American Institute of Mathematical Sciences

Advanced
May  2019, 18(3): 1227-1246. doi: 10.3934/cpaa.2019059

Reaction of the fluid flow on time-dependent boundary perturbation

Eduard Marušić-Paloka and  Igor Pažanin

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

Received  March 2018 Revised  July 2018 Published  November 2018

The aim of this paper is to investigate the effects of time-dependent boundary perturbation on the flow of a viscous fluid via asymptotic analysis. We start from a simple rectangular domain and then perturb the upper part of its boundary by the product of a small parameter $\varepsilon$ and some smooth function $h(x, t)$. The complete asymptotic expansion (in powers of $\varepsilon$) of the solution of the evolutionary Stokes system has been constructed. The convergence of the expansion has been proved providing the rigorous justification of the formally derived asymptotic model.

Keywords: time-dependent boundary perturbation, evolutionary Stokes system, moving coordinates, asymptotic expansion, error estimates.
Mathematics Subject Classification: 76D07, 35B25, 35B40, 76M45.
Citation: Eduard Marušić-Paloka, Igor Pažanin. Reaction of the fluid flow on time-dependent boundary perturbation. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1227-1246. doi: 10.3934/cpaa.2019059
References:
[1]

Y. Achdou, O. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Computer. Phys, 147 (1998), 187-218.  doi: 10.1006/jcph.1998.6088.  Google Scholar

[2]

K. Amedodji, G. Bayada and M. Chambat, On the unsteady Navier-Stokes equations in a time-moving domain with velocity-pressure boundary conditions, Nonlinear Anal. TMA, 49 (2002), 565-587.  doi: 10.1016/S0362-546X(01)00123-7.  Google Scholar

[3]

G. Bayada and M. Chambat, New models in the theory of the hydrodynamic lubrication of rough surfaces, J. Tribol., 110 (1988), 402-407.   Google Scholar

[4]

N. Benhaboucha, M. Chambat and I. Ciuperca, Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary, Quart. Appl. Math., 63 (2005), 369-400.  doi: 10.1090/S0033-569X-05-00963-3.  Google Scholar

[5]

J. M. Bernard, Time-dependent Stokes and Navier-Stokes problems with boundary conditions, Nonlinear Anal. RWA, 4 (2003), 805-839.  doi: 10.1016/S1468-1218(03)00016-6.  Google Scholar

[6]

D. Bresch, C. Choquet, L. Chupin, T. Colin and M. Gisclon, Roughness-induced effect at main order on the Reynolds approximation, SIAM Multiscale Model. Simul., 8 (2010), 997-1017.  doi: 10.1137/090754996.  Google Scholar

[7]

S. Čanić, A. Mikelić, D. Lamponi and J. Tambača, Self-consistent effective equations modeling the blood flow in medium-to-large compliant arteries, SIAM Multiscale Model. Simul., 3 (2005), 559-596.  doi: 10.1137/030602605.  Google Scholar

[8]

L. Chupin and S. Martin, Rigorous derivation of the thin film approximation with roughness-induced correctors, SIAM J. Math. Anal., 44 (2012), 3041-3070.  doi: 10.1137/110824371.  Google Scholar

[9]

O. Damak and E. Hadj-Taieb, Waterhammer in flexible pipes, in Design and modeling of mechanical systems, Springer, 373-380, 2013. Google Scholar

[10]

D. Henry, Perturbation of the Boundary in Boundary-value Problems, London mathematical society lecture notes series, 318, Cambridge university press, 2005. doi: 10.1017/CBO9780511546730.  Google Scholar

[11]

Jäger and A. W. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Diff. Equations, 170 (2001), 96-122.  doi: 10.1006/jdeq.2000.3814.  Google Scholar

[12]

H. Le Dret, R. Lewandowski, D. Priour and F. Changenau, Numerical simulations of a cod end net Part 1: equilibrium in a uniform flow, Elasticity J., 76 (2004), 139-162.  doi: 10.1007/s10659-004-6668-2.  Google Scholar

[13]

E. Marušić-Paloka, Effects of small boundary perturbation on flow of viscous fluid, ZAMM - J. Appl. Math. Mech., 96 (2016), 1103-1118.  doi: 10.1002/zamm.201500195.  Google Scholar

[14]

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

[15]

E. Marušić-Paloka and I. Pažanin, On the Darcy-Brinkman flow through a channel with slightly perturbed boundary, Transp. Porous. Med., 117 (2017), 27-44.  doi: 10.1007/s11242-016-0818-4.  Google Scholar

[16]

I. Pažanin, A note on the solute dispersion in a porous medium, B. Malays. Math. Sci. So., (2017).  doi: 10.1007/s40840-017-00508-6.  Google Scholar

[17]

I. Pažanin and F. J. Suárez-Grau, Analysis of the thin film flow in a rough thin domain filled with micropolar fluid, Comput. Math. Appl., 68 (2014), 1915-1932.  doi: 10.1016/j.camwa.2014.10.003.  Google Scholar

[18]

C. Peskin, Flow patterns around heart valves, J. Comput. Phys., 10 (1972), 252-271.   Google Scholar

[19]

