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

  • Received: 01 March 2014 Revised: 01 September 2014
  • Primary: 35Q92, 35R37; Secondary: 35K57, 74G10, 74S05.

  • 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[J]. Networks and Heterogeneous Media, 2014, 9(4): 635-654. doi: 10.3934/nhm.2014.9.635

    Related Papers:

    [1] Amira Bouhali, Walid Ben Aribi, Slimane Ben Miled, Amira Kebir . Impact of immunity loss on the optimal vaccination strategy for an age-structured epidemiological model. Mathematical Biosciences and Engineering, 2024, 21(6): 6372-6392. doi: 10.3934/mbe.2024278
    [2] Qiuyi Su, Jianhong Wu . Impact of variability of reproductive ageing and rate on childhood infectious disease prevention and control: insights from stage-structured population models. Mathematical Biosciences and Engineering, 2020, 17(6): 7671-7691. doi: 10.3934/mbe.2020390
    [3] Zhisheng Shuai, P. van den Driessche . Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences and Engineering, 2012, 9(2): 393-411. doi: 10.3934/mbe.2012.9.393
    [4] Xinyu Bai, Shaojuan Ma . Stochastic dynamical behavior of COVID-19 model based on secondary vaccination. Mathematical Biosciences and Engineering, 2023, 20(2): 2980-2997. doi: 10.3934/mbe.2023141
    [5] Mostafa Adimy, Abdennasser Chekroun, Claudia Pio Ferreira . Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase. Mathematical Biosciences and Engineering, 2020, 17(2): 1329-1354. doi: 10.3934/mbe.2020067
    [6] Ying He, Junlong Tao, Bo Bi . Stationary distribution for a three-dimensional stochastic viral infection model with general distributed delay. Mathematical Biosciences and Engineering, 2023, 20(10): 18018-18029. doi: 10.3934/mbe.2023800
    [7] Tong Guo, Jing Han, Cancan Zhou, Jianping Zhou . Multi-leader-follower group consensus of stochastic time-delay multi-agent systems subject to Markov switching topology. Mathematical Biosciences and Engineering, 2022, 19(8): 7504-7520. doi: 10.3934/mbe.2022353
    [8] Pengyan Liu, Hong-Xu Li . Global behavior of a multi-group SEIR epidemic model with age structure and spatial diffusion. Mathematical Biosciences and Engineering, 2020, 17(6): 7248-7273. doi: 10.3934/mbe.2020372
    [9] Jinliang Wang, Hongying Shu . Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences and Engineering, 2016, 13(1): 209-225. doi: 10.3934/mbe.2016.13.209
    [10] Shuang-Hong Ma, Hai-Feng Huo . Global dynamics for a multi-group alcoholism model with public health education and alcoholism age. Mathematical Biosciences and Engineering, 2019, 16(3): 1683-1708. doi: 10.3934/mbe.2019080
  • 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.


    [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
    [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
    [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
    [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
    [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, Technical University of Eindhoven, 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), 261-276. 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), 326-344. 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), 295-318.
    [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
    [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
    [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
    [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), 153-169. 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), 261-282. 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), 4012-4030. 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, DOI 10.1007/s10665-014-9743-0.
    [19] J. L. Schnoor, Enviromental Modeling, Fate and transport of pollutants in water, air, and soil, John Willey and Sons, Inc., 1996.
  • This article has been cited by:

    1. Jinhu Xu, Yan Geng, A nonstandard finite difference scheme for a multi-group epidemic model with time delay, 2017, 2017, 1687-1847, 10.1186/s13662-017-1415-8
    2. Zhijun Liu, Jing Hu, Lianwen Wang, Modelling and analysis of global resurgence of mumps: A multi-group epidemic model with asymptomatic infection, general vaccinated and exposed distributions, 2017, 37, 14681218, 137, 10.1016/j.nonrwa.2017.02.009
    3. Xue Ran, Lin Hu, Lin-Fei Nie, Zhidong Teng, Effects of stochastic perturbation and vaccinated age on a vector-borne epidemic model with saturation incidence rate, 2021, 394, 00963003, 125798, 10.1016/j.amc.2020.125798
    4. Yan Geng, Jinhu Xu, Stability preserving NSFD scheme for a multi-group SVIR epidemic model, 2017, 01704214, 10.1002/mma.4357
    5. Yan Liu, Pinrui Yu, Dianhui Chu, Huan Su, Stationary distribution of stochastic multi-group models with dispersal and telegraph noise, 2019, 33, 1751570X, 93, 10.1016/j.nahs.2019.01.007
    6. Xinyou Meng, Qingling Zhang, Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation, 2015, 2015, 1026-0226, 1, 10.1155/2015/302494
    7. Suxia Zhang, Hongbin Guo, Global analysis of age-structured multi-stage epidemic models for infectious diseases, 2018, 337, 00963003, 214, 10.1016/j.amc.2018.05.020
    8. Junyuan Yang, Yuming Chen, Theoretical and numerical results for an age-structured SIVS model with a general nonlinear incidence rate, 2018, 12, 1751-3758, 789, 10.1080/17513758.2018.1528393
    9. Ying Guo, Wei Zhao, Xiaohua Ding, Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, 2019, 343, 00963003, 114, 10.1016/j.amc.2018.07.058
    10. Lan Meng, Wei Zhu, Analysis of SEIR epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies, 2022, 200, 03784754, 489, 10.1016/j.matcom.2022.04.027
    11. Zhen Cao, Lin-Fei Nie, DYNAMICS OF A STOCHASTIC VECTOR-HOST EPIDEMIC MODEL WITH AGE-DEPENDENT OF VACCINATION AND DISEASE RELAPSE, 2023, 13, 2156-907X, 1274, 10.11948/20220099
    12. Han Ma, Yanyan Du, Zong Wang, Qimin Zhang, Positivity and Boundedness Preserving Numerical Scheme for a Stochastic Multigroup Susceptible-Infected-Recovering Epidemic Model with Age Structure, 2024, 1557-8666, 10.1089/cmb.2023.0443
  • Reader Comments
  • © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3890) PDF downloads(64) Cited by(5)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog