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The convective viscous Cahn–Hilliard equation: Exact solutions

Published online by Cambridge University Press:  30 June 2015

P. O. MCHEDLOV-PETROSYAN
P. O. MCHEDLOV-PETROSYAN*
Affiliation:
A.I.Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institite of Physics & Technology”, 1, Akademicheskaya Str., Kharkov, Ukraine61108 email: peter.mchedlov@free.fr
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Abstract

In this paper, we give exact solutions for the convective viscous Cahn--Hilliard equation. This equation with a general symmetric double-well potential and Burgers-type convective term was introduced by T. P. Witelski (1996 Studies in Applied Mathematics96, 277–300) to study the joint effects of nonlinear convection and viscosity. We consider this equation with a polynomial, generally asymmetric potential. We also consider both Burgers-type and cubic convective terms. We obtained exact travelling-wave solutions for both cases. For the former case, with an additional constraint on nonlinearity and viscosity, we also obtained an exact two-wave solution.

Keywords

Cahn–Hilliard equationconvective termviscositytravelling-wave solution
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 1 , February 2016 , pp. 42 - 65
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Bai, F., Elliott, C. M., Gardiner, A., Spence, A. & Stuart, A. M. (1995) The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity 8, 131–160.CrossRefGoogle Scholar
[2] Benjamin, T. B., Bona, J. L. & Mahony, J. J. (1972) Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. London. Series A, Math. Phys. Sci. 272(1220), 4778.Google Scholar
[3] Bonfoh, A. (2011) The viscous Cahn-Hilliard equation with inertial term. Nonlinear Anal. 74, 946–964.CrossRefGoogle Scholar
[4] Caginalp, G. & Chen, X. (1998) Convergence of the phase field model to its sharp interface limits. Europ. J. Appl. Math. 9, 417–445.CrossRefGoogle Scholar
[5] Cahn, J. W. & Hilliard, J. E. (1958) Free energy of nonuniform systems. I. Interfacial free energy. J. Chem. Phys. 28, 258.CrossRefGoogle Scholar
[6] Cross, M. C. & Hohenberg, P. C. (1993) Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
[7] De Groot, S. R. & Mazur, P. (1984). Non-Equilibrium Thermodynamics, Dover Pubns, New York.Google Scholar
[8] Emmott, C. L. & Bray, A. J. (1996) Coarsening dynamics of a one-dimensional driven Cahn-Hilliard system, Phys. Rev. E 54 (5), 4568–4575.Google Scholar
[9] Fife, P. C. (2000) Models for phase separation and their mathematics. Electron. J. Differ. Equ. 2000 (48), 1–26.Google Scholar
[10] Fort, J. & Mendez, V. (2002) Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment. Rep. Progr. Phys. 65, 895–954.Google Scholar
[11] Galenko, P. (2001) Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system. Phys. Lett. A 287, 190–197.Google Scholar
[12] Galenko, P. & Jou, D. (2005) Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys. Rev. E 71, 046125.CrossRefGoogle ScholarPubMed
[13] Galenko, P. & Lebedev, V. (2008) Non-equilibrium effects in spinodal decomposition of a binary system. Phys. Lett. A 372 985–989.Google Scholar
[14] Gatti, S., Grasselli, M., Miranville, A. & Pata, V. (2005) On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation. J. Math. Anal. Appl. 312, 230–247.CrossRefGoogle Scholar
[15] Gatti, S., Grasselli, M., Miranville, A. & Pata, V. (2005) Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D. Math. Models Methods Appl. Sci. 15, 165–198.CrossRefGoogle Scholar
[16] Gelens, L. & Knobloch, E. (2010) Coarsening and frozen faceted structures in the supercritical complex Swift-Hohenberg equation. Eur. Phys. J. D 59 (1), 23–26.CrossRefGoogle Scholar
[17] Gelens, L. & Knobloch, E. (2011) Traveling waves and defects in the complex Swift-Hohenberg equation. Phys. Rev. E 84, 056203.CrossRefGoogle ScholarPubMed
[18] Golovin, A. A., Davis, S. H. & Nepomnyashchy, A. A. (1998) A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth. Physica D 118, 202–230.Google Scholar
