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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

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