Abstract
General conservation laws are considered, and the concept of a weak solution is extended to the case of an equation involving three space variables and time. Four-dimensional vector calculus is used to develop general jump conditions at a shock wave in the material. To illustrate the use of this result, jump conditions at a shock in unsteady three-dimensional compressible gas flow are presented. It is then proved rigorously that these reduce to the commonly assumed conditions in coordinates normal and tangential to the shock face. A similar calculation is also outlined for an unsteady three-dimensional shock in magnetohydrodynamics, and in a chemically reactive fluid. The technique is available for determining shock–jump conditions in quite general continuous media.
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Acknowledgments
This work is associated with Australian Research Council Grant DP140100094. We are indebted to three anonymous referees for constructive comments on this paper.
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Communicated by Chih-Yung Wen.
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Forbes, L.K., Krzysik, O.A. Shock–jump conditions in a general medium: weak-solution approach. Shock Waves 27, 457–466 (2017). https://doi.org/10.1007/s00193-016-0695-3
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DOI: https://doi.org/10.1007/s00193-016-0695-3