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December  2018, 23(10): 4519-4540. doi: 10.3934/dcdsb.2018174

Numerical study of phase transition in van der Waals fluid

1. 

School of Mathematics, Sichuan University, Chengdu, 610064, China

2. 

Department of Mathematics, School of Science, Beijing University of Chemical Technology, Beijing, 100029, China

* Corresponding author: Chang Liu

Received  August 2017 Revised  December 2017 Published  June 2018

Fund Project: The first author is supported by NSFC grant No.11201322, the third author is supported by NSFC grant No.11671027

In this article, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter $ε$ and a new variable. We solve the relaxation system in Lagrangian coordinates for one dimension and solve the system in Eulerian coordinates for two dimension. A second order TVD Runge-Kutta splitting scheme is used in time discretization and upwind or MUSCL scheme is used in space discretization. The long time behavior of the fluid is numerically investigated. If the initial data belongs to elliptic region, the solution converges to two Maxwell states. When the initial data lies in metastable region, the solution either remains in the same phase or converges to the Maxwell states depending to the initial perturbation. If the initial state is in the stable region, the solution remains in that region for all time.

Citation: Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174
References:
[1]

M. Affouf and R. Caflisch, A numerical study of Riemann problem solutions and stability for a system of viscous conservation laws of mixed type, SIAM J. Appl. Math., 51 (1991), 605-634.  doi: 10.1137/0151031.  Google Scholar

[2]

N. AlikakosP. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential. Equations, 90 (1991), 81-135.  doi: 10.1016/0022-0396(91)90163-4.  Google Scholar

[3]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008.  doi: 10.1137/0153049.  Google Scholar

[4]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation. Ⅱ. Layer dynamics and slow invariant manifold, J. Differential. Equations, 117 (1995), 165-216.  doi: 10.1006/jdeq.1995.1052.  Google Scholar

[5]

L. Bronsard and R. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math, 43 (1990), 983-997.  doi: 10.1002/cpa.3160430804.  Google Scholar

[6]

J. CarrM. Gurtin and M. Slemrod, Structured phase transition on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.  doi: 10.1007/BF00280031.  Google Scholar

[7]

B. Cockburn and H. Gau, A model numerical scheme for the propagation of phase transitions in solids, SIAM J. Sci. Comp., 17 (1996), 1092-1121.  doi: 10.1137/S106482759426688X.  Google Scholar

[8]

E. Fermi, Thermodynamics, 1$^{nd}$ edition, Dover, New York, 1956. Google Scholar

[9]

M. Gander, M. Mei and G. Schmidt, Phase transition for a relaxation model of mixed type with periodic boundary condition, Applied Mathematics Research Express, 2007 (2007), Art. ID abm006, 34 pp. doi: 10.1093/amrx/abm006.  Google Scholar

[10]

D. Hoff and M. Khodja, Stability of coexisting phases for compressible van der Waals fluids, SIAM J. Appl. Math., 53 (1993), 1-14.  doi: 10.1137/0153001.  Google Scholar

[11]

D.-Y. Hsieh and X.-P. Wang, Phase transition in Van der Waals fluid, SIAM J. Appl. Math., 57 (1997), 871-892.  doi: 10.1137/S0036139995295165.  Google Scholar

[12]

S. Jin and Z. P. Xin, The relaxation schemes for systems of hyperbolic conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[13]

M. MeiL. Liu and Y. Wong, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition (Ⅰ): Existence and uniform boundedness, Discrete Cont. Dyn. Syst B, 7 (2007), 825-837.  doi: 10.3934/dcdsb.2007.7.825.  Google Scholar

[14]

M. MeiL. Liu and Y. Wong, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition (Ⅱ): Convergence, Discrete Cont. Dyn. Syst B, 7 (2007), 839-857.  doi: 10.3934/dcdsb.2007.7.839.  Google Scholar

[15]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[16]

C. W. Shu, A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting, J. Comp. Phys, 100 (1992), 424-429.  doi: 10.1016/0021-9991(92)90249-X.  Google Scholar

[17]

M. Slemrod and J. E. Flaherty, Numerical integration of a Riemann problem for a van der Waals fluids, Phase Transition C. A. Elias and G. John, eds., Elsevier, New York, (1986), 203-212. Google Scholar

[18]

A. Sommerfeld, Thermodynamics and Statistical Mechanics, 1$^{nd}$ edition, Academic Press, New York and London, 1964. Google Scholar

[19]

P. R. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Num. Anal., 21 (1984), 995-1011.  doi: 10.1137/0721062.  Google Scholar

show all references

References:
[1]

