaller than the Knudsen number, i.e., it is of higher order in Knudsen number Kn.
	www.math.odu.edu /colloquium/ohwada-2.html (252 words)

	Quanhua Sun's Home Page (Site not responding. Last check: 2007-10-21)
		The small scale of aircrafts also indicates the flight of such vehicle will be in the high Knudsen number (Kn, the ratio of the mean free path of air molecules to the characteristic length of the vehicle) flow regime.
		As a result, a low Reynolds number flow is slowed down to a large extent and the boundary layer is thickened, while the leading and trailing edges affect a significant fraction of the plate length.
		Figure 4 shows that the density variation increases when the Reynolds number decreases, and the variation can be as high as 16% when the Reynolds number is around 1, which may also indicate that the isentropic assumption is invalid because the flow is rarefied.
	hpcc.engin.umich.edu /CFD/research/NGPD/MURI (1508 words)

	Piano: Knudsen, J. (Site not responding. Last check: 2007-10-21)
		The serial number can be found stamped somewhere on the frame or soundboard.
		It is usually four to six digits long, the numbers being about 2cm high.
		On Grand Pianos the number may be found on the key stop rail and / or the soundboard.
	www.pianos.co.uk /info/pianos/k/knudsen_j.php (133 words)

	Shear Driven Micro-Flows
		The flow is driven by moving the upper plate at a constant velocity U_o; the upper plate temperature is at T_o while the bottom plate is assumed to be adiabatic.
		The variation of friction coefficient as a function of Mach number and corresponding Knudsen number is shown below.
		In our simulations the Reynolds number is fixed to be Re=5.0, and the Knudsen number is increased by decreasing the channel gap.
	www.cfm.brown.edu /grad/beskok/shear.html (839 words)

	Henning Struchtrup's Abstracts (Site not responding. Last check: 2007-10-21)
		This paper presents a systematic approach to the calculation of heat transfer in rarefied gases (Knudsen numbers between 0.01 and 1) by means of Grad's moment method with high moment numbers.
		The projected moment formalism is extended to an arbitrary number of moments and moment equations and it is shown that the non-relativistic limit of moments and moments equations leads to the so-called central moments of non-relativistic theory.
		In particular we are interested in the number of moments one has to choose in order to have a satisfactory agreement between solutions of the moment equations and solutions of the radiative transfer equation.
	www.me.uvic.ca /~struchtr/abstract.html (6501 words)

	Welcome to the Micro-Flow Web Pages
		The very well known "no-slip" boundary conditions for velocity and tempearture of the fluid on the walls are subject to modifications in order to incorporate the reduction of momentum and energy exchange of the molecules with the surroundings.
		Deviation from continuum hypothesis is identified with Knudsen number, which is the ratio of the mean-free-path of the molecules to a characteristic length scale.
		As Knudsen number is larger than 0.1 the Navier-Stokes equations break-down and a higher level of approximation is obtained by carrying second order terms (in Kn) in the Chapman-Enskog expansion.
	www.cfm.brown.edu /people/beskok/muflow.html (543 words)

	[No title]
		We are now in House Annex No. 2 in Washington, D.C. and the date is August 11th, 1978 and the time is 9:35 a.m.
		Knudsen, pursuant to House Resolution 222 and Committee Rule 4, I am a designated counsel, empowered to take statements under oath, and you have been previously sworn here today.
		Robert L. Knudsen, 3712 Woodburn Road, Anabel, Virginia.
	karws.gso.uri.edu /Marsh/Scans/Knudsen.txt (1479 words)

	Mazur Group Publication
		A simple model is presented that describes the complex behavior of the magnetic field dependence of the thermal conductivity of a polyatomic gas as a function of the Knudsen number.
		The results provide an interpolation scheme between the continuum and Knudsen regimes, in the form of scaling rules in which the Knudesen number appears in the coefficients and arguments of the functions that describe the limiting cases.
		Good agreement is obtained with the available experimental data, and leads to the conclusion that different surface accommodation coefficients are needed for the translational and internal energies of the gas molecules.
	mazur-www.harvard.edu /publications.php?function=display&rowid=67 (151 words)

