Nonstandard Analysis and Shock Wave Jump Conditions in a OneDimensional Compressible Gas
Abstract
Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. This report presents a fairly comprehensive tutorial on nonstandard analysis for physicists and engineers with many examples applicable to generalized functions. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a onedimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservations laws, smooth predistributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, smooth predistributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in nonconservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical RankineHugoniot jump conditions for an inviscid shock wave. Moreover, nonmonotonic entropy jump conditions are obtained for both inviscid and viscous flows. The report shows that products of generalized functions may be defined consistently usingmore »
 Authors:
 Publication Date:
 Research Org.:
 Los Alamos National Laboratory (LANL), Los Alamos, NM
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 913104
 Report Number(s):
 LA14334
TRN: US200802%%351
 DOE Contract Number:
 DEAC5206NA25396
 Resource Type:
 Technical Report
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING; ENTROPY; EQUATIONS OF MOTION; SHOCK WAVES; THERMODYNAMICS; THICKNESS; VISCOUS FLOW; COMPRESSED GASES; ONEDIMENSIONAL CALCULATIONS; FLUID MECHANICS
Citation Formats
Roy S. Baty, F. Farassat, John A. Hargreaves. Nonstandard Analysis and Shock Wave Jump Conditions in a OneDimensional Compressible Gas. United States: N. p., 2007.
Web. doi:10.2172/913104.
Roy S. Baty, F. Farassat, John A. Hargreaves. Nonstandard Analysis and Shock Wave Jump Conditions in a OneDimensional Compressible Gas. United States. doi:10.2172/913104.
Roy S. Baty, F. Farassat, John A. Hargreaves. Fri .
"Nonstandard Analysis and Shock Wave Jump Conditions in a OneDimensional Compressible Gas". United States.
doi:10.2172/913104. https://www.osti.gov/servlets/purl/913104.
@article{osti_913104,
title = {Nonstandard Analysis and Shock Wave Jump Conditions in a OneDimensional Compressible Gas},
author = {Roy S. Baty, F. Farassat, John A. Hargreaves},
abstractNote = {Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. This report presents a fairly comprehensive tutorial on nonstandard analysis for physicists and engineers with many examples applicable to generalized functions. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a onedimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservations laws, smooth predistributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, smooth predistributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in nonconservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical RankineHugoniot jump conditions for an inviscid shock wave. Moreover, nonmonotonic entropy jump conditions are obtained for both inviscid and viscous flows. The report shows that products of generalized functions may be defined consistently using nonstandard analysis; however, physically meaningful products of generalized functions must be determined from the physics of the problem and not the mathematical form of the governing equations.},
doi = {10.2172/913104},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri May 25 00:00:00 EDT 2007},
month = {Fri May 25 00:00:00 EDT 2007}
}

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Nonstandard jump functions for radially symmetric shock waves
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