February  2014, 7(1): 95-111. doi: 10.3934/dcdss.2014.7.95

Upscaling of reactive flows in domains with moving oscillating boundaries

1. 

CASA, Technische Universiteit Eindhoven, Eindhoven, Netherlands, Netherlands

2. 

Department Mathematik, Chair of Applied Mathematics 1, Martensstr. 3, 91058 Erlangen, Germany

Received  March 2012 Revised  November 2012 Published  July 2013

We consider the flow and transport of chemically reactive substances (precursors) in a channel over substrates having complex geometry. In particular, these substrates are in the form of trenches forming oscillating boundaries. The precursors react at the boundaries and get deposited. The deposited layers lead to changes in the geometry and are explicitly taken into account. Consequently, the system forms a free boundary problem. Using formal asymptotic techniques, we obtain the upscaled equations for the system where these equations are defined on a domain with flat boundaries. This provides a huge gain in computational time. Numerical experiments show the effectiveness of the upscaling process.
Citation: Kundan Kumar, Tycho van Noorden, Iuliu Sorin Pop. Upscaling of reactive flows in domains with moving oscillating boundaries. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 95-111. doi: 10.3934/dcdss.2014.7.95
References:
[1]

G. Allaire and M. Amar, Boundary layer tails in periodic homogenization,, ESAIM Control Optim. Calc. Var., 4 (1999), 209. doi: 10.1051/cocv:1999110. Google Scholar

[2]

J. M. Arrieta and S. M. Bruschi, Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 327. doi: 10.3934/dcdsb.2010.14.327. Google Scholar

[3]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary,, J. Math. Pures Appl. (9), 96 (2011), 29. doi: 10.1016/j.matpur.2011.02.003. Google Scholar

[4]

K. Baber, K. Mosthaf, B. Flemisch, R. Helmig, S. Müthing and B. Wohlmuth, Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow,, IMA J. Appl. Math., 77 (2012), 887. doi: 10.1093/imamat/hxs048. Google Scholar

[5]

J. Bogers, K. Kumar, P. H. L. Notten, J. F. M. Oudenhoven and I. S. Pop, A multiscale domain decomposition approach for chemical vapor deposition,, J. Comput. Appl. Math., 246 (2013), 65. doi: 10.1016/j.cam.2012.10.018. Google Scholar

[6]

G. A. Chechkin, A. Friedman and A. L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary,, J. Math. Anal. Appl., 231 (1999), 213. doi: 10.1006/jmaa.1998.6226. Google Scholar

[7]

J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodriguez-Ferran, Arbitrary Lagrangian-Eulerian methods,, The Encyclopedia of Computational Mechanics, 1 (2004), 413. Google Scholar

[8]

C. J. van Duijn and P. Knabner, Travelling wave behaviour of crystal dissolution in porous media flow,, European J. Appl. Math., 8 (1997), 49. Google Scholar

[9]

C. J. van Duijn, A. Mikelic, C. Rosier and I. S. Pop, Effective dispersion equations for reactive flows with dominant peclet and damkohler numbers,, Advances in Chemical Engineering, 34 (2008), 1. Google Scholar

[10]

C. J. van Duijn and I. S. Pop, Crystal dissolution and precipitation in porous media: pore scale analysis,, J. Reine Angew. Math., 577 (2004), 171. doi: 10.1515/crll.2004.2004.577.171. Google Scholar

[11]

A. Friedman and B. Hu, A non-stationary multi-scale oscillating free boundary for the Laplace and heat equations,, J. Differential Equations, 137 (1997), 119. doi: 10.1016/S0022-0396(06)80006-9. Google Scholar

[12]

A. Friedman, B. Hu and Y. Liu, A boundary value problem for the Poisson equation with multi-scale oscillating boundary,, J. Differential Equations, 137 (1997), 54. doi: 10.1006/jdeq.1997.3257. Google Scholar

[13]

M. K. Gobbert and C. A. Ringhofer, An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface,, SIAM J. Appl. Math., 58 (1998), 737. doi: 10.1137/S0036139999528467. Google Scholar

[14]

F. Golfier, B. D. Wood, L. Orgogozo, M. Quintard and M. Buès, Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium,, Adv. Water Res., 32 (2009), 463. doi: 10.1016/j.advwatres.2008.11.012. Google Scholar

[15]

E. Hairer and G. Wanner, On the instability of the BDF formulas,, SIAM J. Numer. Anal., 20 (1983), 1206. doi: 10.1137/0720090. Google Scholar

[16]

