Mathematical modelling of a mushy region formation during sulphation of calcium carbonate

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December 2014, 9(4): 635-654. doi: 10.3934/nhm.2014.9.635

Mathematical modelling of a mushy region formation during sulphation of calcium carbonate

1.	Department of Mathematics, University of the Aegean, GR-832 00 Karlovassi, Samos, Greece

Received March 2014 Revised September 2014 Published December 2014

Abstract
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The subject of the present paper is the derivation and asymptotic analysis of a mathematical model for the formation of a mushy region during sulphation of calcium carbonate. The model is derived by averaging, with the use of the multiple scales method, applied on microscopic moving - boundary problems. The latter problems describe the transformation of calcium carbonate into gypsum on the microscopic scale. The derived macroscopic model is solved numerically with the use of a finite element method. The results of some simulations and a relevant discussion are also presented.

Keywords: moving boundary problems, asymptotic analysis., perturbation methods, Monument corrosion, sulphation.

Mathematics Subject Classification: Primary: 35Q92, 35R37; Secondary: 35K57, 74G10, 74S0.

Citation: Christos V. Nikolopoulos. Mathematical modelling of a mushy region formation during sulphation of calcium carbonate. Networks & Heterogeneous Media, 2014, 9 (4) : 635-654. doi: 10.3934/nhm.2014.9.635

References:

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[10]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction - diffusion system predicting concrete corrosion,, Applicable Analysis, 91 (2012), 1129. doi: 10.1080/00036811.2011.625016. Google Scholar
[11]	F. R. Guarguaglini and R. Natalini, Fast reaction limit and large time behavior of solutions to a nonlinear model of sulphation phenomena,, Commun. Partial Differ. Equations, 32 (2007), 163. doi: 10.1080/03605300500361438. Google Scholar
[12]	F. R. Guarguaglini and R. Natalini, Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones,, Nonlinear Analysis: Real World Applications, 6 (2005), 477. doi: 10.1016/j.nonrwa.2004.09.007. Google Scholar
[13]	E. J. Hinch, Perturbation Methods,, Cambridge University Press, (1991). doi: 10.1017/CBO9781139172189. Google Scholar
[14]	A. A. Lacey and L. A. Herraiz, Macroscopic models for melting derived from averaging microscopic Stefan problems I: Simple geometries with kinetic undercooling or surface tension,, Euro. Jnl. of Applied Mathematics, 11 (2002), 153. doi: 10.1017/S0956792599004027. Google Scholar
[15]	A. A. Lacey and L. A. Herraiz, Macroscopic models for melting derived from averaging microscopic Stefan problems II: Effect of varying geometry and composition,, Euro. Jnl. of Applied Mathematics, 13 (2002), 261. doi: 10.1017/S0956792501004818. Google Scholar
[16]	R. J. Leveque, Finite Volume Methods for Hyperbolic Problems,, Caimbridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar
[17]	C. V. Nikolopoulos, A mushy region in concrete corrosion,, Applied Mathematical Modelling, 34 (2010), 4012. doi: 10.1016/j.apm.2010.04.005. Google Scholar
[18]	C. V. Nikolopoulos, Macroscopic models for a mushy region in concrete corrosion,, Journal of Engineering Mathematics, (2014), 10665. Google Scholar
[19]	J. L. Schnoor, Enviromental Modeling, Fate and transport of pollutants in water, air, and soil,, John Willey and Sons, (1996). Google Scholar

show all references

References:

[1]	G. Ali, V. Furuholt, R. Natalini and I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I. Modeling and qualitative analysis,, Transport in Porous Media, 69 (2007), 109. doi: 10.1007/s11242-006-9067-2. Google Scholar
[2]	G. Ali, V. Furuholt, R. Natalini and I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II: Numerical approximation,, Transport in Porous Media, 69 (2007), 175. doi: 10.1007/s11242-006-9068-1. Google Scholar
[3]	D. Aregba-Driollet, F. Diele and R. Natalini, A Mathematical Model for the SO2 Aggression to Calcium Carbonate Stones: Numerical Approximation and Asymptotic Analysis,, SIAM J. APPL. MATH. , 64 (2004), 1636. doi: 10.1137/S003613990342829X. Google Scholar
[4]	F. Clareli, A. Fasano and R. Natalini, Mathematics and monument conservation: Free boundary models of marble sulfation,, SIAM Journal on Applied Mathematics, 69 (2008), 149. doi: 10.1137/070695125. Google Scholar
[5]	A. Fasano and R. Natalini, Lost Beauties of the Acropolis: What Mathematics Can Say,, SIAM news, (2006). Google Scholar
[6]	T. Fatima, Multiscale Reaction Diffusion Systems Describing Concrete Corrosion: Modelling and Analysis,, Ph.D thesis, (2013). Google Scholar
[7]	T. Fatima, N. Arab, E. P. Zemskov and A. Muntean, Homogenization of a reaction - diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains,, Journal of Engineering Mathematics, 69 (2011), 261. doi: 10.1007/s10665-010-9396-6. Google Scholar
[8]	T. Fatima and A. Muntean, Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence,, Nonlinear Analysis: Real World Applications, 15 (2014), 326. doi: 10.1016/j.nonrwa.2012.01.019. Google Scholar
[9]	T. Fatima, A. Muntean and T. Aiki, Distributed space scales in a semilinear reaction-diffusion system including a parabolic variational inequality: A well-posedness study,, Adv. Math. Sci. Appl., 22 (2012), 295. Google Scholar
[10]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction - diffusion system predicting concrete corrosion,, Applicable Analysis, 91 (2012), 1129. doi: 10.1080/00036811.2011.625016. Google Scholar
[11]	F. R. Guarguaglini and R. Natalini, Fast reaction limit and large time behavior of solutions to a nonlinear model of sulphation phenomena,, Commun. Partial Differ. Equations, 32 (2007), 163. doi: 10.1080/03605300500361438. Google Scholar
[12]	F. R. Guarguaglini and R. Natalini, Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones,, Nonlinear Analysis: Real World Applications, 6 (2005), 477. doi: 10.1016/j.nonrwa.2004.09.007. Google Scholar
[13]	E. J. Hinch, Perturbation Methods,, Cambridge University Press, (1991). doi: 10.1017/CBO9781139172189. Google Scholar
[14]	A. A. Lacey and L. A. Herraiz, Macroscopic models for melting derived from averaging microscopic Stefan problems I: Simple geometries with kinetic undercooling or surface tension,, Euro. Jnl. of Applied Mathematics, 11 (2002), 153. doi: 10.1017/S0956792599004027. Google Scholar
[15]	A. A. Lacey and L. A. Herraiz, Macroscopic models for melting derived from averaging microscopic Stefan problems II: Effect of varying geometry and composition,, Euro. Jnl. of Applied Mathematics, 13 (2002), 261. doi: 10.1017/S0956792501004818. Google Scholar
[16]	R. J. Leveque, Finite Volume Methods for Hyperbolic Problems,, Caimbridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar
[17]	C. V. Nikolopoulos, A mushy region in concrete corrosion,, Applied Mathematical Modelling, 34 (2010), 4012. doi: 10.1016/j.apm.2010.04.005. Google Scholar
[18]	C. V. Nikolopoulos, Macroscopic models for a mushy region in concrete corrosion,, Journal of Engineering Mathematics, (2014), 10665. Google Scholar
[19]	J. L. Schnoor, Enviromental Modeling, Fate and transport of pollutants in water, air, and soil,, John Willey and Sons, (1996). Google Scholar

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