Abstract
We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition. The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase interface. We apply the characteristic method and hodograph transform of Riemann to reduce the nonlinear PDEs to a concise form. Specially for the case of incident rarefaction wave, reduced linear equations are convenient to solve by Laplace transform. Then an integral formula in wave interaction region is derived in this paper, instead of the hypergeometric functions solutions for non-isothermal polytropic gases. It is also observed that when incident waves travel from the vapor phase to the liquid phase, the refracted waves must be accelerated and move forward.
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References
Amadori, D., Baiti, P., Corli, A., Santo, E. Global weak solutions for a model of two-phase flow with a single interface. J. Evol. Equ., 15(3): 699–726 (2015)
Amadori, D, Corli, A. On a model of multiphase flow. SIAM J. Math. Anal, 40(1): 134–166 (2008)
Amadori, D., Corli, A. Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow. Nonlinear Anal, 72(5): 2527–2541 (2010)
Courant, R., Friedrichs K. Supersonic Flow and Shock Waves. Interscience, New York, 1948
Courant. R., Hilbert, D. Methods of Mathematical Physics: Volume II. Wiley, New York, 1989
Debnath L. Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser, Boston, 2012
Fan H. On a model of the dynamics of liquid/vapor phase transitions. SIAM J. Appl. Math., 60 (4): 1270–1301 (2000)
Gosse, L. Existence of L∞ entropy solutions for a reacting Euler system. Port. Math., 58(4): 473–484 (2001)
Jenssen, H. On exact solutions of rarefaction-rarefaction interactions in compressible isentropic flow. J. Math. Fluid Mech., 19(4): 685–708 (2017)
Nishida, T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc. Japan Acad., 44: 642–646 (1968)
Nishida T, Smoller J. Solutions in the large for some nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math., 26(2):183–200 (1973)
Yosida, K. Lectures on Differential and Integral Equations. Interscience Publishers, New York-London, 1960
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Supported by the National Natural Science Foundation of China (No. 11901475), China Postdoctoral Science Foundation(No.2019M653815XB), Chongqing Special Postdoctoral Science Foundation (No. XmT2018045).
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Hu, K., Kan, H., Tang, Cl. et al. Reflection and Refraction of Waves Across an Interface of Two-phase Flow. Acta Math. Appl. Sin. Engl. Ser. 37, 137–147 (2021). https://doi.org/10.1007/s10255-021-0992-1
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DOI: https://doi.org/10.1007/s10255-021-0992-1