, COMSOL Inc.,, , (). Google Scholar

[17]

W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 403. Google Scholar

[18]

W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111. doi: 10.1137/S003613999833678X. Google Scholar

[19]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow,, J. Differential Equations, 170 (2001), 96. doi: 10.1006/jdeq.2000.3814. Google Scholar

[20]

K. Kumar, M. van Helvoort and I. S. Pop, Rigorous upscaling of rough boundaries for reactive flows,, CASA Report 12-37, (2012), 12. Google Scholar

[21]

K. Kumar, T. L. van Noorden and I. S. Pop, Effective dispersion equations for reactive flows involving free boundaries at the microscale,, Multiscale Model. Simul., 9 (2011), 29. doi: 10.1137/100804553. Google Scholar

[22]

, MATLAB,, , (). Google Scholar

[23]

K. Mosthaf, K. Baber, B. Flemisch, R. Helmig, A. Leijnse, I. Rybak and B. Wohlmuth, A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow,, Water Resour. Res., 47 (2011). doi: 10.1029/2011WR010685. Google Scholar

[24]

N. Neuss, M. Neuss-Radu and A. Mikelić, Effective laws for the Poisson equation on domains with curved oscillating boundaries,, Appl. Anal., 85 (2006), 479. doi: 10.1080/00036810500340476. Google Scholar

[25]

T. L. van Noorden, Crystal precipitation and dissolution in a porous medium: Effective equations and numerical experiments,, Multiscale Model. Simul., 7 (2008), 1220. doi: 10.1137/080722096. Google Scholar

[26]

T. L. van Noorden, Crystal precipitation and dissolution in a thin strip,, European J. Appl. Math., 20 (2009), 69. doi: 10.1017/S0956792508007651. Google Scholar

[27]

T. L. van Noorden, I. S. Pop, A. Ebigbo and R. Helmig, An upscaled model for biofilm growth in a thin strip,, Water Resour. Res., 46 (2010). Google Scholar

[28]

P. H. L. Notten, F. Roozeboom, R. A. H. Niessen and L. Baggetto, 3-d integrated all-solid-state rechargeable batteries,, Advanced Materials, 19 (2007), 4564. doi: 10.1002/adma.200702398. Google Scholar

[29]

J. F. M. Oudenhoven, L. Bagetto and P. H. L. Notten, All-solid-state lithium-ion microbatteries: A review of vaious three-dimensional concepts,, Advanced Energy Materials, 1 (2011), 10. Google Scholar

[30]

J. F. M. Oudenhoven, T. van Dongen, R. A. H. Niessen, M. H. J. M. de Croon and P. H. L. Notten, Low-pressure chemical vapor deposition of licoo2 thin films: A systematic investigation of the deposition parameters,, Journal of the Electrochemical Society, 156 (2009). Google Scholar

show all references

References:
[1]

G. Allaire and M. Amar, Boundary layer tails in periodic homogenization,, ESAIM Control Optim. Calc. Var., 4 (1999), 209. doi: 10.1051/cocv:1999110. Google Scholar

[2]

J. M. Arrieta and S. M. Bruschi, Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 327. doi: 10.3934/dcdsb.2010.14.327. Google Scholar

[3]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary,, J. Math. Pures Appl. (9), 96 (2011), 29. doi: 10.1016/j.matpur.2011.02.003. Google Scholar

[4]

K. Baber, K. Mosthaf, B. Flemisch, R. Helmig, S. Müthing and B. Wohlmuth, Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow,, IMA J. Appl. Math., 77 (2012), 887. doi: 10.1093/imamat/hxs048. Google Scholar

[5]

J. Bogers, K. Kumar, P. H. L. Notten, J. F. M. Oudenhoven and I. S. Pop, A multiscale domain decomposition approach for chemical vapor deposition,, J. Comput. Appl. Math., 246 (2013), 65. doi: 10.1016/j.cam.2012.10.018. Google Scholar

[6]

G. A. Chechkin, A. Friedman and A. L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary,, J. Math. Anal. Appl., 231 (1999), 213. doi: 10.1006/jmaa.1998.6226. Google Scholar

[7]

J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodriguez-Ferran, Arbitrary Lagrangian-Eulerian methods,, The Encyclopedia of Computational Mechanics, 1 (2004), 413. Google Scholar

[8]

C. J. van Duijn and P. Knabner, Travelling wave behaviour of crystal dissolution in porous media flow,, European J. Appl. Math., 8 (1997), 49. Google Scholar

[9]

