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| The '''Stiffened equation of state''' is a simplified form of the Grüneisen equation of state <ref>[http://catalog.lanl.gov/F/YJK296TXG13BGDB17QSADFQLGAAKTSYRFYFSA24M48868X7V31-25707?func=item-global&doc_library=LNL01&doc_number=000518513&year=&volume=&sub_library=LANL Francis H. Harlow and Anthony A. Amsden "Fluid Dynamics", Los Alamos Report Number LA-4700 page 3 (1971)]</ref>.
| | ===Stiffened equation of state=== |
| When considering water under very high pressures (typical applications are underwater explosions, extracorporeal shock wave lithotripsy, and sonoluminescence) the stiffened [[Equations of state|equation of state]] is often used:
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| :<math> p=\rho(\gamma-1)e- p^* </math>
| | When considering water under very high pressures (typical applications are underwater explosions, extracorporeal shock wave lithotripsy, and sonoluminescence) the stiffened equation of state is often used: |
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| where <math>e</math> is the internal energy per unit mass, given by (Eq. 15 in <ref>[http://dx.doi.org/10.1016/S0045-7930(02)00021-X H. Paillère, C. Corre, and J. R. Garcı́a Cascales "On the extension of the AUSM+ scheme to compressible two-fluid models", Computers & Fluids '''32''' pp. 891-916 (2003)]</ref>):
| | :<math> p=\rho(\gamma-1)e-\gamma p^0 \,</math> |
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| :<math> e = \frac{C_p}{\gamma}T + \frac{p^*}{\gamma \rho} </math>
| | where <math>e</math> is the internal energy per unit mass, <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^0</math> is another constant, representing the molecular attraction between water molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres). |
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| where <math>C_p</math> is the [[heat capacity]] at constant [[pressure]]. <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^*</math> is another constant, representing the molecular attraction between [[water]] molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres).
| | The equation is stated in this form because the speed of sound in water is given by <math>c^2=\gamma(p+p^0)/\rho</math>. |
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| It can be shown, by linearizing the [[Euler equations]], that the [[speed of sound]] in water is given by
| | Thus water behaves as though it is an ideal gas that is ''already'' under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa). |
| :<math>c^2=\frac{\gamma p+p^* }{\rho_0}</math>, | |
| from which the value of <math>p^*</math> may be computed given all the other variables. | |
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| Thus water behaves as though it is an [[ideal gas]] that is ''already'' under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).
| | This equation mispredicts the [[specific heat capacity]] of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks. |
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| This equation mispredicts the heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks. | |
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| It is useful to notice that, given this equation of state, the [[adiabatic]] law is modified from its ideal form:
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| :<math> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma , </math>
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| which, rearranged, is known as the [[Cole equation of state]].
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| ==References==
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| <references/>
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| [[Category:equations of state]] | | [[Category:equations of state]] |