Deltahedron: /* Zeros of the Dirichlet L-functions */ notation

2014-08-19T21:16:37Z

Zeros of the Dirichlet L-functions: notation

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	~~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 (ratio)\|scale]]~~. ~~This length scale could be, for example, the [[radius]] of the body in a fluid~~. ~~The number is named after [[Denmark\|Danish]] physicist [[Martin Knudsen]] (1871–1949).~~

	~~==Definition==~~
	~~The Knudsen number is a dimensionless number defined as:~~

	~~:<math>\mathrm{Kn} = \frac {\lambda}{L}</math>~~

	~~where~~

	* <math>\lambda</math> = [[mean free path]] [L<~~sup~~>1<~~/sup>]~~
	* <math>L</math> = representative physical length scale [L<sup>1</sup>].

	~~For a [[Boltzmann]] gas, the [[mean free path]] may be readily calculated so that:~~

	~~:<math>\mathrm{Kn} = \frac {k_B T}{\sqrt{2}\pi\sigma^2 p L}</math>~~

	~~where~~
	* <math>k_B</math> is the [[Boltzmann constant]] (1.3806504(24) × 10<sup>−23</sup> J/K in [[SI]] units), [M<sup>1</sup> L<sup>2</sup> T<sup>-~~2</sup> θ<sup>~~-~~1</sup>]~~
	* <math>T</math> is the [[thermodynamic temperature]], [θ<sup>1</sup>]
	* <math>\sigma</math> is the particle hard shell diameter, [L<sup>1</sup>]
	* <math>p</math> is the total pressure, [M<sup>1</sup> L<sup>-1</sup> T<sup>-2</sup>].

	~~For particle dynamics in the [[atmosphere]], and assuming [[standard temperature and pressure]], i.e. 25 °C and 1 atm, we have <math>\lambda</math> ≈ 8 × 10<sup>−8</sup> m.~~

	~~==Relationship to Mach and Reynolds numbers in gases==~~
	~~The Knudsen number can be related to the [[Mach number]] and the [[Reynolds number]]:~~

	~~Noting the following:~~

	~~[[Dynamic viscosity]],~~
	~~:<math>\mu =\frac{1}{2}\rho \bar{c} \lambda.</math>~~

	~~Average molecule speed~~ (~~from [~~[~~Maxwell-Boltzmann distribution]]),~~
	~~:<math>\bar{c} = \sqrt{\frac{8 k_BT}{\pi m}}</math>~~

	~~thus the mean free path,~~
	~~:<math>\lambda =\frac{\mu }{\rho }\sqrt{\frac{\pi m}{2 k_BT}}</math>~~

	~~dividing through by ''L'' (some characteristic length) the Knudsen number is obtained~~:
	~~:<math>\frac{\lambda }{L}=\frac{\mu }{\rho L}\sqrt{\frac{\pi m}{2 k_BT}}</math>~~

	~~where~~
	* ''<math>\bar{c}</math>'' is the average molecular speed from the [[Maxwell–Boltzmann distribution]], [L<sup>1</sup> T<sup>-1</sup>]
	* ''T'' is the [[thermodynamic temperature]], [θ<sup>1</sup>]
	* ''μ'' is the [[dynamic viscosity]], [M<sup>1</sup> L<sup>-1</sup> T<sup>-1</sup>]
	* ''m'' is the [[molecular mass]], [M<sup>1</sup>]
	* ''k<sub>B</sub>'' is the [[Boltzmann constant]], [M<sup>1</sup> L<sup>2</sup> T<sup>-2</sup> θ<sup>-1</sup>]
	* ''ρ'' is the density, [M<sup>1</~~sup> L<sup>-3<~~/~~sup>]~~.

