# American Institute of Mathematical Sciences

March  2018, 17(2): 579-592. doi: 10.3934/cpaa.2018031

## On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption

 1 Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30,10000, Zagreb, Croatia 2 Department of Applied Mathematics, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, Brazil

Partially supported by CNPq 302960/2014-7 and 471210/2013-7, FAPESP 2013/22275-1, Brazil

Received  March 2017 Revised  July 2017 Published  March 2018

Fund Project: Partially supported by the Croatian Science Foundation (Grant No. 3955) and University of Zagreb (Grant No. 202778)

Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convection-diffusion-reaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a two-dimensional region can be approximated with one which is regular, one-dimensional and captures the effects of all physical processes which are relevant for the original problem.

Citation: Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031
##### References:
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##### References:
 [1] G. Allaire and A.-L. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium, C. R. Acad. Sci. Ser. I, 344 (2007), 523-528. Google Scholar [2] G. S. Aragão, A. L. Pereira and M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Meth. Appl. Sci., 35 (2012), 1110-1116. Google Scholar [3] G. S. Aragão, A. L. Pereira and M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating in the boundary, J. Dyn. Differ. Equ., 26 (2014), 871-888. Google Scholar [4] R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. London Sect. A, 235 (1956), 67-77. Google Scholar [5] J. M. Arrieta, A. Jiménez-Casas and A. Rodríguez-Bernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Revista Matemática Iberoamericana, 24 (2008), 183-211. Google Scholar [6] J. M. Arrieta and M. C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, J. Math. Anal. Appl., 404 (2013), 86-104. Google Scholar [7] V. Balasubramanian, G. Jayaraman and S. R. K. Iyengar, Effect of secondary flows in contaminant dispersion with weak boundary absorption Appl. Math. Model. 21 (1997), 275-285.Google Scholar [8] S. R. M. Barros and M. C. Pereira, Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary, J. Math. Anal. Appl., 441 (2016), 375-392. Google Scholar [9] L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, 19. American Mathematical Society, 2010. Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman Advanced Publishing Program, 1985. Google Scholar [11] J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 71 (1992), 33-95. Google Scholar [12] A. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. Appl., 379 (2011), 567-588. Google Scholar [13] M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis Springer-Verlag, New York, 1984.Google Scholar [14] O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Ellipitic Equations Academic Press, 1968. Google Scholar [15] E. Marušić-Paloka and I. Pažanin, On the reactive solute transport through a curved pipe, Appl. Math. Lett., 24 (2011), 878-882. Google Scholar [16] A. Mikelić, V. Devigne and C. J. van Duijn, Rigorous upscaling of the reactive flow through a pore, under dominant Péclet and Damkohler numbers, SIAM J. Math. Anal., 38 (2006), 1262-1287. Google Scholar [17] I. Pažanin, Modelling of solute dispersion in a circular pipe filled with a micropolar fluid, Math. Comp. Model., 57 (2013), 2366-2373. Google Scholar [18] M. C. Pereira, Remarks on Semilinear Parabolic Systems with terms concentrating in the boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1921-1930. Google Scholar [19] P. G. Siddheshwar and S. Manjunath, Unsteady convective-diffusion with heterogeneous chemical reaction in a plane-Poseuille flow of a micropolar fluid, Int. J. Engng. Sci., 38 (2000), 765-783. Google Scholar [20] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London Sect. A, 219 (1953), 186-203. Google Scholar [21] G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647-687. Google Scholar [22] H. F. Woolard, J. Billingham, O. E. Jensen and G. Lian, A multi-scale model for solute transport in a wavy-walled channel, J. Eng. Math., 64 (2009), 25-48. Google Scholar
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