Cole equation of state

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The Cole equation of state [1][2] has the form

p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma  -1 \right]

In it, \rho_0 is a reference density around which the density varies, \gamma is the adiabatic index, and B 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 B is large, in the following sense. The fluctuations of the density are related to the speed of sound as

\frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,

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

c^2 = \frac{\gamma B}{\rho_0}.

Therefore, if B=100 \rho_0 v^2 / \gamma, 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:

p = B \gamma \left[
\frac{\rho-\rho_0}{\rho_0}
 \right]


References

  1. R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227
  2. G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962