Polytropic solutions to the problem of spherically symmetric flow of an ideal gas

Details Polytropic solutions to the problem of spherically symmetric flow of an ideal gas Polytropic solutions to the problem of spherically symmetric flow of an ideal gas

Aug 1980

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1980cajph..58.1085s&link_type=abstract

Canadian Journal of Physics, vol. 58, Aug. 1980, p. 1085-1092. Natural Sciences and Engineering Research Council of Canada

Astronomy and Astrophysics

Astrophysics

2

Compressible Flow, Flow Equations, Ideal Gas, Polytropic Processes, Radial Flow, Steady Flow, Astrophysics, Asymptotic Methods, Bernoulli Theorem, Differential Equations, Points (Mathematics), Singularity (Mathematics)

Scientific paper

An ideal, compressible gas is considered in steady, spherically symmetric, purely radial motion in the presence of a gravitating point mass situated at the origin of coordinates. The gas pressure and mass density are assumed to satisfy a simple polytrope law, with pressure proportional to density to the beta power, where beta is the polytropic index which is assumed to take on any real value; the self-gravitation of the gas is neglected. The model equations, which are expressed in a form appropriate to both the expansion and accretion cases, reduce to two nonlinear ordinary differential equations, for which Bernoulli integrals are readily found. Singular points of the differential equations are analyzed, and the complete set of asymptotic solutions for the velocity, temperature and Mach number are given for lambda approaching 0 and infinity where lambda is proportional to the inverse of the radial coordinate, as well as a special class of solutions valid for lambda approaching constant (not equal to 0). Families of velocity profiles are sketched which are representative of the complete range of beta. The polytropic model, special cases of which have been used successfully in astrophysics in stellar wind and accretion problems, is here cast in general fluid-dynamic terms so that the complete set of solutions obtained may be applicable to a wide class of gas expansion and accretion problems.

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