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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

Christos V. Nikolopoulos 1,

1. 

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

Received  March 2014 Revised  September 2014 Published  December 2014

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:
[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), 109-122. 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), 175-188. 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), 1636-1667. 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), 149-168. 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, Technical University of Eindhoven, 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), 261-276. 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), 326-344. 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), 295-318.  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), 1129-1154. 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), 163-189. 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), 477-494. 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), 153-169. 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), 261-282. 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), 4012-4030. 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, DOI 10.1007/s10665-014-9743-0. Google Scholar

[19]

J. L. Schnoor, Enviromental Modeling, Fate and transport of pollutants in water, air, and soil, John Willey and Sons, Inc., 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), 109-122. 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), 175-188. 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), 1636-1667. 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), 149-168. 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, Technical University of Eindhoven, 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), 261-276. 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), 326-344. 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), 295-318.  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), 1129-1154. 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), 163-189. 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), 477-494. 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), 153-169. 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), 261-282. 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), 4012-4030. 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, DOI 10.1007/s10665-014-9743-0. Google Scholar

[19]

J. L. Schnoor, Enviromental Modeling, Fate and transport of pollutants in water, air, and soil, John Willey and Sons, Inc., 1996. Google Scholar

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