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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:
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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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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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