×

Motion of uniformly advancing piston. (English) Zbl 07453767

Summary: We study, in a unified manner the motion of a uniformly advancing piston in the planar \((m=0)\), cylindrically symmetric \((m=1)\) and spherically symmetric \((m=2)\) isentropic flow governed by Euler’s equations. In the three dimensional space the piston is replaced by an expanding cylinder or a sphere which produces a motion in the medium outside. Under the hypothesis of self symmetry pair of ordinary differential equations is derived from Euler’s equations of fluid flow. Based on the analysis of critical solutions of this system of ordinary differential equations, unified treatment of motion of uniformly advancing piston is given with different geometries viz. planar, cylindrically symmetric and spherically symmetric flow configuration. In case of nonplanar flows, solutions in the neighbourhood of nontransitional critical points approach these critical points in finite time. Transitional solutions exist in nonplanar cases which doesn’t correspond to the motion of a piston. In the planar case, it is found that solutions take infinite time to reach critical line.

MSC:

35Q31 Euler equations
35L65 Hyperbolic conservation laws
35J70 Degenerate elliptic equations
35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Courant, R.; Friedrichs, KO, Supersonic Flow and Shock Waves (1948), New York: Interscience Publishers Inc, New York · Zbl 0041.11302
[2] Korobeinikov, VP, Problems in the Theory of Point Explosions in Gases (1973), Steklov: Trudy Mat Inst, Steklov
[3] Yuxi Zheng. System of Conservation LawsTwo Dimensional Riemann Problems. Birkhauser Boston, 2001. · Zbl 0971.35002
[4] Yuxi Zheng. Absorption of characteristics by sonic curve of the two dimensional euler equations, , 23, number 1 & 2 january and february 2009. Discrete and Continuous Dynamical Systems, 13(1 and 2), 2009. · Zbl 1154.35402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.