[19] Golovin, A. A., Davis, S. H. & Nepomnyashchy, A. A. (1999) Modeling the formation of facets and corners using a convective Cahn-Hilliard model. J. Cryst. Growth 198/199, 1245–1250.Google Scholar
[20] Golovin, A. A., Nepomnyashchy, A. A., Davis, S. H. & Zaks, M. A. (2001) Convective Cahn-Hilliard models: From coarsening to roughening. Phys. Rev. Lett. 86, 1550–1553.Google Scholar
[21] Grasselli, M., Petzeltova, H. & Schimperna, G. (2007) Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term. J. Differ. Equ. 239, 38–60.Google Scholar
[22] Hirota, R. (1980) Direct methods in soliton theory. In: Bullough, R. K. and Caudrey, P. J. (editors), Solitons, Springer, Berlin.Google Scholar
[23] Kuramoto, Y. & Tsuzuki, T. (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theoret. Phys. 55, 356–369.Google Scholar
[24] Leung, K. (1990) Theory on morphological instability in driven systems. J. Stat. Phys. 61 (1/2), 345–364.CrossRefGoogle Scholar
[25] Lifshitz, I. M. & Slyozov, V. V. (1961) The kinetic of precipitation from supersaturated solid solution. J. Phys. Chem. Solids 19, 35–50.Google Scholar
[26] Mchedlov-Petrosyan, P. O. & Davydov, L. N. (2015) Travelling wave solutions for the Penrose-Fife phase field model, submitted.Google Scholar
[27] Mchedlov-Petrosyan, P. O. & Kopiychenko, D. (2013) Exact solutions for some modifications of the nonlinear Cahn-Hilliard equation. Rep. Natl. Acad. Sci. Ukraine (12), 88–93.Google Scholar
[28] Novick-Cohen, A. (1988) On the viscous Cahn-Hilliard equation. In: Ball, J. M. (editor), Material Instabilities in Continuum Mechanics and Related Mathematical Problems, Oxford University Press, Oxford, pp. 329–342.Google Scholar
[29] Novick-Cohen, A. (2008) The Cahn-Hilliard equation. In: Dafermos, C. M. and Feireisl, E. (editors), Handbook of Differential Equations, Evolutionary Equations, Vol. 4, Elsevier B.V. Google Scholar
[30] Novick-Cohen, A. & Segel, L. A. (1984) Nonlinear aspects of the Cahn-Hilliard equation. Physica D 10, 277–298.CrossRefGoogle Scholar
[31] Peletier, L. A. & Troy, W. C. (2001) Spatial patterns: Higher order models in physics and mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol. 45, Birkhauser, Boston.Google Scholar
[32] Penrose, O. & Fife, P. (1990) Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43, 44–62.Google Scholar
[33] Penrose, O. & Fife, P. (1993) On the relation between the standard phase-field model and a “thermodynamically consistent'' phase-field model. Physica D 69, 107–113.Google Scholar
[34] Podolny, A., Zaks, M. A., Rubinstein, B. Y., Golovin, A. A. & Nepomnyashchy, A. A. (2005) Dynamics of domain walls governed by the convective Cahn-Hilliard equation. Physica D 201, 291–305.Google Scholar
[35] Sivashinsky, G. (1977) Nonlinear analysis of hydrodynamic instability in laminar flames I. Derivation of basic equations. Acta Astron. 4, 1177–1206.CrossRefGoogle Scholar
[36] Van der Waals, J. D. (1979) The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. (1893). Translated by J. S. Rowlinson. J. Stat. Phys. 20, 197–244.Google Scholar
[37] Watson, S. J., Otto, F., Rubinstein, B. Y. & Davis, S. H. (2002) Coarsening dynamics for the convective Cahn-Hilliard equation, Max-Plank-Institut für Mathematik in den Naturwissenschaften, Leipzig, Preprint no. 35.Google Scholar
[38] Watson, S. J., Otto, F., Rubinstein, B. Y. & Davis, S. H. (2003) Coarsening dynamics of the convective Cahn-Hilliard equation. Physica D 178 127–148.Google Scholar
[39] Witelski, T. P. (1995) Shocks in nonlinear diffusion. Appl. Math. Lett. 8 (8), 27–32.CrossRefGoogle Scholar
[40] Witelski, T. P. (1996) The structure of internal layers for unstable nonlinear diffusion equations. Stud. Appl. Math. 96, 277–300.Google Scholar
[41] Xiaopeng, Z. & Bo, L. (2012) The existence of global attractor for convective Cahn-Hilliard equation. J. Korean Math. Soc. 49 (2), 357–378.Google Scholar