M. Affouf and R. Caflisch, A numerical study of Riemann problem solutions and stability for a system of viscous conservation laws of mixed type, SIAM J. Appl. Math., 51 (1991), 605-634.  doi: 10.1137/0151031.  Google Scholar

[2]

N. AlikakosP. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential. Equations, 90 (1991), 81-135.  doi: 10.1016/0022-0396(91)90163-4.  Google Scholar

[3]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008.  doi: 10.1137/0153049.  Google Scholar

[4]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation. Ⅱ. Layer dynamics and slow invariant manifold, J. Differential. Equations, 117 (1995), 165-216.  doi: 10.1006/jdeq.1995.1052.  Google Scholar

[5]

L. Bronsard and R. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math, 43 (1990), 983-997.  doi: 10.1002/cpa.3160430804.  Google Scholar

[6]

J. CarrM. Gurtin and M. Slemrod, Structured phase transition on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.  doi: 10.1007/BF00280031.  Google Scholar

[7]

B. Cockburn and H. Gau, A model numerical scheme for the propagation of phase transitions in solids, SIAM J. Sci. Comp., 17 (1996), 1092-1121.  doi: 10.1137/S106482759426688X.  Google Scholar

[8]

E. Fermi, Thermodynamics, 1$^{nd}$ edition, Dover, New York, 1956. Google Scholar

[9]

M. Gander, M. Mei and G. Schmidt, Phase transition for a relaxation model of mixed type with periodic boundary condition, Applied Mathematics Research Express, 2007 (2007), Art. ID abm006, 34 pp. doi: 10.1093/amrx/abm006.  Google Scholar

[10]

D. Hoff and M. Khodja, Stability of coexisting phases for compressible van der Waals fluids, SIAM J. Appl. Math., 53 (1993), 1-14.  doi: 10.1137/0153001.  Google Scholar

[11]

D.-Y. Hsieh and X.-P. Wang, Phase transition in Van der Waals fluid, SIAM J. Appl. Math., 57 (1997), 871-892.  doi: 10.1137/S0036139995295165.  Google Scholar

[12]

S. Jin and Z. P. Xin, The relaxation schemes for systems of hyperbolic conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[13]

M. MeiL. Liu and Y. Wong, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition (Ⅰ): Existence and uniform boundedness, Discrete Cont. Dyn. Syst B, 7 (2007), 825-837.  doi: 10.3934/dcdsb.2007.7.825.  Google Scholar

[14]

M. MeiL. Liu and Y. Wong, Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition (Ⅱ): Convergence, Discrete Cont. Dyn. Syst B, 7 (2007), 839-857.  doi: 10.3934/dcdsb.2007.7.839.  Google Scholar

[15]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[16]

C. W. Shu, A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting, J. Comp. Phys, 100 (1992), 424-429.  doi: 10.1016/0021-9991(92)90249-X.  Google Scholar

[17]

M. Slemrod and J. E. Flaherty, Numerical integration of a Riemann problem for a van der Waals fluids, Phase Transition C. A. Elias and G. John, eds., Elsevier, New York, (1986), 203-212. Google Scholar

[18]

A. Sommerfeld, Thermodynamics and Statistical Mechanics, 1$^{nd}$ edition, Academic Press, New York and London, 1964. Google Scholar

[19]

P. R. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Num. Anal., 21 (1984), 995-1011.  doi: 10.1137/0721062.  Google Scholar

Figure 1.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.1 e^{-4x^{2}})^{T}$, $\epsilon = 0.0025$, 1025 grid points. $V(x)$ at various time T. Elliptic region is between the solid lines, and dashed lines represent the Maxwell states.
Figure 2.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T} $, $\epsilon = 10^{-8}$, 1025 grid points.
Figure 3.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T} $, $\epsilon = 0.005$, 1025 grid points.
Figure 4.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T} $, $\epsilon = 0.025$, 1025 grid points.
Figure 5.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 0.05$, 1025 grid points.
Figure 6.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.6}, 0.1 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.
Figure 7.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.6}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-4}$, 1025 grid points.
Figure 8.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.6+0.4\sin(x)}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.
Figure 9.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.7}, 0.5 e^{-4x^{2}})^{T} $, $\epsilon = 10^{-8}$, 1025 grid points.
Figure 10.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.7+0.1\cos(x)}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.
Figure 11.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.0, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256 $ grid points.
Figure 12.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.6, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256 $ grid points.
Figure 13.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.6, 0.5 e^{-4(x^{2}+y^{2})}, 0.5 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256 $ grid points.
Figure 14.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.7, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T} $, $\epsilon = 0.002$, $256 \times 256 $ grid points.
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