# American Institute of Mathematical Sciences

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

## Solidification and separation in saline water

 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.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Solidification and separation in saline water. Discrete and 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), 13-21. doi: 10.1016/j.physd.2007.07.009. [2] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Berlin, 1996. doi: 10.1007/978-1-4612-4048-8. [3] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [4] J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [5] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [6] M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902, 13 pp. doi: 10.1063/1.2992478. [7] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [8] M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids, Eur. J. Mech., 30 (2011), 281-287. doi: 10.1016/j.euromechflu.2010.12.003. [9] E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68 (1993), 326-343. doi: 10.1016/0167-2789(93)90128-N. [10] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [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), 815-831. doi: 10.1142/S0218202596000341. [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), 2257-2263. doi: 10.1016/S0022-0248(01)02240-0. [13] B. Kutschan, K. Morawetz and S. Gemming, Modeling the morphogenesis of brine channels in sea ice, Phys. Rev. E, 81 (2010), 036106. doi: 10.1103/PhysRevE.81.036106. [14] R. A. Lake and E. L. Lewis, Salt rejection by sea ice during growth, J. Geophys. Research, 75 (1970), 583-597. doi: 10.1029/JC075i003p00583. [15] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.

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##### 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), 13-21. doi: 10.1016/j.physd.2007.07.009. [2] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Berlin, 1996. doi: 10.1007/978-1-4612-4048-8. [3] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [4] J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [5] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907. [6] M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902, 13 pp. doi: 10.1063/1.2992478. [7] M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [8] M. Fabrizio, C. Giorgi and A. Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids, Eur. J. Mech., 30 (2011), 281-287. doi: 10.1016/j.euromechflu.2010.12.003. [9] E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68 (1993), 326-343. doi: 10.1016/0167-2789(93)90128-N. [10] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [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), 815-831. doi: 10.1142/S0218202596000341. [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), 2257-2263. doi: 10.1016/S0022-0248(01)02240-0. [13] B. Kutschan, K. Morawetz and S. Gemming, Modeling the morphogenesis of brine channels in sea ice, Phys. Rev. E, 81 (2010), 036106. doi: 10.1103/PhysRevE.81.036106. [14] R. A. Lake and E. L. Lewis, Salt rejection by sea ice during growth, J. Geophys. Research, 75 (1970), 583-597. doi: 10.1029/JC075i003p00583. [15] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.
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