# Difference between revisions of "Cole equation of state"

Carl McBride (talk | contribs) m (Slight tidy: replaced underscores with spaces.) |
(Derivation --- work in progress) |
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The '''Cole equation of state''' | The '''Cole equation of state''' | ||

<ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref> | <ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref> | ||

− | is the adiabatic version of the [[stiffened equation of state]]. | + | is the adiabatic version of the [[stiffened equation of state]]. (See ''Derivation'', below.) |

It has the form | It has the form | ||

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It is quite common that the name "[[Tait equation of state]]" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39). | It is quite common that the name "[[Tait equation of state]]" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39). | ||

+ | ==Derivation== | ||

+ | |||

+ | Let us write the stiffened EOS as | ||

+ | |||

+ | :<math>p+ p^* = (\gamma -1) e \rho = (\gamma -1) E / V ,</math> | ||

+ | |||

+ | where ''E'' is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the | ||

+ | first law reads | ||

+ | |||

+ | :<math> dW= -p dV = dE</math> | ||

+ | |||

+ | ... | ||

==References== | ==References== | ||

<references/> | <references/> | ||

[[category: equations of state]] | [[category: equations of state]] |

## Revision as of 15:10, 6 March 2015

The **Cole equation of state**
^{[1]}^{[2]}^{[3]}
is the adiabatic version of the stiffened equation of state. (See *Derivation*, below.)
It has the form

In it, is a reference density around which the density varies, is the adiabatic index, and is a pressure parameter.

Usually, the equation is used to model a nearly incompressible system. In this case, the exponent is often set to a value of 7, and is large, in the following sense. The fluctuations of the density are related to the speed of sound as

where is the largest velocity, and is the speed of sound (the ratio is Mach's number). The speed of sound can be seen to be

Therefore, if , the relative density fluctuations will be about 0.01.

If the fluctuations in the density are indeed small, the equation of state may be approximated by the simpler:

It is quite common that the name "Tait equation of state" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39).

## Derivation

Let us write the stiffened EOS as

where *E* is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the
first law reads

...

## References

- ↑ Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)
- ↑ G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962
- ↑ Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)