	~~The dimensionless Mach number can be written:~~
	~~:<math>\mathrm{Ma} = \frac {U_\infty}{c_s}</math>~~

	~~where the speed of sound is given by~~
	~~:<math>c_s=\sqrt{\frac{\gamma R T}{M}}=\sqrt{\frac{\gamma k_BT}{m}}</math>~~

	~~where~~
	* ''U<sub>∞</sub>'' is the freestream speed, [L<sup>1</sup> T<sup>-1</sup>]
	* ''R'' is the Universal [[gas constant]], (in [[SI]], 8.~~314 47215 J K<sup>−1<~~/~~sup> mol<sup>−1<~~/~~sup>), [M<sup>1<~~/~~sup> L<sup>2</sup> T<sup>-2</sup> θ<sup>-1</sup> 'mol'<sup>-1</sup>]~~
	* ''M'' is the [[molar mass]], [M<sup>1</sup> 'mol'<sup>-~~1</sup>]~~
	* <math>\gamma</math> is the [[ratio of specific heats]], and is dimensionless.

	~~The dimensionless [[Reynolds number]] can be written:~~
	~~:<math>\mathrm{Re} = \frac {\rho U_\infty L}{\mu}~~.~~</math>~~

	~~Dividing the Mach number by the Reynolds number,~~

	:<math>\frac{\mathrm{Ma}}{\mathrm{Re}}=\frac{U_\infty / c_s}{\rho U_\infty L / \mu }=\frac{\mu }{\rho L c_s}=\frac{\mu }{\rho L \sqrt{\frac{\gamma k_BT}{m}}}=\frac{\mu }{\rho L }\sqrt{\frac{m}{\gamma k_BT}}</math>

	~~and by multiplying by <math>\sqrt{\frac{\gamma \pi }{2}}</math>,~~

	:<math>\frac{\mu }{\rho L }\sqrt{\frac{m}{\gamma k_BT}}\sqrt{\frac{\gamma \pi }{2}}=\frac{\mu }{\rho L }\sqrt{\frac{\pi m}{2k_BT}}=\frac{1}{\nu L }\sqrt{\frac{\pi m}{2k_BT}} = \mathrm{Kn}</math>

	~~yields the Knudsen number~~.

	~~The Mach, Reynolds and Knudsen numbers are therefore related by:~~

	~~:<math>\mathrm{Kn} = \frac{\mathrm{Ma}}{\mathrm{Re}} \; \sqrt{ \frac{\gamma \pi}{2}}.</math>~~

	~~==Application==~~

	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. In this case statistical methods must be used.

	Problems with high Knudsen numbers include the calculation of the motion of a [[dust]] particle through the lower [[Earth's atmosphere\|atmosphere]], or the motion of a [[satellite]] through the [[exosphere]]. ~~One of the most widely used applications for the Knudsen number is in [[microfluidics]] and [[MEMS]] device design~~. ~~The solution of the flow around an [[aircraft]] has a low Knudsen number, making it firmly in the realm of continuum mechanics~~. ~~Using the Knudsen number an adjustment for [[Stokes' Law]~~] ~~can be used in the [[Cunningham correction factor]], this is a drag force correction due to slip in small particles (i.e. ''d''<sub>''p''</sub> < 5 µm~~).

	~~==See also==~~
	* [[Cunningham correction factor]]
	* [[Fluid dynamics]]
	* [[Mach number]]
	* [[Knudsen Flow]]
	* [[Knudsen diffusion]]

	~~== References ==~~
	* {{cite book\|last=Cussler\|first=E. L.\|title=Diffusion: Mass Transfer in Fluid Systems\|publisher=Cambridge University Press\|year=1997\|isbn=0-521-45078-0}}

	~~{{NonDimFluMech}}~~

	~~[[Category:Dimensionless numbers]]~~
	~~[[Category:Fluid dynamics]]~~

en>Spectral sequence: /* Zeros of the Dirichlet L-functions */ example zero-free region, cite Montgomery

2013-06-17T20:33:17Z

Zeros of the Dirichlet L-functions: example zero-free region, cite Montgomery

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Dirichlet L-function - Revision history

Deltahedron: /* Zeros of the Dirichlet L-functions */ notation

en>Spectral sequence: /* Zeros of the Dirichlet L-functions */ example zero-free region, cite Montgomery

76.120.181.41: edit