# American Institute of Mathematical Sciences

September  2017, 12(3): 371-380. doi: 10.3934/nhm.2017016

## 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

* Corresponding author: Michael Herty

Received  May 2016 Revised  July 2017 Published  September 2017

Fund Project: The first author is supported by the DFG Collaborative Research Centre SFB-TR-154, C03. The second author is supported by NSF RNMS grant No. 1107444, DFG HE5386/13, 14, 15-1 and the DAAD{MIUR project.The third author is supported by the DFG Collaborative Research Center SFB-TR-40, TP A1

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.

Citation: Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states. Networks & Heterogeneous Media, 2017, 12 (3) : 371-380. doi: 10.3934/nhm.2017016
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##### References:
Junction of $n+m=|\delta^-|+|\delta^+|$ connected pipes
The supply function $\rho {\,\mapsto\,} s(\rho;\bar{U})$ in red for given data $\bar{U}$ indicated by a cross for $\bar{\rho }>\rho^*$ (left) and $\bar{\rho }<\rho^*$ (right). Also shown in blue is the curve $L_2^-.$ The state $U^*$ is indicated by a circle.
The demand function $\rho {\,\mapsto\,} d(\rho; \bar{U})$ in red for given data $\bar{U}$. Also shown in blue is the curve $L_1.$
Random initial states $U_{0,k}$ on incoming pipes $k<0$ and outgoing pipes $k>0.$ The initial difference in the sum of the mass fluxes is given by $\Delta = 9.162e+00.$
 Pipe $k$ $(\rho_{0,k},q_{0,k})$ $p(\rho_{0,k})$ -1 (5.151e-01, 2.519e+00) 2.653e-01 -2 (6.317e-01, 2.794e+00) 3.991e-01 -3 (6.642e-01, 3.905e+00) 4.412e-01 1 (5.730e-01, -2.648e-01) 3.283e-01 2 (7.460e-01, -1.523e-01) 5.565e-01 3 (5.931e-01, -1.280e-01) 3.518e-01 4 (5.849e-01, 6.020e-01) 3.421e-01
 Pipe $k$ $(\rho_{0,k},q_{0,k})$ $p(\rho_{0,k})$ -1 (5.151e-01, 2.519e+00) 2.653e-01 -2 (6.317e-01, 2.794e+00) 3.991e-01 -3 (6.642e-01, 3.905e+00) 4.412e-01 1 (5.730e-01, -2.648e-01) 3.283e-01 2 (7.460e-01, -1.523e-01) 5.565e-01 3 (5.931e-01, -1.280e-01) 3.518e-01 4 (5.849e-01, 6.020e-01) 3.421e-01
Terminal states $U_{k}(t,0\pm)$ for $k\in\delta^\pm.$ The difference in the sum of the mass fluxes is zero
 Pipe $k$ $(\rho_{k},q_{k})$ $p(\rho_{k})$ -1 (2.089e+00, 5.101e+00) 4.364e+00 -2 (2.089e+00, 4.868e+00) 4.364e+00 -3 (2.089e+00, 8.090e+00) 4.364e+00 1 (2.089e+00, 3.757e+00) 4.364e+00 2 (2.089e+00, 3.357e+00) 4.364e+00 3 (2.089e+00, 4.147e+00) 4.364e+00 4 (2.089e+00, 6.798e+00) 4.364e+00
 Pipe $k$ $(\rho_{k},q_{k})$ $p(\rho_{k})$ -1 (2.089e+00, 5.101e+00) 4.364e+00 -2 (2.089e+00, 4.868e+00) 4.364e+00 -3 (2.089e+00, 8.090e+00) 4.364e+00 1 (2.089e+00, 3.757e+00) 4.364e+00 2 (2.089e+00, 3.357e+00) 4.364e+00 3 (2.089e+00, 4.147e+00) 4.364e+00 4 (2.089e+00, 6.798e+00) 4.364e+00
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