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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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