	Aerothermodynamic Characteristics in the Hypersonic Continuum-Rarefied Transitional Regime, AIAA 2001-2962, June 11-14, ...
		Abstract: Stagnation pressure and heating for cylinders and spheres representative of wing leading edges and nose sections for small-scale space transportation vehicles are presented for various Knudsen numbers at Earth entry conditions that bridge from the free molecular to continuum regimes.
		CFD, DSMC, and collisionless DSMC are used to span Knudsen numbers from 0.001 to 100 with the CFD/DSMC overlap at a Knudsen number of 0.01 and the DSMC/collisionless DSMC overlap at 10.
		The present results include density contours, density and temperature along the stagnation streamline for all cases, and stagnation pressure and heat transfer coefficient as a function of Knudsen number.
	techreports.larc.nasa.gov /ltrs/refer/2001/aiaa/NASA-aiaa-2001-2962.refer.html (133 words)

	Technical Program Paper Detail - AIChE
		In addition, due to the small dynamic viscosity of air, the particle velocities may remain correlated during the time that it takes the particles to move distances on the order of their radii.
		We consider non-continuum hydrodynamic interactions and particle inertia in order to predict the rate of coagulation of Brownian particles over the full range of values of the Knudsen number (ratio of mean-free path to particle radius, Kn) and Stokes number (ratio of persistence distance for particle motion to particle radius, St).
		For typical experimental conditions, the Stokes number will be large whenever the Knudsen number is large.
	www.aiche.org /conferences/techprogram/paperdetail.asp?PaperID=466&DSN=annual01 (294 words)

	[No title]
		HSCA DEPOSITION OF ROBERT KNUDSEN, 8/11/78 Robert Knudsen developed the fl and white autopsy photographs.
		Let me ask you about photograph number 44F.
		Like I say, this number seven -- I do not now, but it sticks in the back of my mind.
	karws.gso.uri.edu /Marsh/autopsy/knudsen1.txt (9082 words)

	Rarefied Gas Turbulence in the Knudsen Transition Regime (Site not responding. Last check: 2007-10-21)
		Future hypersonic vehicles may have a flight profile in which the Mach number can range from zero to about 25 and fly in conditions ranging from a conventional continuum at low altitudes to free-molecule flow at high altitudes.
		Because of the low to moderate flight Reynolds numbers at high altitude, the boundary layer may be sufficiently thick that it represents a substantial portion of the inlet flow.
		In this figure, dark regions (representing a high degree of rarefaction) are seen to coincide with regions in the flow where the Navier-Stokes equations are invalid (light grey surfaces).
	www.nearinc.com /near/project/rarefied.htm (439 words)

	Slip-flow heat transfer from cylinders in subsonic airstreams (Site not responding. Last check: 2007-10-21)
		Over 1000 measured convective heat-transfer coefficients for normal cylinders in subsonic slip flow have been correlated by using Nusselt number as a function of Reynolds and Knudsen (or Mach) numbers.
		The experimental range corresponds to the following dimensionless groups: Mach number, M, 0.05 to 0.80; Reynolds number Re, 1 to 75; Knudsen number Kn, 0.009 to 0.077.
		Air temperatures between 0 and 280 degrees F and cylinder temperatures between 34 and 620 degrees F were used.
	naca.larc.nasa.gov /reports/1958/naca-tn-4369 (103 words)

	PCSE (Site not responding. Last check: 2007-10-21)
		The underlying continuum assumption is valid when the characteristic dimensions of the flow system are large compared to molecular length scales, or, in the case of gas flows, when the ratio of mean the mean free path of the molecules and a characteristic dimension of the system is small.
		A measure for the latter is the Knudsen number Kn.
		The Knudsen number is small for all macroscopic systems at atmospheric or elevated pressures, but this is no longer the case at very low pressure conditions (such as in vacuum equipment) or in very small systems (such as MEMS and micro-reactors).
	pcse.tudelft.nl /index.php?page=Abstr_Kleijn (486 words)

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