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On the Darcy–Brinkman Flow Through a Channel with Slightly Perturbed Boundary

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Abstract

The goal of this paper is to study the effects of a slightly perturbed boundary on the Darcy–Brinkman flow through a porous channel. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter \(\epsilon \) and arbitrary smooth function h. Using asymptotic analysis with respect to \(\epsilon \), the effective model has been formally derived. Being in the form of the explicit formulae for the velocity and pressure, the asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution.

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References

  • Achdou, Y., Pironneau, O., Valentin, F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147, 187–218 (1998)

    Article  Google Scholar 

  • Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes I. Abstract Framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113, 209–259 (1991)

    Article  Google Scholar 

  • Ammari, H., Kang, H., Lee, H., Lim, J.: Boundary perturbations due to the presence of small linear cracks in an elastic body. J. Elast. 113, 75–91 (2013)

    Article  Google Scholar 

  • Beretta, E., Francini, E.: An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities. SIAM J. Math. Anal. 38, 1249–1261 (2006)

    Article  Google Scholar 

  • Bresch, D., Choquet, C., Chupin, L., Colin, T., Gisclon, M.: Roughness-induced effect at main order on the Reynolds approximation. SIAM Multiscale Model. Simul. 8, 997–1017 (2010)

    Article  Google Scholar 

  • Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1947)

    Google Scholar 

  • Darcy, H.: Les fontaines publiques de la ville de Dijon. Victor Darmon, Paris (1856)

    Google Scholar 

  • Ekneligoda, T.C., Zimmerman, R.W.: Boundary perturbation solution for nearly circular holes and rigid inclusions in an infinite elastic medium. J. Appl. Mech. 75, 011015 (2008)

    Article  Google Scholar 

  • Galdi, G.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I. Springer, New York (1994)

    Google Scholar 

  • Gipouloux, O., Marušić-Paloka, E.: Asymptotic behavior of the incompressible Newtonian flow through thin constricted fracture. In: Antonić, N., van Duijn, C.J., Jager, W., Mikelić, A. (eds.) Multiscale Problems in Science and Technology, pp. 189–202. Springer, Berlin (2002)

  • Gray, D.D., Ogretim, E., Bromhal, G.S.: Darcy flow in a wavy channel filled with a porous medium. Transp. Porous Media 98, 743–753 (2013)

    Article  Google Scholar 

  • Hannukainen, A., Juntunen, M., Stenberg, R.: Computations with finite element methods for the Brinkman problem. Comput. Geosci. 15, 155–166 (2011)

    Article  Google Scholar 

  • Jäger, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001)

    Article  Google Scholar 

  • Levy, T.: Fluid flow through an array of fixed particles. Int. J. Eng. Sci. 21, 11–23 (1983)

    Article  Google Scholar 

  • Marušić-Paloka, E.: Effects of small boundary perturbation on flow of viscous fluid. ZAMM J. Appl. Math. Mech. 96, 1103–1118 (2016)

    Article  Google Scholar 

  • Marušić-Paloka, E., Pažanin, I., Marušić, S.: Comparison between Darcy and Brinkman laws in a fracture. Appl. Math. Comput. 218, 7538–7545 (2012)

    Google Scholar 

  • Marušić-Paloka, E., Pažanin, I., Radulović, M.: Flow of a micropolar fluid through a channel with small boundary perturbation. Z. Naturforsch. A 71, 607–619 (2016)

    Google Scholar 

  • Ng, C.-O., Wang, C.Y.: Darcy–Brinkman flow through a corrugated channel. Transp. Porous Media 85, 605–618 (2010)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media, 2nd edn. Springer, New York (1999)

    Book  Google Scholar 

  • Pažanin, I., Suárez-Grau, F.J.: Analysis of the thin film flow in a rough thin domain filled with micropolar fluid. Comput. Math. Appl. 68, 1915–1932 (2014)

    Article  Google Scholar 

  • Sanchez-Palencia, E.: On the asymptotics of the fluid flow past an array of fixed obstacles. Int. J. Eng. Sci. 20, 1291–1301 (1982)

    Article  Google Scholar 

  • Sisavath, S., Jing, X., Zimmerman, R.W.: Creeping flow through a pipe of varying radius. Phys. Fluids 12, 2762–2772 (2001)

    Article  Google Scholar 

  • Xu, X., Zhang, S.: A new divergence-free interpolation operator with applications to the Darcy–Stokes–Brinkman equations. SIAM J. Sci. Comput. 32, 855–874 (2010)

    Article  Google Scholar 

  • Yu, L.H., Wang, C.Y.: Darcy–Brinkman flow through a bumpy channel. Transp. Porous Media 97, 281–294 (2013)

    Article  Google Scholar 

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Acknowledgements

The authors have been supported by the Croatian Science Foundation (Project 3955: Mathematical modeling and numerical simulations of processes in thin or porous domains). The authors would like to thank the referee for his/her comments and suggestions that helped to significantly improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Eduard Marušić-Paloka & Igor Pažanin

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  1. Eduard Marušić-Paloka
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  2. Igor Pažanin
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Correspondence to Igor Pažanin.

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Marušić-Paloka, E., Pažanin, I. On the Darcy–Brinkman Flow Through a Channel with Slightly Perturbed Boundary. Transp Porous Med 117, 27–44 (2017). https://doi.org/10.1007/s11242-016-0818-4

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  • DOI: https://doi.org/10.1007/s11242-016-0818-4

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