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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
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| :<math> \mathrm{Ca} = \frac{\rho v^2}{K}</math>,
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| where
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| : <math>\rho</math> = density of fluid, ([[SI]] units: [[kilograms|kg]]/[[meters|m]]<sup>3</sup>)
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| : ''v'' = local fluid velocity, (SI units: m/[[seconds|s]])
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| : ''K'' = [[Bulk modulus|bulk modulus of elasticity]], (SI units: [[Pascal (unit)|Pa]])
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| == Relation between Cauchy number and Mach number ==
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| 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.
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| If the fluid obeys the [[ideal gas law]], we have
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| :<math> K_s = \gamma p = \gamma \rho R T = \,\rho a^2</math>,
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| where
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| : <math>a = \sqrt{\gamma RT}</math> = speed of sound, (SI units: m/s)
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| : ''R'' = [[gas constant|characteristic gas constant]], (SI units: [[Joule (unit)|J]]/(kg [[Kelvin (unit)|K]]) )
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| : ''T'' = temperature, (SI units: K)
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| Substituting ''K'' (''K<sub>s</sub>'') in the equation for Ca yields
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| :<math> \mathrm{Ca} = \frac{v^2}{a^2} = \mathrm{M}^2</math>.
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| Thus, the Cauchy number is square of the Mach number for [[Isentropic process#Isentropic flow|isentropic flow]] of a [[Ideal gas|perfect gas]].
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| ==References==
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| *{{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 }}
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| {{NonDimFluMech}}
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| [[Category:Dimensionless numbers of fluid mechanics]]
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| [[Category:Fluid dynamics]]
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My name's Leonora Golding but everybody calls me Leonora. I'm from Austria. I'm studying at the high school (2nd year) and I play the Cello for 8 years. Usually I choose songs from my famous films :D.
I have two brothers. I love Leaf collecting and pressing, watching TV (Arrested Development) and Radio-Controlled Car Racing.
my web page wordpress backup plugin