-
Previous Article
Symmetry of solutions to a class of Monge-Ampère equations
- CPAA Home
- This Issue
-
Next Article
Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian
Reaction of the fluid flow on time-dependent boundary perturbation
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia |
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.
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. |
[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. |
[3] |
G. Bayada and M. Chambat,
New models in the theory of the hydrodynamic lubrication of rough surfaces, J. Tribol., 110 (1988), 402-407.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
O. Damak and E. Hadj-Taieb, Waterhammer in flexible pipes, in Design and modeling of mechanical systems, Springer, 373-380, 2013. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[18] |
C. Peskin,
Flow patterns around heart valves, J. Comput. Phys., 10 (1972), 252-271.
|
[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.
|
[20] |
A. Szeri, Fluid Film Lubrication, Cambridge university press, 2nd edition 2010. |
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. |
[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. |
[3] |
G. Bayada and M. Chambat,
New models in the theory of the hydrodynamic lubrication of rough surfaces, J. Tribol., 110 (1988), 402-407.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
O. Damak and E. Hadj-Taieb, Waterhammer in flexible pipes, in Design and modeling of mechanical systems, Springer, 373-380, 2013. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[18] |
C. Peskin,
Flow patterns around heart valves, J. Comput. Phys., 10 (1972), 252-271.
|
[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.
|
[20] |
A. Szeri, Fluid Film Lubrication, Cambridge university press, 2nd edition 2010. |
[1] |
Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 |
[4] |
Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039 |
[5] |
Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 |
[6] |
Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178 |
[7] |
Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044 |
[8] |
Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313 |
[9] |
Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1127-1141. doi: 10.3934/dcdss.2021090 |
[10] |
Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 |
[11] |
Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065 |
[12] |
Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127 |
[13] |
Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077 |
[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 and Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002 |
[15] |
Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 |
[16] |
Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 |
[17] |
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 |
[18] |
Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 |
[19] |
Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081 |
[20] |
Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic and Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]