Coupling conditions for the transition from supersonic to subsonic fluid states

Martin Gugat; Michael  Herty; Siegfried  Müller; Martin Gugat; Michael  Herty; Siegfried  Müller

doi:10.3934/nhm.2017016

Networks and Heterogeneous Media

2017, Volume 12, Issue 3: 371-380. doi: 10.3934/nhm.2017016

Previous Article Next Article

Coupling conditions for the transition from supersonic to subsonic fluid states

1.
Lehrstuhl für Angewandte Mathematik 2, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, D-91058 Erlangen Germany
2.
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, D-52064 Aachen, Germany

Received: 01 May 2016 Revised: 01 July 2017
Primary: 35L60, 35L04; Secondary: 35R02

We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas flow is governed by a general p-system. By extending the notion of demand and supply known from traffic flow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.
- p-system,
- coupling,
- conditions networks
Citation: Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states[J]. Networks and Heterogeneous Media, 2017, 12(3): 371-380. doi: 10.3934/nhm.2017016

Related Papers:

Abstract

We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas flow is governed by a general p-system. By extending the notion of demand and supply known from traffic flow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.

References

[1]	Coupling conditions for gas networks governed by the isothermal Euler equations. Netw. Heterog. Media (2006) 1: 295-314 (electronic).
[2]	Gas flow in pipeline networks. Netw. Heterog. Media (2006) 1: 41-56.
[3]	A. Bressan, Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.
[4]	Flow on networks: recent results and perspectives. European Mathematical Society-Surveys in Mathematical Sciences (2014) 1: 47-111.
[5]	The Cauchy problem for the Euler equations for compressible fluids. Handbook of Mathematical Fluid Dynamics (2002) 1: 421-543.
[6]	Traffic flow on a road network. SIAM J. Math. Anal. (2005) 36: 1862-1886 (electronic).
[7]	Optimal control in networks of pipes and canals. SIAM J. Control Optim. (2009) 48: 2032-2050.
[8]	A well posed Riemann problem for the p-system at a junction. Netw. Heterog. Media (2006) 1: 495-511.
[9]	On the Cauchy problem for the p-system at a junction. SIAM J. Math. Anal. (2008) 39: 1456-1471.
[10]	On 2×2 conservation laws at a junction. SIAM J. Math. Anal. (2008) 40: 605-622.
[11]	Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l'introduction des marèes dans leur lit.. C.R. Acad. Sci. Paris (1871) 73: 147-154.
[12]	M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models.
[13]	E. Godlewski and P. -A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9
[14]	Coupling conditions for a class of second-order models for traffic flow. SIAM J. Math. Anal. (2006) 38: 595-616.
[15]	Assessment of coupling conditions in water way intersections. Internat. J. Numer. Methods Fluids (2013) 71: 1438-1460.
[16]	A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. (1995) 26: 999-1017.
[17]	Riemann problems with a kink. SIAM J. Math. Anal. (1999) 30: 497-515 (electronic).
[18]	S. Joana, M. Joris and T. Evangelos, Technical and Economical Characteristics of Co2 Transmission Pipeline Infrastructure, Technical report, JRC Scientic and Technical Reports, European Commission.
[19]	Les modeles macroscopiques du traffic. Annales des Ponts. (1993) 67: 24-45.
[20]	Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps. Experiments in Fluids (2008) 45: 513-521.
[21]	Simulation of transient flow in gas networks. Int. Journal for Numerical Methods in Fluid Dynamics (1984) 4: 13-23.
[22]	B. Sultanian, Fluid Mechanics: An Intermediate Approach, CRC Press, 2015.
[23]	R. Ugarelli and V. D. Federico, Transition from supercritical to subcritical regime in free surface flow of yield stress fluids Geophys. Res. Lett. , 34 (2007), L21402. doi: 10.1029/2007GL031487

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)