G. D. Rigby, L. Strezov, C. D. Rilley, S. D. Sciffer, J. A. Lucas and G. M. Evans, Hydrodynamics of fluid flow approaching a moving bounday, Metall. Mater. Trans. B, 31 (2000), 1117-1123.   Google Scholar

[20]

A. Szeri, Fluid Film Lubrication, Cambridge university press, 2nd edition 2010. Google Scholar

show all references

References:
[1]

Y. Achdou, O. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Computer. Phys, 147 (1998), 187-218.  doi: 10.1006/jcph.1998.6088.  Google Scholar

[2]

K. Amedodji, G. Bayada and M. Chambat, On the unsteady Navier-Stokes equations in a time-moving domain with velocity-pressure boundary conditions, Nonlinear Anal. TMA, 49 (2002), 565-587.  doi: 10.1016/S0362-546X(01)00123-7.  Google Scholar

[3]

G. Bayada and M. Chambat, New models in the theory of the hydrodynamic lubrication of rough surfaces, J. Tribol., 110 (1988), 402-407.   Google Scholar

[4]

N. Benhaboucha, M. Chambat and I. Ciuperca, Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary, Quart. Appl. Math., 63 (2005), 369-400.  doi: 10.1090/S0033-569X-05-00963-3.  Google Scholar

[5]

J. M. Bernard, Time-dependent Stokes and Navier-Stokes problems with boundary conditions, Nonlinear Anal. RWA, 4 (2003), 805-839.  doi: 10.1016/S1468-1218(03)00016-6.  Google Scholar

[6]

D. Bresch, C. Choquet, L. Chupin, T. Colin and M. Gisclon, Roughness-induced effect at main order on the Reynolds approximation, SIAM Multiscale Model. Simul., 8 (2010), 997-1017.  doi: 10.1137/090754996.  Google Scholar

[7]

S. Čanić, A. Mikelić, D. Lamponi and J. Tambača, Self-consistent effective equations modeling the blood flow in medium-to-large compliant arteries, SIAM Multiscale Model. Simul., 3 (2005), 559-596.  doi: 10.1137/030602605.  Google Scholar

[8]

L. Chupin and S. Martin, Rigorous derivation of the thin film approximation with roughness-induced correctors, SIAM J. Math. Anal., 44 (2012), 3041-3070.  doi: 10.1137/110824371.  Google Scholar

[9]

O. Damak and E. Hadj-Taieb, Waterhammer in flexible pipes, in Design and modeling of mechanical systems, Springer, 373-380, 2013. Google Scholar

[10]

D. Henry, Perturbation of the Boundary in Boundary-value Problems, London mathematical society lecture notes series, 318, Cambridge university press, 2005. doi: 10.1017/CBO9780511546730.  Google Scholar

[11]

Jäger and A. W. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Diff. Equations, 170 (2001), 96-122.  doi: 10.1006/jdeq.2000.3814.  Google Scholar

[12]

H. Le Dret, R. Lewandowski, D. Priour and F. Changenau, Numerical simulations of a cod end net Part 1: equilibrium in a uniform flow, Elasticity J., 76 (2004), 139-162.  doi: 10.1007/s10659-004-6668-2.  Google Scholar

[13]

E. Marušić-Paloka, Effects of small boundary perturbation on flow of viscous fluid, ZAMM - J. Appl. Math. Mech., 96 (2016), 1103-1118.  doi: 10.1002/zamm.201500195.  Google Scholar

[14]

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

[15]

E. Marušić-Paloka and I. Pažanin, On the Darcy-Brinkman flow through a channel with slightly perturbed boundary, Transp. Porous. Med., 117 (2017), 27-44.  doi: 10.1007/s11242-016-0818-4.  Google Scholar

[16]

I. Pažanin, A note on the solute dispersion in a porous medium, B. Malays. Math. Sci. So., (2017).  doi: 10.1007/s40840-017-00508-6.  Google Scholar

[17]

I. Pažanin and F. J. Suárez-Grau, Analysis of the thin film flow in a rough thin domain filled with micropolar fluid, Comput. Math. Appl., 68 (2014), 1915-1932.  doi: 10.1016/j.camwa.2014.10.003.  Google Scholar

[18]

C. Peskin, Flow patterns around heart valves, J. Comput. Phys., 10 (1972), 252-271.   Google Scholar

[19]

G. D. Rigby, L. Strezov, C. D. Rilley, S. D. Sciffer, J. A. Lucas and G. M. Evans, Hydrodynamics of fluid flow approaching a moving bounday, Metall. Mater. Trans. B, 31 (2000), 1117-1123.   Google Scholar

[20]

A. Szeri, Fluid Film Lubrication, Cambridge university press, 2nd edition 2010. Google Scholar

[1]

Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
[2]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659
[3]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[4]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[5]

Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313
[6]

Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127
[7]

Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077
[8]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[9]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[10]

Masahiro Kubo, Noriaki Yamazaki. Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. Conference Publications, 2007, 2007 (Special) : 614-623. doi: 10.3934/proc.2007.2007.614
[11]

Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591
[12]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
[13]

Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011
[14]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
[15]

Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
[16]

Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
[17]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[18]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
[19]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
[20]

Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (61)
  • HTML views (178)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences