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The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term

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Abstract

In this paper, we investigate the limits of Riemann solutions to the Euler equations of compressible fluid flow with a source term as the adiabatic exponent tends to one. The source term can represent friction or gravity or both in Engineering. For instance, a concrete physical model is a model of gas dynamics in a gravitational field with entropy assumed to be a constant. The body force source term is presented if there is some external force acting on the fluid. The force assumed here is the gravity. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a Coulomb-like friction term, and the intermediate density between the two shocks tends to a weighted \(\delta \)-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a Coulomb-like friction term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

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Acknowledgements

The authors are very grateful to the four anonymous referees for their valuable comments and corrections, which have improved the original manuscript greatly. This work is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2019J01642).

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  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Shouqiong Sheng & Zhiqiang Shao

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  1. Shouqiong Sheng
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  2. Zhiqiang Shao
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Correspondence to Zhiqiang Shao.

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Sheng, S., Shao, Z. The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term. J Eng Math 125, 1–22 (2020). https://doi.org/10.1007/s10665-020-10066-3

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