Replacement of the Euler Fluid and Navier-Stokes Equations

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Scientific Paper
Title Replacement of the Euler Fluid and Navier-Stokes Equations
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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

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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.