Abstract
Basic properties of a reduced viscous compressible gas–liquid two-fluid model are explored. The model is composed of two conservation laws representing mass balance for gas and liquid coupled to two elliptic equations (Stokes system) for the two fluid velocities and obtained by ignoring acceleration terms in the full momentum equations. First, we present a result that shows existence and uniqueness of regular solutions for a fixed time T 0 > 0 which depends on the initial data and the constant viscosity coefficients. Moreover, T 0 can be large when the viscosity coefficients are large. However, for a fixed set of viscosity coefficients, we conjecture that the smooth solution might blow up, at least, as time tends to infinity. This result is backed up by considering a numerical example for a fixed set of viscosity coefficients demonstrating that for smooth and small initial data with no single-phase regions, the solution may tend to produce both single-phase regions and blow-up of mass gradients as time becomes large.
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References
Brennen C.E.: Fundamentals of Multiphase Flow. Cambridge University Press, New York (2005)
Bresch, D., Desjardins, B., Ghidaglia, J.-M., Grenier, E.: Global weak solutions to a generic two-fluid model. Arch. Ration. Mech. Anal. 196, 599C629 (2010)
Bresch D., Huang X.D., Li J.: Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system. Commun. Math. Phys. 309, 737–755 (2012)
Byrne H.M., Owen M.R.: A new interpretation of the Keller–Segel model based on multiphase modelling. J. Math. Biol. 49, 604–626 (2004)
Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83, 243–275 (2004)
Delhaye J.M., Giot M., Riethmuller M.L.: Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. Von Karman Institute, McGraw-Hill, New York (1981)
Evje S., Flåtten T.: On the wave structure of two-phase model. SIAM J. Appl. Math. 67(2), 487–511 (2007)
Evje S.: Weak solution for a gas–liquid model relevant for describing gas-kick oil wells. SIAM J. Math. Anal. 43, 1887–1922 (2011)
Evje S., Karlsen K.H.: Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ. 245(9), 2660–2703 (2008)
Evje S., Karlsen K.H.: Global weak solutions for a viscous liquid–gas model with singular pressure law. Commun. Pure Appl. Anal. 8, 1867–1894 (2009)
Evje S., Flåtten T., Friis H. A.: Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal. TMA 70, 3864–3886 (2009)
Gavrilyuk S.L., Fabre J.: Lagrangian coordinates for a drift-flux model of a gas–liquid mixture. Int. J. Multiph. Flow 22(3), 453–460 (1996)
Hao C.C., Li H.L.: Well-posedness for a multidimensional viscous liquid–gas flow model. SIAM J. Math. Anal. 44(3), 1304–1332 (2012)
Ladyzhenskaya O.A., Solonnikov V.A.: Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids. J. Sov. Math. 59, 697–749 (1978)
Liu Q.Q., Zhu C.J.: Asymptotic behavior of a viscous liquid–gas model with mass-dependent viscosity and vacuum. J. Differ. Equ. 252, 2492–2519 (2012)
Masella J.M., Tran Q.H., Ferre D., Pauchon C.: Transient simulation of two-phase flows in pipes. Int. J. Multiph. Flow 24, 739–755 (1998)
Prosperetti A., Tryggvason G.: Computational Methods for Multiphase Flow. Cambridge University Press, New York (2007)
Shoham, O.: Mechanistic modeling of gas–liquid two-phase flow in pipes. SPE (2006)
Yao L., Zhang T., Zhu C.-J.: Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid–gas two-phase flow model. SIAM J. Math. Anal. 42(4), 1874–1897 (2010)
Yao L., Zhu C.-J.: Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J. Differ. Equ. 247(10), 2705–2739 (2009)
Yao L., Zhu C.J.: Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math. Ann. 349, 903–928 (2010)
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Communicated by G.-Q. Chen
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Evje, S., Wen, H. Analysis of a Compressible Two-Fluid Stokes System with Constant Viscosity. J. Math. Fluid Mech. 17, 423–436 (2015). https://doi.org/10.1007/s00021-015-0215-8
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DOI: https://doi.org/10.1007/s00021-015-0215-8