C. J. van Duijn, A. Mikelic, C. Rosier and I. S. Pop, Effective dispersion equations for reactive flows with dominant peclet and damkohler numbers,, Advances in Chemical Engineering, 34 (2008), 1. Google Scholar

[10]

C. J. van Duijn and I. S. Pop, Crystal dissolution and precipitation in porous media: pore scale analysis,, J. Reine Angew. Math., 577 (2004), 171. doi: 10.1515/crll.2004.2004.577.171. Google Scholar

[11]

A. Friedman and B. Hu, A non-stationary multi-scale oscillating free boundary for the Laplace and heat equations,, J. Differential Equations, 137 (1997), 119. doi: 10.1016/S0022-0396(06)80006-9. Google Scholar

[12]

A. Friedman, B. Hu and Y. Liu, A boundary value problem for the Poisson equation with multi-scale oscillating boundary,, J. Differential Equations, 137 (1997), 54. doi: 10.1006/jdeq.1997.3257. Google Scholar

[13]

M. K. Gobbert and C. A. Ringhofer, An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface,, SIAM J. Appl. Math., 58 (1998), 737. doi: 10.1137/S0036139999528467. Google Scholar

[14]

F. Golfier, B. D. Wood, L. Orgogozo, M. Quintard and M. Buès, Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium,, Adv. Water Res., 32 (2009), 463. doi: 10.1016/j.advwatres.2008.11.012. Google Scholar

[15]

E. Hairer and G. Wanner, On the instability of the BDF formulas,, SIAM J. Numer. Anal., 20 (1983), 1206. doi: 10.1137/0720090. Google Scholar

[16]

, COMSOL Inc.,, , (). Google Scholar

[17]

W. Jäger and A. Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 403. Google Scholar

[18]

W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman,, SIAM J. Appl. Math., 60 (2000), 1111. doi: 10.1137/S003613999833678X. Google Scholar

[19]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow,, J. Differential Equations, 170 (2001), 96. doi: 10.1006/jdeq.2000.3814. Google Scholar

[20]

K. Kumar, M. van Helvoort and I. S. Pop, Rigorous upscaling of rough boundaries for reactive flows,, CASA Report 12-37, (2012), 12. Google Scholar

[21]

K. Kumar, T. L. van Noorden and I. S. Pop, Effective dispersion equations for reactive flows involving free boundaries at the microscale,, Multiscale Model. Simul., 9 (2011), 29. doi: 10.1137/100804553. Google Scholar

[22]

, MATLAB,, , (). Google Scholar

[23]

K. Mosthaf, K. Baber, B. Flemisch, R. Helmig, A. Leijnse, I. Rybak and B. Wohlmuth, A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow,, Water Resour. Res., 47 (2011). doi: 10.1029/2011WR010685. Google Scholar

[24]

N. Neuss, M. Neuss-Radu and A. Mikelić, Effective laws for the Poisson equation on domains with curved oscillating boundaries,, Appl. Anal., 85 (2006), 479. doi: 10.1080/00036810500340476. Google Scholar

[25]

T. L. van Noorden, Crystal precipitation and dissolution in a porous medium: Effective equations and numerical experiments,, Multiscale Model. Simul., 7 (2008), 1220. doi: 10.1137/080722096. Google Scholar

[26]

T. L. van Noorden, Crystal precipitation and dissolution in a thin strip,, European J. Appl. Math., 20 (2009), 69. doi: 10.1017/S0956792508007651. Google Scholar

[27]

T. L. van Noorden, I. S. Pop, A. Ebigbo and R. Helmig, An upscaled model for biofilm growth in a thin strip,, Water Resour. Res., 46 (2010). Google Scholar

[28]

P. H. L. Notten, F. Roozeboom, R. A. H. Niessen and L. Baggetto, 3-d integrated all-solid-state rechargeable batteries,, Advanced Materials, 19 (2007), 4564. doi: 10.1002/adma.200702398. Google Scholar

[29]

J. F. M. Oudenhoven, L. Bagetto and P. H. L. Notten, All-solid-state lithium-ion microbatteries: A review of vaious three-dimensional concepts,, Advanced Energy Materials, 1 (2011), 10. Google Scholar

[30]

J. F. M. Oudenhoven, T. van Dongen, R. A. H. Niessen, M. H. J. M. de Croon and P. H. L. Notten, Low-pressure chemical vapor deposition of licoo2 thin films: A systematic investigation of the deposition parameters,, Journal of the Electrochemical Society, 156 (2009). Google Scholar

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