Difference between revisions of "Replacement of the Euler Fluid and Navier-Stokes Equations"
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The Euler fluid equations, which can be expressed as the vector equation (<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">)</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">g</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">=</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">(</span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">)</span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">t</span></span>'', are shown to be missing three important terms and to contain a simplistic version of a fourth. Whenever the fluid is significantly affected by an external thrust and by either a gravitational field or a set of gravitational fields, the Euler fluid equations must be replaced by <span style="FONT-FAMILY: SymbolPS">( ) ( ) </span><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS">( </span></span><span style="FONT-SIZE: xx-small"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Times New Roman">2 </span></span><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS">) </span></span><span style="FONT-FAMILY: SymbolPS">( ) ( ) </span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e e e e e </span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p </span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+??</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">2 </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">f</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">3 </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">3</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">??</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">R</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">- </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">M G r R</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">F</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">= </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">(</span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">t</span></span>'', where two of the three missing terms are combined into the first term times the unit vector <span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">R </span></span>''(which points radially away from the effective center of gravity), ? ''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">f </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">? </span></span>is numerically the degrees of freedom of the fluid, and ? <span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">r </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span> is the distance from the effective center of gravity to the differential volume. The third missing term, F, is the non-gravitational force. The simplistic gravitational acceleration term, g, is replaced by a more general expression that takes into account the ubiquitous nature of gravity. The Euler fluid equations are used in the derivation of the Navier-Stokes equations, so the foregoing developments cause these equations to change. | The Euler fluid equations, which can be expressed as the vector equation (<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">)</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">g</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">=</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">(</span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">)</span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">t</span></span>'', are shown to be missing three important terms and to contain a simplistic version of a fourth. Whenever the fluid is significantly affected by an external thrust and by either a gravitational field or a set of gravitational fields, the Euler fluid equations must be replaced by <span style="FONT-FAMILY: SymbolPS">( ) ( ) </span><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS">( </span></span><span style="FONT-SIZE: xx-small"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Times New Roman">2 </span></span><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: large; FONT-FAMILY: SymbolPS">) </span></span><span style="FONT-FAMILY: SymbolPS">( ) ( ) </span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e e e e e </span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p </span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+??</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">2 </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">f</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">3 </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">p</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">3</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">??</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">R</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">- </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">M G r R</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">-</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">F</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: Symbol,Italic">r</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">= </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">+</span></span><span style="FONT-SIZE: small; FONT-FAMILY: SymbolPS">(</span><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">v </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS"><span style="FONT-SIZE: xx-small; FONT-FAMILY: SymbolPS">?</span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">t</span></span>'', where two of the three missing terms are combined into the first term times the unit vector <span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">R </span></span>''(which points radially away from the effective center of gravity), ? ''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">f </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">? </span></span>is numerically the degrees of freedom of the fluid, and ? <span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">e </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman,Italic">r </span></span>''<span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman"><span style="FONT-SIZE: xx-small; FONT-FAMILY: TimesNewRoman">?</span></span> is the distance from the effective center of gravity to the differential volume. The third missing term, F, is the non-gravitational force. The simplistic gravitational acceleration term, g, is replaced by a more general expression that takes into account the ubiquitous nature of gravity. The Euler fluid equations are used in the derivation of the Navier-Stokes equations, so the foregoing developments cause these equations to change. | ||
− | [[Category:Scientific Paper]] | + | [[Category:Scientific Paper|replacement euler fluid navier-stokes equations]] |
[[Category:Gravity]] | [[Category:Gravity]] |
Revision as of 11:01, 1 January 2017
Scientific Paper | |
---|---|
Title | Replacement of the Euler Fluid and Navier-Stokes Equations |
Read in full | Link to paper |
Author(s) | Donald G Carpenter |
Keywords | Euler, Navier-Stokes, fluid equations, fluid motion, atmosphere, atmospheric particle motion, barometric, barometric equation, degrees of freedom, hydrostatic, Newton |
Published | 2000 |
Journal | Apeiron |
Volume | 7 |
Number | 3-4 |
No. of pages | 7 |
Pages | 148-154 |
Read the full paper here
Abstract
The Euler fluid equations, which can be expressed as the vector equation (-?pr)-g=(v??)v+?v ?t, are shown to be missing three important terms and to contain a simplistic version of a fourth. Whenever the fluid is significantly affected by an external thrust and by either a gravitational field or a set of gravitational fields, the Euler fluid equations must be replaced by ( ) ( ) ( 2 ) ( ) ( ) e e e e e -?p r +??2 f-3 p3rr??R- M G r R-Fr= v??v+(?v ?t, where two of the three missing terms are combined into the first term times the unit vector e R (which points radially away from the effective center of gravity), ? f ? is numerically the degrees of freedom of the fluid, and ? e r ? is the distance from the effective center of gravity to the differential volume. The third missing term, F, is the non-gravitational force. The simplistic gravitational acceleration term, g, is replaced by a more general expression that takes into account the ubiquitous nature of gravity. The Euler fluid equations are used in the derivation of the Navier-Stokes equations, so the foregoing developments cause these equations to change.