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2013-05-28T23:29:36Z

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[[File:Thin fluid film with particles flowing down an inclined plane.jpg|right|frame|A thin layer of liquid mixed with particles flowing down an inclined plane.]]

In [[fluid dynamics]], '''lubrication theory''' describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above [[air hockey]] tables, where the thickness of the air layer beneath the puck is much smaller than the dimensions of the puck itself.

Internal flows are those where the fluid is fully bounded. Internal flow lubrication theory has many industrial applications because of its role in the design of [[fluid bearing]]s. Here a key goal of lubrication theory is to determine the pressure distribution in the fluid volume, and hence the forces on the bearing components. The working fluid in this case is often termed a [[lubricant]].

Free film lubrication theory is concerned with the case in which one of the surfaces containing the fluid is a free surface. In that case the position of the free surface is itself unknown, and one goal of lubrication theory is then to determine this. [[Surface tension]] may then be significant, or even dominant. Issues of [[wetting]] and [[dewetting]] then arise. For very thin films (thickness less than one [[micrometre]]), additional intermolecular forces, such as [[Van der Waals force]]s or [[disjoining force]]s, may become significant.

== Theoretical basis ==

Mathematically, lubrication theory can be seen as exploiting the disparity between two length scales. The first is the characteristic film thickness, <math>H</math>, and the second is a characteristic substrate length scale <math>L</math>. The key requirement for lubrication theory is that the ratio <math>\epsilon = H/L</math> is small, that is, <math>\epsilon \ll 1</math>.
The [[Navier-Stokes equations]] (or [[Stokes flow|Stokes equations]], when fluid inertia may be neglected) are expanded in this small parameter, and the [[leading-order]] equations are then
:<math>\frac{\partial p}{\partial z} = 0</math>
:<math>\frac{\partial p}{\partial x} = \mu\frac{\partial^2 u}{\partial z^2}</math>

where <math>x</math> and <math>z</math> are coordinates in the direction of the substrate and perpendicular to it respectively. Here <math>p</math> is the fluid pressure, and <math>u</math> is the fluid velocity component parallel to the substrate; <math>\mu</math> is the fluid viscosity. The equations show, for example, that pressure variations across the gap are small, and that those along the gap are proportional to the fluid viscosity. A more general formulation of the lubrication approximation would include a third dimension, and the resulting differential equation is known as the [[Reynolds equation]]. 

Further details can be found in the literature<ref name="odb97">Oron, A; Davis S. H., and S. G. Bankoff, "[http://prola.aps.org/abstract/RMP/v69/i3/p931_1 Long-scale evolution of thin liquid films]", Rev. Mod. Phys. 69, 931 - 980 (1997) </ref> or in the textbooks given in the bibliography.

==Applications==
An important application area is [[lubrication]] of machinery components such as [[fluid bearing]]s and [[seal (mechanical)|mechanical seals]]. [[Coating]] is another major application area including the preparation of [[thin films]], [[printing]], [[painting]] and [[adhesive]]s.

Biological applications have included studies of [[red blood cells]] in narrow capillaries and of liquid flow in the lung and eye.

==Notes==
{{reflist}}

== References ==
*Batchelor, G. K. (1976), An introduction to fluid mechanics, Cambridge: CUP. ISBN 978-0-521-09817-5.
*Panton, R. L. (2005), Incompressible Flow (3rd ed.), New York: Wiley . ISBN 978-0-471-26122-3.
*San Andres, L., MEEN334 Mechanical Systems Course Notes [http://rotorlab.tamu.edu/me626].
[[Category:Fluid dynamics]]

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2012-12-27T21:26:46Z

83.255.56.75:

{{lowercase}}
:''For the acceleration-related quantity in mechanics, see [[g-force|''g''-force]].''

A '''''g''-factor''' (also called '''''g'' value''' or '''dimensionless magnetic moment''') is a dimensionless quantity which characterizes the [[magnetic moment]] and [[gyromagnetic ratio]] of a particle or [[atomic nucleus|nucleus]]. It is essentially a proportionality constant that relates the observed magnetic moment μ of a particle to the appropriate [[angular momentum]] [[quantum number]] and the appropriate fundamental quantum unit of magnetism, usually the [[Bohr magneton]] or [[nuclear magneton]].

