# American Institute of Mathematical Sciences

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

## Reaction of the fluid flow on time-dependent boundary perturbation

 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.

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. [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.

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##### 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.
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