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

[2]

G. S. AragãoA. L. Pereira and M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Meth. Appl. Sci., 35 (2012), 1110-1116. 

[3]

G. S. AragãoA. 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. 

[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. 

[5]

J. M. ArrietaA. 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. 

[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. 

[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.

[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. 

[9]

L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, 19. American Mathematical Society, 2010.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman Advanced Publishing Program, 1985.

[11]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 71 (1992), 33-95. 

[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. 

[13]

M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis Springer-Verlag, New York, 1984.

[14]

O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Ellipitic Equations Academic Press, 1968.

[15]

E. Marušić-Paloka and I. Pažanin, On the reactive solute transport through a curved pipe, Appl. Math. Lett., 24 (2011), 878-882. 

[16]

A. MikelićV. 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. 

[17]

I. Pažanin, Modelling of solute dispersion in a circular pipe filled with a micropolar fluid, Math. Comp. Model., 57 (2013), 2366-2373. 

[18]

M. C. Pereira, Remarks on Semilinear Parabolic Systems with terms concentrating in the boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1921-1930. 

[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. 

[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. 

[21]

G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647-687. 

[22]

H. F. WoolardJ. BillinghamO. E. Jensen and G. Lian, A multi-scale model for solute transport in a wavy-walled channel, J. Eng. Math., 64 (2009), 25-48. 

show all references

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. 

[2]

G. S. AragãoA. L. Pereira and M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Meth. Appl. Sci., 35 (2012), 1110-1116. 

[3]

G. S. AragãoA. 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. 

[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. 

[5]

J. M. ArrietaA. 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. 

[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. 

[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.

[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. 

[9]

L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, 19. American Mathematical Society, 2010.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman Advanced Publishing Program, 1985.

[11]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 71 (1992), 33-95. 

[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. 

[13]

M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis Springer-Verlag, New York, 1984.

[14]

O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Ellipitic Equations Academic Press, 1968.

[15]

E. Marušić-Paloka and I. Pažanin, On the reactive solute transport through a curved pipe, Appl. Math. Lett., 24 (2011), 878-882. 

[16]

A. MikelićV. 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. 

[17]

I. Pažanin, Modelling of solute dispersion in a circular pipe filled with a micropolar fluid, Math. Comp. Model., 57 (2013), 2366-2373. 

[18]

M. C. Pereira, Remarks on Semilinear Parabolic Systems with terms concentrating in the boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1921-1930. 

[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. 

[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. 

[21]

G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647-687. 

[22]

H. F. WoolardJ. BillinghamO. E. Jensen and G. Lian, A multi-scale model for solute transport in a wavy-walled channel, J. Eng. Math., 64 (2009), 25-48. 

Figure 1.  The domain under consideration
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