American Institute of Mathematical Sciences

Advanced
April  2016, 36(4): 1927-1955. doi: 10.3934/dcds.2016.36.1927

Global solutions to a one-dimensional non-conservative two-phase model

Steinar Evje 1, , Huanyao Wen 2, and  Lei Yao 3,

1. 

University of Stavanger (UiS), 4036 Stavanger
2. 

University of Stavanger, NO-4036 Stavanger, Norway
3. 

School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127

Received  March 2014 Revised  November 2014 Published  September 2015

In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation. The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid). We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem). We demonstrate that under appropriate smallness conditions on initial data we can obtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.
Keywords: uniqueness, long-time behavior., Navier-Stokes, Two-fluid model, existence.
Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 65M12, 35L6.
Citation: Steinar Evje, Huanyao Wen, Lei Yao. Global solutions to a one-dimensional non-conservative two-phase model. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1927-1955. doi: 10.3934/dcds.2016.36.1927
References:
[1]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8.

[2]

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[3]

D. Bresch and B. Desjardins, Stabilité de solutions faibles globales pour leséquations de Navier-Stokes compressible avec température, C. R. Math. Acad. Sci., 343 (2006), 219-224. doi: 10.1016/j.crma.2006.05.016.

[4]

D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629. doi: 10.1007/s00205-009-0261-6.

[5]

D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755. doi: 10.1007/s00220-011-1379-6.

[6]

F. Coquel, K. El Amine, E. Godlewki, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys., 136 (1997), 272-288. doi: 10.1006/jcph.1997.5730.

[7]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.

[8]

S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210. doi: 10.1016/j.jcp.2003.07.001.

[9]

S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput., 26 (2005), 1449-1484. doi: 10.1137/030600631.

[10]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932.

[11]

S. Evje and H. Y. Wen, Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators, Discrete Contin. Dyn. Syst., 33 (2013), 4497-4530. doi: 10.3934/dcds.2013.33.4497.

[12]

A. C. Fowler and P. E. Lisseter, Flooding and flow reversal in annular two-phase flow, SIAM J. Appl. Math., 52 (1992), 15-33. doi: 10.1137/0152002.

[13]

Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333.

[14]

B.L. Keyfitz, R. Sanders, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 541-563. doi: 10.3934/dcdsb.2003.3.541.

[15]

H.-L. Li, J. Li and Z. P. Xin, Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations, Commun. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4.

[16]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[17]

A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2008), 1344-1365. doi: 10.1137/060658199.

[18]

A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow, Corrected paperback reprint of the 2007 original. Cambridge University Press, Cambridge, 2009.

[19]

M. D. Thanh, D. Kröner and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61 (2011), 702-721. doi: 10.1016/j.apnum.2011.01.004.

[20]

L. Yao, H. L. Guo and Z. H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 2323-2342. doi: 10.1016/j.nonrwa.2012.02.001.

show all references

References:
[1]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8.

[2]

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[3]

D. Bresch and B. Desjardins, Stabilité de solutions faibles globales pour leséquations de Navier-Stokes compressible avec température, C. R. Math. Acad. Sci., 343 (2006), 219-224. doi: 10.1016/j.crma.2006.05.016.

[4]

D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629. doi: 10.1007/s00205-009-0261-6.

[5]

D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755. doi: 10.1007/s00220-011-1379-6.

[6]

F. Coquel, K. El Amine, E. Godlewki, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys., 136 (1997), 272-288. doi: 10.1006/jcph.1997.5730.

[7]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.

[8]

S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210. doi: 10.1016/j.jcp.2003.07.001.

[9]

S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput., 26 (2005), 1449-1484. doi: 10.1137/030600631.

[10]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932.

[11]

S. Evje and H. Y. Wen, Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators, Discrete Contin. Dyn. Syst., 33 (2013), 4497-4530. doi: 10.3934/dcds.2013.33.4497.

[12]

A. C. Fowler and P. E. Lisseter, Flooding and flow reversal in annular two-phase flow, SIAM J. Appl. Math., 52 (1992), 15-33. doi: 10.1137/0152002.

[13]

Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427. doi: 10.1137/070680333.

[14]

B.L. Keyfitz, R. Sanders, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 541-563. doi: 10.3934/dcdsb.2003.3.541.

[15]

H.-L. Li, J. Li and Z. P. Xin, Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations, Commun. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4.

[16]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[17]

A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2008), 1344-1365. doi: 10.1137/060658199.

[18]

A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow, Corrected paperback reprint of the 2007 original. Cambridge University Press, Cambridge, 2009.

[19]

M. D. Thanh, D. Kröner and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61 (2011), 702-721. doi: 10.1016/j.apnum.2011.01.004.

[20]

L. Yao, H. L. Guo and Z. H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 2323-2342. doi: 10.1016/j.nonrwa.2012.02.001.

[1]

Oscar Jarrín, Manuel Fernando Cortez. On the long-time behavior for a damped Navier-Stokes-Bardina model. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022028
[2]

Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397
[3]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[4]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
[5]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[6]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[7]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[8]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427
[9]

Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237
[10]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[11]

Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105
[12]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112
[13]

Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021231
[14]

Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437
[15]

Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
[16]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[17]

Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084
[18]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 37-53. doi: 10.3934/dcdss.2009.2.37
[19]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[20]

Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2495-2532. doi: 10.3934/dcdsb.2020020

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (146)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy