• Previous Article
    Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity
  • NHM Home
  • This Issue
  • Next Article
    A one dimensional free boundary problem for adsorption phenomena
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

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.
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. doi: 10.1007/s11242-006-9067-2.

[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.

[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.

[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.

[5]

A. Fasano and R. Natalini, Lost Beauties of the Acropolis: What Mathematics Can Say,, SIAM news, (2006).

[6]

T. Fatima, Multiscale Reaction Diffusion Systems Describing Concrete Corrosion: Modelling and Analysis,, Ph.D thesis, (2013).

[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.

[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.

[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.

[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.

[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.

[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.

[13]

E. J. Hinch, Perturbation Methods,, Cambridge University Press, (1991). doi: 10.1017/CBO9781139172189.

[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.

[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.

[16]

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems,, Caimbridge University Press, (2002). doi: 10.1017/CBO9780511791253.

[17]

C. V. Nikolopoulos, A mushy region in concrete corrosion,, Applied Mathematical Modelling, 34 (2010), 4012. doi: 10.1016/j.apm.2010.04.005.

[18]

C. V. Nikolopoulos, Macroscopic models for a mushy region in concrete corrosion,, Journal of Engineering Mathematics, (2014), 10665.

[19]

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

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.

[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.

[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.

[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.

[5]

A. Fasano and R. Natalini, Lost Beauties of the Acropolis: What Mathematics Can Say,, SIAM news, (2006).

[6]

T. Fatima, Multiscale Reaction Diffusion Systems Describing Concrete Corrosion: Modelling and Analysis,, Ph.D thesis, (2013).

[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.

[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.

[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.

[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.

[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.

[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.

[13]

E. J. Hinch, Perturbation Methods,, Cambridge University Press, (1991). doi: 10.1017/CBO9781139172189.

[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.

[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.

[16]

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems,, Caimbridge University Press, (2002). doi: 10.1017/CBO9780511791253.

[17]

C. V. Nikolopoulos, A mushy region in concrete corrosion,, Applied Mathematical Modelling, 34 (2010), 4012. doi: 10.1016/j.apm.2010.04.005.

[18]

C. V. Nikolopoulos, Macroscopic models for a mushy region in concrete corrosion,, Journal of Engineering Mathematics, (2014), 10665.

[19]

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

[1]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185

[2]

Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619

[3]

Ziran Yin, Liwei Zhang. Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-11. doi: 10.3934/jimo.2018100

[4]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669

[5]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577

[6]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573

[7]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[8]

Giovanni Russo, Francis Filbet. Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinetic & Related Models, 2009, 2 (1) : 231-250. doi: 10.3934/krm.2009.2.231

[9]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799

[10]

Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735

[11]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355

[12]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241

[13]

Jijun Liu, Gen Nakamura. Recovering the boundary corrosion from electrical potential distribution using partial boundary data. Inverse Problems & Imaging, 2017, 11 (3) : 521-538. doi: 10.3934/ipi.2017024

[14]

Xiao-Wen Chang, Ren-Cang Li. Multiplicative perturbation analysis for QR factorizations. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 301-316. doi: 10.3934/naco.2011.1.301

[15]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371

[16]

Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial & Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181

[17]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389

[18]

Abdon Atangana, Zakia Hammouch, Kolade M. Owolabi, Gisele Mephou. Preface: New trends on numerical analysis and analytical methods with their applications to real world problems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : ⅰ-ⅰ. doi: 10.3934/dcdss.201903i

[19]

P. Lima, L. Morgado. Analysis of singular boundary value problems for an Emden-Fowler equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 321-336. doi: 10.3934/cpaa.2006.5.321

[20]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]