Cole equation of state - Revision history

103.15.217.222: space inserted between 'the' and 'EOS' for friendly reading

2015-12-05T12:16:25Z

space inserted between 'the' and 'EOS' for friendly reading

← Older revision		Revision as of 14:16, 5 December 2015
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	:<math> dW= -p dV = dE .</math>		:<math> dW= -p dV = dE .</math>

	Taking differences on ~~theEOS~~,		Taking differences on the EOS,

	:<math> dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , </math>		:<math> dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , </math>

Dduque at 22:16, 6 March 2015

2015-03-06T22:16:45Z

← Older revision		Revision as of 00:16, 7 March 2015
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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]]. (See ''Derivation'', below.)		is the adiabatic version of the [[stiffened equation of state]] for liquids. (See ''Derivation'', below.)
	It has the form		It has the form

Dduque: Minor details

2015-03-06T22:16:23Z

Minor details

← Older revision		Revision as of 00:16, 7 March 2015
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	:<math> B = p^* / \gamma </math>		:<math> B = p^* / \gamma </math>

	Moreover, the Cole EOS is slightly ~~incorrect~~, as it should read (as indeed does in e.g. the book by Courant)		Moreover, the Cole EOS differs slightly, as it should read (as indeed does in e.g. the book by Courant)

	:<math>p = A \left( \frac{\rho}{\rho_0} \right)^\gamma - B ,</math>		:<math>p = A \left( \frac{\rho}{\rho_0} \right)^\gamma - B ,</math>
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	:<math> p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>		:<math> p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>


	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Dduque: Derivation complete. Formulas a bit ugly, will make them nicer soon

2015-03-06T22:11:34Z

Derivation complete. Formulas a bit ugly, will make them nicer soon

← Older revision		Revision as of 00:11, 7 March 2015
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	which can be written as		which can be written as

	:<math> p = ( p_0 + p^* / \gamma) (V_0/V)^\gamma - ~~p^* / \gamma~~ . </math>		:<math> p = p_0 + ( p_0 + p^* / \gamma) ( (V_0/V)^\gamma - 1 ) . </math>

	Going back to densities, instead of volumes,		Going back to densities, instead of volumes,

	:<math> p = ( p_0 + p^* / \gamma) (\rho/\rho_0)^\gamma - p^* / \gamma . </math>		:<math> p = p_0 + ( p_0 + p^* / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>

			Comparing with the Cole EOS, we can readily identify

			:<math> B = p^* / \gamma </math>

			Moreover, the Cole EOS is slightly incorrect, as it should read (as indeed does in e.g. the book by Courant)

			:<math>p = A \left( \frac{\rho}{\rho_0} \right)^\gamma - B ,</math>

			with

			:<math> A = p^* / \gamma + p_0 . </math>

			This difference is negligible for liquids but for an ideal gas <math>p^*=0</math> and there is a huge
			difference, ''B'' being zero and ''A'' being equal to the reference pressure.

	Now, the speed of sound is given by		Now, the speed of sound is given by

	:<math> c^2=\frac{dp}{d\rho} , </math>		:<math> c^2=\frac{dp}{d\rho} </math>

	with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain		with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain

			:<math> c^2= ( p_0 + p^* / \gamma) \gamma /\rho_0 . </math>

			From this expression a value of <math>p^</math> can be deduced. For water, <math>p^\approx 23000</math> bar,
			from which <math>B\approx 3000</math> bar. If the speed of sound is used in the EOS one obtains the rather
			elegant expression


			:<math> p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>


	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Dduque: /* Derivation */

2015-03-06T21:58:43Z

Derivation

@@ Line 43: / Line 43: @@
 first law reads
-:<math>  dW= -p dV  = dE</math>
+:<math>  dW= -p dV  = dE .</math>
-...
+Taking differences on theEOS,
 ==References==
 <references/>
 [[category: equations of state]]

Dduque: Derivation --- work in progress

2015-03-06T13:10:27Z

Derivation --- work in progress

← Older revision		Revision as of 15:10, 6 March 2015
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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]]

Carl McBride: Slight tidy: replaced underscores with spaces.

2015-03-06T11:42:51Z

Slight tidy: replaced underscores with spaces.

← Older revision		Revision as of 13:42, 6 March 2015
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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]].
	It has the form		It has the form

Dduque: added link to the stiffened EOS

2015-03-05T10:55:55Z

added link to the stiffened EOS

← Older revision		Revision as of 12:55, 5 March 2015
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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>
	has the form		is the adiabatic version of the [[stiffened equation of state]].
			It has the form

	:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>		:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>

Dduque: Added ref and a link to Tait EOS

2015-03-05T10:55:04Z

Added ref and a link to Tait EOS

← Older revision		Revision as of 12:55, 5 March 2015
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	The '''Cole equation of state''' <ref>R. H. Cole "Underwater ~~Explosions~~", Princeton University Press (1948) ~~ISBN 9780691069227~~</ref><ref>		The '''Cole equation of state'''
	G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962</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>
	has the form		has the form

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	\frac{\rho-\rho_0}{\rho_0}		\frac{\rho-\rho_0}{\rho_0}
	\right]</math>		\right]</math>


			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).

Dduque: minor syntax

2012-10-17T11:54:08Z

minor syntax

← Older revision		Revision as of 13:54, 17 October 2012
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	:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>		:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>

	In it, <math>\rho_0</math> is a reference density around which the density varies		In it, <math>\rho_0</math> is a reference density around which the density varies,
	<math>\gamma</math> is the [[Heat capacity#Adiabatic index \| adiabatic index]] and <math>B</math> is a pressure parameter.		<math>\gamma</math> is the [[Heat capacity#Adiabatic index \| adiabatic index]], and <math>B</math> is a pressure parameter.

	Usually, the equation is used to model a nearly incompressible system. In this case,		Usually, the equation is used to model a nearly incompressible system. In this case,
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	Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations		Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations
	will be of about 0.01.		will be about 0.01.

	If the fluctuations in the density are indeed small, the		If the fluctuations in the density are indeed small, the