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On the existence of a solution for an adsorption dynamic model with the Langmuir isotherm

Published online by Cambridge University Press:  30 July 2014

J. R. FERNÁNDEZ ,
M. C. MUÑIZ  and
C. NÚÑEZ
J. R. FERNÁNDEZ
Affiliation:
Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain email: jose.fernandez@uvigo.es
M. C. MUÑIZ
Affiliation:
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Vida s/n, 15782 Santiago de Compostela, Spain email: mcarmen.muniz@usc.es
C. NÚÑEZ
Affiliation:
Departamento de Didáctica de las Ciencias Experimentales, Facultad de Ciencias de la Educación, Campus Norte, 15782 Santiago de Compostela, Spain email: cristina.nunez.garcia@usc.es
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Abstract

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

Keywords

76T1076A2076M2076M3092C45
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 5 , October 2014 , pp. 629 - 653
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Brézis, H. (1985) Análisis Funcional: Teoría y Aplicaciones, Madrid, Alianza.Google Scholar
[2]Chang, C. H. & Franses, E. I. (1995) Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data and mechanisms. Colloids Surf. 100, 1–45.CrossRefGoogle Scholar
[3]Chipot, M. (2000) Elements of Nonlinear Analysis, Birkhäuser Verlag, Basel.CrossRefGoogle Scholar
[4]Duvaut, G. & Lions, J. L. (1972) Les Inéquations en Mécanique et en Physique, Paris, Dunod.Google Scholar
[5]Eastoe, J. & Dalton, J. S. (2000) Dynamic surface tension and adsorption mechanisms of surfactants at the air-water interface. Adv. Colloid Interface Sci. 85, 103–144.CrossRefGoogle ScholarPubMed
[6]Egry, I. & Ricci, E., Novakovic, R. & Ozawa, S. (2010) Surface tension of liquid metals and alloys: Recent developments. Adv. Colloid Interface Sci. 159, 198–212.CrossRefGoogle ScholarPubMed
[7]Evans, L. C. (1998) Partial differential equations, Graduate studies in mathematics. 19. Providence. American Mathematical Society.Google Scholar
[8]Fernández, J. R., Kalita, P., Migórski, S., Muñiz, M. C. & Núñez, C. (2014) Variational analysis of the Langmuir Hinshelwood dynamic mixed-kinetic adsorption model. Nonlinear Anal.: Real World Appl. 15, 205–220.CrossRefGoogle Scholar
[9]Fernández, J. R. & Muñiz, M. C. (2011) Numerical analysis of surfactant dynamics at air-water interface using the Henry isotherm. J. Math. Chem. 49, 1624–1645.CrossRefGoogle Scholar
[10]Fernández, J. R., Muñiz, M. C. & Núñez, C. (2012) A mixed kinetic-diffusion surfactant model for the Henry isotherm. J. Math. Anal. Appl. 389, 670–684.CrossRefGoogle Scholar
[11]Fürst, T. & Vodák, R. (2009) Diffusion with nonlinear adsorption. Acta Applicandae Math. 105, 303–321.CrossRefGoogle Scholar
[12]Galiano, G. & Velasco, J. (2006) A dynamic boundary value problem arising in the ecology of mangroves. Nonlinear Anal. Real World Appl. 7, 1129–1144.CrossRefGoogle Scholar
[13]Gundabala, V. R., Zimmerman, W. B. & Routh, A. F. (2004) A model for surfactant distribution in latex coatings. Langmuir 20, 8721–8727.CrossRefGoogle Scholar
[14]McCoy, B. J. (1983) Analytical solutions for diffusion-controlled adsorption kinetics with nonlinear adsorption isotherms. Colloid Polym. Sci. 261, 535–539.CrossRefGoogle Scholar
[15]Miller, R. (1981) On the solution of diffusion controlled adsorption kinetics for any adsorption isotherms. Colloid Poly. Sci. 259, 375–381.CrossRefGoogle Scholar
[16]Miller, R., Joos, P. & Fainerman, V. B. (1994) Dynamic surface and interfacial tensions of surfactant and polymer solutions. Adv. Colloid Interface Sci. 49, 249–302.CrossRefGoogle Scholar
[17]Rodrigues, J. F. (1987) Obstacle Problems in Mathematical Physics, Amsterdam, North Holland.Google Scholar
[18]Roubíček, T. (2005) Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel.Google Scholar
[19]Showalter, R. E. (1997) Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence.Google Scholar
[20]Vrábel, V. & Slodicka, M. (2013) Nonlinear parabolic equation with a dynamical boundary condition of diffusive type. Appl. Math. Comput. 222, 372–380.Google Scholar