==Calculation==

=== Electron ''g''-factors ===
There are three magnetic moments associated with an electron: One from its [[Spin (physics)|spin angular momentum]], one from its [[azimuthal quantum number|orbital angular momentum]], and one from its total angular momentum (the quantum-mechanical sum of those two components). Corresponding to these three moments are three different ''g''-factors:

==== Electron spin ''g''-factor ====
The most famous of these is the ''electron spin g-factor'' (more often called simply the ''electron g-factor''), ''ge'', defined by

:<math> \boldsymbol{\mu}_S = \frac{g_e\mu_\mathrm{B}}{\hbar}\boldsymbol{S}</math>

where '''''μS''''' is the total magnetic moment resulting from the spin of an electron, '''''S''''' is its [[spin (physics)|spin]] angular momentum, and ''μ''B is the [[Bohr magneton]]. In atomic physics, the electron spin ''g''-factor is often defined as the ''absolute value'' or ''negative'' of ''ge'':

:<math>g_S = |g_e| = -g_e.</math>

The ''z''-component of the magnetic moment then becomes

:<math> \mu_z=-g_S \mu_\mathrm{B} m_s</math>

The value ''gS'' is roughly equal to 2.002319, and is known to extraordinary precision.<ref>{{cite journal|url=http://cerncourier.com/main/article/46/8/20 |title=Precision pins down the electron's magnetism|journal=CERN Courier|volume=46|issue=8|month=October|pages=35–37|first1=Gerald|last1= Gabrielse |first2= David|last2= Hanneke|year=2006}}</ref><ref>
{{cite journal
| title = New measurement of the electron magnetic moment using a one-electron quantum cyclotron
| year = 2006
| journal = [[Physical Review Letters]]
| volume = 97
| issue = 3
| pages = 030801
| doi = 10.1103/PhysRevLett.97.030801
| pmid=16907490
| bibcode=2006PhRvL..97c0801O
| last1 = Odom
| first1 = B.
| last2 = Hanneke
| first2 = D.
| last3 = d’Urso
| first3 = B.
| last4 = Gabrielse
| first4 = G.
}}</ref> The reason it is not ''precisely'' two is explained by [[quantum electrodynamics]] calculation of the [[anomalous magnetic dipole moment]].<ref>
{{cite journal
| title = A nonperturbative calculation of the electron's magnetic moment
| year = 2004
| journal = [[Nuclear Physics B]]
| volume = 703
| issue = 1–2
| pages = 333–362
| doi = 10.1016/j.nuclphysb.2004.10.027
|arxiv = hep-ph/0406325 |bibcode = 2004NuPhB.703..333B
| last1 = Brodsky
| first1 = S
| last2 = Franke
| first2 = V
| last3 = Hiller
| first3 = J
| last4 = McCartor
| first4 = G
| last5 = Paston
| first5 = S
| last6 = Prokhvatilov
| first6 = E }}</ref>

==== Electron orbital ''g''-factor ====
Secondly, the ''electron orbital g-factor'', ''gL'', is defined by

:<math> \boldsymbol{\mu}_L = -\frac{g_L \mu_\mathrm{B}}{\hbar}\boldsymbol{L}</math>

where '''''μL''''' is the total magnetic moment resulting from the orbital angular momentum of an electron, '''''L''''' is the magnitude of its orbital angular momentum, and ''μ''B is the [[Bohr magneton]]. The value of ''gL'' is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the [[magnetogyric ratio|classical magnetogyric ratio]]. For an electron in an orbital with a [[magnetic quantum number]] ml, the ''z''-component of the orbital angular momentum is

:<math>\mu_z=g_L \mu_\mathrm{B} m_l</math>

which, since ''gL'' = 1, is just ''μ''Bml

====Total angular momentum (Landé) ''g''-factor ====
Thirdly, the ''[[Landé g-factor]]'', ''gJ'', is defined by
:<math> \boldsymbol{\mu} = -\frac{g_J \mu_\mathrm{B} }{\hbar}\boldsymbol{J}</math>

where '''''μ''''' is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, '''''J''''' = '''''L'''''+'''''S''''' is its total angular momentum, and ''μ''B is the [[Bohr magneton]]. The value of ''gJ'' is related to ''gL'' and ''gS'' by a quantum-mechanical argument; see the article [[Landé g-factor]].

