UVW mapping - Revision history

en>Enoshd: Additional external link to UVW Mapping Basics

2014-06-23T07:27:37Z

Additional external link to UVW Mapping Basics

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en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

2014-02-24T03:26:05Z

inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

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	The ~~'''Cauchy number''' ('''Ca''')~~ is a ~~[[dimensionless number]] in [[fluid dynamics]] used in~~ the ~~study~~ of ~~[[compressible flow]]s~~. It is ~~named after the French mathematician~~ [~~[Augustin Louis Cauchy]~~]. ~~When the compressibility is important the elastic forces must be considered along with inertial forces for dynamic similarity~~. ~~Thus~~, the ~~Cauchy Number is defined as the ratio between inertial and the compressibility force (elastic force)~~ in ~~a flow and~~ can be ~~expressed~~ as		\ո
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	:<~~math~~> ~~\mathrm{Ca} = \frac{\rho v^2}{K}~~<~~/math~~>,

	~~where~~

	: <~~math~~>~~\rho~~<~~/math~~> ~~= density~~ of ~~fluid~~, ~~([[SI]] units: [[kilograms\|kg]]/[[meters\|m]]~~<~~sup~~>3<~~/sup~~>)
	: '~~'v'' = local fluid velocity, (SI units~~: m/~~[[seconds\|s~~]])
	: ''~~K'' = [~~[~~Bulk modulus\|bulk modulus of elasticity]], (SI units~~: ~~[[Pascal (unit)\|Pa~~]])

	~~== Relation between Cauchy number and Mach number ==~~

	~~For [[isentropic process]]es,~~ the ~~Cauchy number may be expressed in terms~~ of ~~[[Mach number]]~~. ~~The isentropic bulk modulus~~ <~~math~~> ~~K_s = \gamma p~~<~~/math~~>, ~~where~~ <~~math~~>~~\gamma~~<~~/math~~> is ~~the~~ [~~[Heat capacity ratio\|specific heat capacity ratio~~]~~] and ''p'' is the fluid pressure~~.
	If the ~~fluid obeys the [[ideal gas law]]~~, ~~we have~~

	~~:<math> K_s = \gamma p = \gamma \rho R T = \~~,~~\rho a^2~~<~~/math~~>,

	~~where~~

	: <~~math~~>~~a = \sqrt{\gamma RT}</math> = speed of sound~~, ~~(SI units: m/s)~~
	~~: ''R'' = [[gas constant\|characteristic gas constant]]~~, ~~(SI units: [[Joule (unit)\|J]]/(kg [[Kelvin (unit)\|K]]) )~~
	~~: ''T'' = temperature~~, ~~(SI units: K)~~

	~~Substituting ''K'' (''K~~<~~sub~~>s<~~/sub~~>~~'')~~ in ~~the equation for Ca yields~~

	:<~~math~~> ~~\mathrm{Ca} = \frac{v^2}{a^2} = \mathrm{M}^2~~<~~/math~~>.

	~~Thus~~, the ~~Cauchy number is square~~ of the ~~Mach number for~~ [~~[Isentropic process#Isentropic flow\|isentropic flow]] of a [[Ideal gas\|perfect gas]]~~.

	~~==References==~~
	*{{cite book \|first=B. S. \|last=Massey \|first2=J. \|last2=Ward-~~Smith \|title=Mechanics of Fluids \|edition=7th \|publisher=Nelson Thornes \|location=Cheltenham \|year=1998 \|isbn=0~~-~~7487~~-~~4043~~-~~0 }}~~

	~~{{NonDimFluMech}}~~

	~~[[Category:Dimensionless numbers of fluid mechanics]]~~
	~~[[Category:Fluid dynamics]~~]

en>Vaxquis: revving an obvious vandalism

2012-05-13T19:06:14Z

revving an obvious vandalism

New page

The '''Cauchy number''' ('''Ca''') is a [[dimensionless number]] in [[fluid dynamics]] used in the study of [[compressible flow]]s. It is named after the French mathematician [[Augustin Louis Cauchy]]. When the compressibility is important the elastic forces must be considered along with inertial forces for dynamic similarity. Thus, the Cauchy Number is defined as the ratio between inertial and the compressibility force (elastic force) in a flow and can be expressed as

:<math> \mathrm{Ca} = \frac{\rho v^2}{K}</math>,

where

: <math>\rho</math> = density of fluid, ([[SI]] units: [[kilograms|kg]]/[[meters|m]]<sup>3</sup>)
: ''v'' = local fluid velocity, (SI units: m/[[seconds|s]])
: ''K'' = [[Bulk modulus|bulk modulus of elasticity]], (SI units: [[Pascal (unit)|Pa]])

== Relation between Cauchy number and Mach number ==

For [[isentropic process]]es, the Cauchy number may be expressed in terms of [[Mach number]]. The isentropic bulk modulus <math> K_s = \gamma p</math>, where <math>\gamma</math> is the [[Heat capacity ratio|specific heat capacity ratio]] and ''p'' is the fluid pressure.
If the fluid obeys the [[ideal gas law]], we have

:<math> K_s = \gamma p = \gamma \rho R T = \,\rho a^2</math>,

where

: <math>a = \sqrt{\gamma RT}</math> = speed of sound, (SI units: m/s)
: ''R'' = [[gas constant|characteristic gas constant]], (SI units: [[Joule (unit)|J]]/(kg [[Kelvin (unit)|K]]) )
: ''T'' = temperature, (SI units: K)

Substituting ''K'' (''K<sub>s</sub>'') in the equation for Ca yields

:<math> \mathrm{Ca} = \frac{v^2}{a^2} = \mathrm{M}^2</math>.

Thus, the Cauchy number is square of the Mach number for [[Isentropic process#Isentropic flow|isentropic flow]] of a [[Ideal gas|perfect gas]].

==References==
*{{cite book |first=B. S. |last=Massey |first2=J. |last2=Ward-Smith |title=Mechanics of Fluids |edition=7th |publisher=Nelson Thornes |location=Cheltenham |year=1998 |isbn=0-7487-4043-0 }}

{{NonDimFluMech}}

[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Fluid dynamics]]