
-
Previous Article
Nonlinear flux-limited models for chemotaxis on networks
- NHM Home
- This Issue
-
Next Article
Traveling waves for degenerate diffusive equations on networks
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 |
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] |
M. K. Banda, M. Herty and A. Klar,
Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314 (electronic).
doi: 10.3934/nhm.2006.1.295. |
[2] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
[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] |
A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli,
Flow on networks: recent results and perspectives, European Mathematical Society-Surveys in Mathematical Sciences, 1 (2014), 47-111.
doi: 10.4171/EMSS/2. |
[5] |
G.-Q. Chen and D. Wang,
The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, 1 (2002), 421-543.
doi: 10.1016/S1874-5792(02)80012-X. |
[6] |
G. M. Coclite, M. Garavello and B. Piccoli,
Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic).
doi: 10.1137/S0036141004402683. |
[7] |
R. M. Colombo, G. Guerra, M. Herty and V. Schleper,
Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.
doi: 10.1137/080716372. |
[8] |
R. M. Colombo and M. Garavello,
A well posed Riemann problem for the p-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.
doi: 10.3934/nhm.2006.1.495. |
[9] |
R. M. Colombo and M. Garavello,
On the Cauchy problem for the p-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.
doi: 10.1137/060665841. |
[10] |
R. M. Colombo, M. Herty and V. Sachers,
On 2×2 conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.
doi: 10.1137/070690298. |
[11] |
A. de Saint-Venant,
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, 73 (1871), 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] |
M. Herty and M. Rascle,
Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[15] |
M. Herty and M. Seaïd,
Assessment of coupling conditions in water way intersections, Internat. J. Numer. Methods Fluids, 71 (2013), 1438-1460.
doi: 10.1002/fld.3719. |
[16] |
H. Holden and N. H. Risebro,
A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[17] |
H. Holden and N. H. Risebro,
Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497-515 (electronic).
doi: 10.1137/S0036141097327033. |
[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] |
J.-P. Lebacque,
Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.
|
[20] |
F. Murzyn and H. Chanson,
Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps, Experiments in Fluids, 45 (2008), 513-521.
doi: 10.1007/s00348-008-0494-4. |
[21] |
A. Osiadacz,
Simulation of transient flow in gas networks, Int. Journal for Numerical Methods in Fluid Dynamics, 4 (1984), 13-23.
doi: 10.1002/fld.1650040103. |
[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. |
show all references
References:
[1] |
M. K. Banda, M. Herty and A. Klar,
Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314 (electronic).
doi: 10.3934/nhm.2006.1.295. |
[2] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
[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] |
A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli,
Flow on networks: recent results and perspectives, European Mathematical Society-Surveys in Mathematical Sciences, 1 (2014), 47-111.
doi: 10.4171/EMSS/2. |
[5] |
G.-Q. Chen and D. Wang,
The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, 1 (2002), 421-543.
doi: 10.1016/S1874-5792(02)80012-X. |
[6] |
G. M. Coclite, M. Garavello and B. Piccoli,
Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic).
doi: 10.1137/S0036141004402683. |
[7] |
R. M. Colombo, G. Guerra, M. Herty and V. Schleper,
Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.
doi: 10.1137/080716372. |
[8] |
R. M. Colombo and M. Garavello,
A well posed Riemann problem for the p-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.
doi: 10.3934/nhm.2006.1.495. |
[9] |
R. M. Colombo and M. Garavello,
On the Cauchy problem for the p-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.
doi: 10.1137/060665841. |
[10] |
R. M. Colombo, M. Herty and V. Sachers,
On 2×2 conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.
doi: 10.1137/070690298. |
[11] |
A. de Saint-Venant,
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, 73 (1871), 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] |
M. Herty and M. Rascle,
Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[15] |
M. Herty and M. Seaïd,
Assessment of coupling conditions in water way intersections, Internat. J. Numer. Methods Fluids, 71 (2013), 1438-1460.
doi: 10.1002/fld.3719. |
[16] |
H. Holden and N. H. Risebro,
A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[17] |
H. Holden and N. H. Risebro,
Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497-515 (electronic).
doi: 10.1137/S0036141097327033. |
[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] |
J.-P. Lebacque,
Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.
|
[20] |
F. Murzyn and H. Chanson,
Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps, Experiments in Fluids, 45 (2008), 513-521.
doi: 10.1007/s00348-008-0494-4. |
[21] |
A. Osiadacz,
Simulation of transient flow in gas networks, Int. Journal for Numerical Methods in Fluid Dynamics, 4 (1984), 13-23.
doi: 10.1002/fld.1650040103. |
[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. |



Pipe | ||
-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 | ||
-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 | ||
-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 | ||
-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 |
[1] |
Yannick Holle, Michael Herty, Michael Westdickenberg. New coupling conditions for isentropic flow on networks. Networks and Heterogeneous Media, 2020, 15 (4) : 605-631. doi: 10.3934/nhm.2020016 |
[2] |
Rinaldo M. Colombo, Francesca Marcellini. Coupling conditions for the $3\times 3$ Euler system. Networks and Heterogeneous Media, 2010, 5 (4) : 675-690. doi: 10.3934/nhm.2010.5.675 |
[3] |
Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295 |
[4] |
Rinaldo M. Colombo, Graziano Guerra. A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case. Networks and Heterogeneous Media, 2016, 11 (2) : 313-330. doi: 10.3934/nhm.2016.11.313 |
[5] |
Chol-Ung Choe, Thomas Dahms, Philipp Hövel, Eckehard Schöll. Control of synchrony by delay coupling in complex networks. Conference Publications, 2011, 2011 (Special) : 292-301. doi: 10.3934/proc.2011.2011.292 |
[6] |
Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715 |
[7] |
Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049 |
[8] |
Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063 |
[9] |
Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks and Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543 |
[10] |
Maoli Chen, Yicheng Liu, Xiao Wang. Delay-dependent flocking dynamics of a two-group coupling system. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022099 |
[11] |
Jens Lang, Pascal Mindt. Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions. Networks and Heterogeneous Media, 2018, 13 (1) : 177-190. doi: 10.3934/nhm.2018008 |
[12] |
Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic and Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027 |
[13] |
Chui-Jie Wu, Hongliang Zhao. Generalized HWD-POD method and coupling low-dimensional dynamical system of turbulence. Conference Publications, 2001, 2001 (Special) : 371-379. doi: 10.3934/proc.2001.2001.371 |
[14] |
Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations and Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008 |
[15] |
Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113 |
[16] |
Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 |
[17] |
Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267 |
[18] |
Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, Cristina Trombetti. Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1. Communications on Pure and Applied Analysis, 2013, 12 (1) : 253-267. doi: 10.3934/cpaa.2013.12.253 |
[19] |
Rinaldo M. Colombo, Mauro Garavello. A Well Posed Riemann Problem for the $p$--System at a Junction. Networks and Heterogeneous Media, 2006, 1 (3) : 495-511. doi: 10.3934/nhm.2006.1.495 |
[20] |
Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 605-629. doi: 10.3934/dcdss.2012.5.605 |
2021 Impact Factor: 1.41
Tools
Metrics
Other articles
by authors
[Back to Top]