=== Nucleon and nucleus ''g''-factors ===
Protons, neutrons, and many nuclei have spin and magnetic moments, and therefore associated ''g''-factors. The formula conventionally used is
:<math> \boldsymbol{\mu} = \frac{g \mu_\mathrm{N}}{\hbar}\boldsymbol{I}</math>

where '''''μ''''' is the magnetic moment resulting from the nuclear spin, '''''I''''' is the nuclear spin angular momentum, ''μ''N is the [[nuclear magneton]], and ''g'' is the effective ''g''-factor.

=== Muon ''g''-factor ===
[[Image:The muon g-2.svg|frame|If [[supersymmetry]] is realized in nature, there will be corrections to g-2 of the muon due to loop diagrams involving the new particles. Amongst the leading corrections are those depicted here: a [[neutralino]] and a [[smuon]] loop, and a [[chargino]] and a muon [[sneutrino]] loop. This represents an example of "beyond the Standard-Model" physics that might contribute to g-2.]]
The muon, like the electron has a ''g''-factor from its spin, given by the equation
:<math>\mathbf{\mu} = \frac{ge}{2m_\mu}\mathbf{S}</math>

where '''''μ''''' is the magnetic moment resulting from the muon’s spin, '''''S''''' is the spin angular momentum, and ''mμ'' is the muon mass.

The fact that the muon g-factor is not quite the same as the electron g-factor is mostly explained by quantum electrodynamics and its calculation of the [[anomalous magnetic dipole moment]]. Almost all of the small difference between the two values (99.96% of it) is due to a well-understood lack of a heavy-particle diagrams contributing to the probability for emission of a photon representing the magnetic dipole field, which are present for muons, but not electrons, in QED theory. These are entirely a result of the mass difference between the particles.

However, not all of the difference between the g-factors for electrons and muons are exactly explained by the quantum electrodynamics [[Standard Model]]. The muon ''g''-factor can, at least in theory, be affected by physics [[beyond the Standard Model]], so it has been measured very precisely, in particular at the [[Brookhaven National Laboratory]]. As of November 2006, the experimentally measured value is {{val|2.0023318416|(13)}}, compared to the theoretical prediction of {{val|2.0023318361|(10)}}.<ref>
{{cite journal
| last1 = Hagiwara
| first1 = K. |last2= Martin,|first2= A. D. |last3= Nomura|first3= Daisuke |last4= Teubner|first4= T.
| title = Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)
| year = 2006
| doi = 10.1016/j.physletb.2007.04.012
| journal = Physics Letters B
| volume = 649
| issue = 2–3
| pages = 173–179
|arxiv = hep-ph/0611102 | bibcode=2007PhLB..649..173H
}}</ref> This is a difference of 3.4 [[standard deviation]]s, suggesting beyond-the-Standard-Model physics may be having an effect. The Brookhaven muon storage ring is being transported to Fermilab where the g-2 experiment will use it to make more precise measurements of muon g-factor.<ref>http://muon-g-2.fnal.gov/</ref>

== Measured ''g''-factor values ==
{| class="wikitable" style="text-align:center"
|-
! Particle !! ''g''-factor !! Uncertainty
|-
! Electron <math>g_\mathrm{e}</math>
| {{gaps|−2.002|319|304|361|53}} || {{gaps|0.000|000|000|000|53}}
|-
! Neutron <math>g_\mathrm{n}</math>
| {{gaps|−3.826|085|45}} || {{gaps|0.000|000|90}}
|-
! Proton <math>g_\mathrm{p}</math>
| {{gaps|5.585|694|713}} || {{gaps|0.000|000|046}}
|-
! Muon <math>g_{\mu}</math>
| {{gaps|−2.002|331|8414}} || {{gaps|0.000|000|0012}}
|-
|+Currently accepted [[NIST]] ''g''-factor values <ref>{{cite web|work=NIST|title=CODATA values of the fundamental constants|url=http://physics.nist.gov/cgi-bin/cuu/Category?view=html&All+values.x=80&All+values.y=11}}</ref>
|}

The electron ''g''-factor is one of the most precisely measured values in physics, with a relative standard uncertainty of 2.6 x 10−13.

==Notes and references==
{{reflist}}

==See also==
*[[anomalous magnetic dipole moment]]
*[[Electron magnetic dipole moment]]
*[http://physics.nist.gov/cuu/Constants/codata.pdf CODATA recommendations 2006]

[[Category:Atomic physics]]
[[Category:Nuclear physics]]
[[Category:Particle physics]]
[[Category:Physical constants]]

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