American Institute of Mathematical Sciences

Advanced
February  2016, 9(1): 139-155. doi: 10.3934/dcdss.2016.9.139

Solidification and separation in saline water

Mauro Fabrizio 1, , Claudio Giorgi 2, and  Angelo Morro 3,

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.Donato 5, 40127 Bologna, Italy
2. 

DICATAM, Università degli studi di Brescia, Via D.Valotti 9, 25133 Brescia
3. 

DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

Motivated by the formation of brine channels, this paper is devoted to a continuum model for salt separation and phase transition in saline water. The mass density and the concentrations of salt and ice are the pertinent variables describing saline water. Hence the balance of mass is considered for the single constituents (salt, water, ice). To keep the model as simple as possible, the balance of momentum and energy are considered for the mixture as a whole. However, due to the internal structure of the mixture, an extra-energy flux is allowed to occur in addition to the heat flux. Also, the mixture is allowed to be viscous. The constitutive equations involve the dependence on the temperature, the mass density of the mixture, the salt concentration and the ice concentration, in addition to the stretching tensor, and the gradient of temperature and concentrations. The balance of mass for the single constituents eventually result in the evolution equations for the concentrations. A whole set of constitutive equations compatible with thermodynamics are established. A free energy function is given which allows for capturing the main feature which occurs during the freezing of the salted water. That is, the salt entrapment in small regions (brine channels) where the cryoscopic effect forbids complete ice formation.
Keywords: evolution equations, Ginzburg-Landau theory., freezing of saline water, solid-liquid phase transition, phase separation, Thermodynamics of solutions.
Mathematics Subject Classification: Primary: 76T99; Secondary: 74F20, 82C26, 82D1.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Solidification and separation in saline water. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 139-155. doi: 10.3934/dcdss.2016.9.139
References:
[1]

V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.  doi: 10.1016/j.physd.2007.07.009.  Google Scholar

[2]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[3]

J. W. Cahn, On spinodal decomposition,, Acta Metall., 9 (1961), 795.   Google Scholar

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.   Google Scholar

[5]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, Arch. Rational Mech. Anal., 88 (1985), 95.  doi: 10.1007/BF00250907.  Google Scholar

[6]

M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model,, J. Math. Phys., 49 (2008).  doi: 10.1063/1.2992478.  Google Scholar

[7]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144.  doi: 10.1016/j.physd.2006.01.002.  Google Scholar

[8]

M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech., 30 (2011), 281.  doi: 10.1016/j.euromechflu.2010.12.003.  Google Scholar

[9]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326.  doi: 10.1016/0167-2789(93)90128-N.  Google Scholar

[10]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[11]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter,, Math. Mod. Meth. Appl. Sci., 6 (1996), 815.  doi: 10.1142/S0218202596000341.  Google Scholar

[12]

C. Himawan, R. J. C. Vaessen, H. J. M. Kramer, M. M. Seckler and G. J. Witkamp, Dynamic modeling and simulation of eutectic freeze crystallization,, Journal of Crystal Growth, 237-239 (2002), 237.  doi: 10.1016/S0022-0248(01)02240-0.  Google Scholar

[13]

B. Kutschan, K. Morawetz and S. Gemming, Modeling the morphogenesis of brine channels in sea ice,, Phys. Rev. E, 81 (2010).  doi: 10.1103/PhysRevE.81.036106.  Google Scholar

[14]

R. A. Lake and E. L. Lewis, Salt rejection by sea ice during growth,, J. Geophys. Research, 75 (1970), 583.  doi: 10.1029/JC075i003p00583.  Google Scholar

[15]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions,, Proc. R. Soc. Lond. A, 454 (1998), 2617.  doi: 10.1098/rspa.1998.0273.  Google Scholar

show all references

References:
[1]

V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.  doi: 10.1016/j.physd.2007.07.009.  Google Scholar

[2]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[3]

J. W. Cahn, On spinodal decomposition,, Acta Metall., 9 (1961), 795.   Google Scholar

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.   Google Scholar

[5]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, Arch. Rational Mech. Anal., 88 (1985), 95.  doi: 10.1007/BF00250907.  Google Scholar

[6]

M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model,, J. Math. Phys., 49 (2008).  doi: 10.1063/1.2992478.  Google Scholar

[7]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144.  doi: 10.1016/j.physd.2006.01.002.  Google Scholar

[8]

M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech., 30 (2011), 281.  doi: 10.1016/j.euromechflu.2010.12.003.  Google Scholar

[9]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter,, Physica D, 68 (1993), 326.  doi: 10.1016/0167-2789(93)90128-N.  Google Scholar

[10]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[11]

M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter,, Math. Mod. Meth. Appl. Sci., 6 (1996), 815.  doi: 10.1142/S0218202596000341.  Google Scholar

[12]

C. Himawan, R. J. C. Vaessen, H. J. M. Kramer, M. M. Seckler and G. J. Witkamp, Dynamic modeling and simulation of eutectic freeze crystallization,, Journal of Crystal Growth, 237-239 (2002), 237.  doi: 10.1016/S0022-0248(01)02240-0.  Google Scholar

[13]

B. Kutschan, K. Morawetz and S. Gemming, Modeling the morphogenesis of brine channels in sea ice,, Phys. Rev. E, 81 (2010).  doi: 10.1103/PhysRevE.81.036106.  Google Scholar

[14]

R. A. Lake and E. L. Lewis, Salt rejection by sea ice during growth,, J. Geophys. Research, 75 (1970), 583.  doi: 10.1029/JC075i003p00583.  Google Scholar

[15]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions,, Proc. R. Soc. Lond. A, 454 (1998), 2617.  doi: 10.1098/rspa.1998.0273.  Google Scholar

[1]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
[2]

Satoshi Kosugi, Yoshihisa Morita. Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 149-168. doi: 10.3934/dcds.2006.14.149
[3]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[4]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[5]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[6]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[7]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[8]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[9]

Alessia Berti, Claudio Giorgi, Angelo Morro. Mathematical modeling of phase transition and separation in fluids: A unified approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1889-1909. doi: 10.3934/dcdsb.2014.19.1889
[10]

Pavel Krejčí, Songmu Zheng. Pointwise asymptotic convergence of solutions for a phase separation model. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 1-18. doi: 10.3934/dcds.2006.16.1
[11]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[12]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[13]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[14]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[15]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[16]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
[17]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
[18]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[19]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
[20]

Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (23)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences