Micropolar fluid-thin elastic structure interaction: variational analysis
Abstract
We consider the non-stationary flow of a micropolar fluid in a thin channel with an impervious wall and an elastic stiff wall, motivated by applications to blood flows through arteries. We assume that the elastic wall is composed of several layers with different elastic characteristics and that the domains occupied by the two media are infinite in one direction and the problem is periodic in the same direction. We provide a complete variational analysis of the two dimensional interaction between the micropolar fluid and the stratified elastic layer. For a suitable data regularity, we prove the existence, the uniqueness and the regularity of the solution to the variational problem associated to the physical system. Increasing the data regularity, we prove that the fluid pressure is unique, we obtain additional regularity for all the unknown functions and we show that the solution to the variational problem is solution for the physical system, as well.
Keyword : fluid-structure interaction, micropolar fluid, stratified elastic layer, periodic flow, existence, uniqueness, regularity
How to Cite
Panasenko, G., Paoli, L., & Stavre, R. (2024). Micropolar fluid-thin elastic structure interaction: variational analysis. Mathematical Modelling and Analysis, 29(4), 641–668. https://doi.org/10.3846/mma.2024.20053
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G.P. Panasenko and R. Stavre. Viscous fluid-thin elastic plate interaction: asymptotic analysis with respect to the rigidity and density of the plate. Appl. Math. Optim., 81:141–194, 2020. https://doi.org/10.1007/s00245-018-9480-2
I. Pažanin and M. Radulović. Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe. Appl. Anal., 99(12):2045–2092, 2020. https://doi.org/10.1080/00036811.2018.1553036
T. Richter. Fluid-structure interactions, Models, analysis and finite elements, volume 118. Springer, 2017.
J. Simon. Compact sets in the space Lp(0,t;b). Ann. Mat. Pura Appl., 146(1):65–96, 1986. https://doi.org/10.1007/BF01762360
R. Stavre. Optimization and numerical approximation for micropolar fluids. Num. Funct. Anal. Optimiz., 24(3–4):223–241, 2003. https://doi.org/10.1081/NFA-120022919
R. Stavre. A boundary control problem for the blood flow in venous insufficiency. the general case. Nonlin. Anal. Real World Appl., 29:98–116, 2016. https://doi.org/10.1016/j.nonrwa.2015.11.003
R. Stavre. Optimization of the blood pressure with the control in coefficients. Evol. Equ. Control Theory, 9(1):131–151, 2020. ttps://doi.org/10.3934/eect.2020019
R. Temam. Navier-Stokes equations; theory & numerical analysis. Amsterdam, North-Holland, 1984.
D. Wang, J. Wu and Z. Ye. Global regularity of the three-dimensional fractional micropolar equations. J. Math. Fluid Mech., 22(28), 2020. https://doi.org/10.1007/s00021-020-0490-x
L. Xue. Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math. Methods Appl. Sci., 34(14):1760–1777, 2011. https://doi.org/10.1002/mma.1491
A. Zaman, N. Ali and O. Anwar Beg. Numerical simulation of the unsteady micropolar hemodynamics in a tapered catheterized artery with a combination of stenosis and aneurysm. Medical & Biological Engineering & Computing, 54:1423 –1436, 2016. https://doi.org/10.1007/s11517-015-1415-3
Y. Zócalo, D. Bia, E.I. Cabrera-Fischer, S. Wray, C. Galli and R.L. Armentano. Structural and functional properties of venous wall: relationship between elastin, collagen, and smooth muscle components and viscoelastic properties. International Scholarly Research Notices, 2013(1):906031, 2013. https://doi.org/10.1155/2013/906031
E.L. Aero and E.V. Kuvshinsky. Main equations of theory of elastic media with rotational interaction of particles. Physics of Solids (Fizika Tverdogo Tela), 2(7):1399 –1409, 1960.
M. Benešs and I. Pažanin. Effective flow of incompressible micropolar fluid through a system of thin pipes. Acta Appl. Math., 143(1):29–43, 2016. https://doi.org/10.1007/s10440-015-0026-1
M. Beneš, I. Pažanin, M. Radulović and B. Rukavina. Nonzero boundary conditions for the unsteady micropolar pipe flow: Wellposedness and asymptotics. Appl. Math. Comput., 427(127184):1–22, 2022. https://doi.org/10.1016/j.amc.2022.127184
L. Bociu, S. Čanić, B. Muha and J.T. Webster. Multilayered poroelasticity interacting with Stokes flow. SIAM J. Math. Anal., 53(6):6243–6279, 2021. https://doi.org/10.1137/20M1382520
M. Boukrouche and L. Paoli. Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary. SIAM J. Math. Anal., 44(2):1211– 1256, 2012. https://doi.org/10.1137/110837772
M. Boukrouche, L. Paoli and F. Ziane. Unsteady micropolar fluid flow in a thin domain with Tresca fluid-solid interface law. Comput. Math. Appl., 77(11):2917– 2932, 2019. https://doi.org/10.1016/j.camwa.2018.08.071
M. Boukrouche, L. Paoli and F. Ziane. Micropolar fluid flow in a thick domain with multiscale oscillating roughness and friction boundary conditions. J. Math. Anal. Appl., 495(1):124688, 2021. https://doi.org/10.1016/j.jmaa.2020.124688
Q. Chen and C. Miao. Global well-posedness for the micropolar fluid system in critical Besov spaces. J. Differ. Equ., 252(3):2698–2724, 2012. https://doi.org/10.1016/j.jde.2011.09.035
B. Dong and Z. Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete Contin. Dyn. Syst., 23(3):765–784, 2009. https://doi.org/10.3934/dcds.2009.23.765
B. Dong, J. Li and J. Wu. Global well-posedness and large-time decay for the 2D micropolar equations. J. Differ. Equ., 262(6):3488–3523, 2017. https://doi.org/10.1016/j.jde.2016.11.029
B. Dong, J. Wu, X. Xu and Z. Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete Contin. Dyn. Syst., 38(8):4133– 4162, 2018. https://doi.org/10.3934/dcds.2018180
B. Dong and Z. Zhang. Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ., 249(1):200–213, 2010. https://doi.org/10.1016/j.jde.2010.03.016
D. Dupuy, G.P. Panasenko and R. Stavre. Asymptotic solution for a micropolar flow in a curvilinear channel. Z. Angew. Math. Mech., 88(10):793–807, 2008. https://doi.org/10.1002/zamm.200700136
A.C. Eringen. Theory of micropolar fluids. J. Math. Mech., 16(1):1–18, 1967. https://doi.org/10.1512/iumj.1967.16.16001
C. Grandmont and F. Vergnet. Existence for a quasi-static interaction problem between a viscous fluid and an active structure. J. Math. Fluid Mech., 23(45), 2021. https://doi.org/10.1007/s00021-020-00552-0
Q. Jiu, J. Liu, J. Wu and H. Yu. On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation. Z. Angew. Math. Phys., 68(107), 2017. https://doi.org/10.1007/s00033-017-0855-z
J. Liu and S. Wang. Initial-boundary value problem for 2D micropolar equations without angular viscosity. Commun. Math. Sci., 16(8):2147–2165, 2018. https://doi.org/10.4310/CMS.2018.v16.n8.a5
G. Lukaszewicz. Micropolar Fluids: Theory and Applications. Birkha¨user, Boston, Basel, Berlin, 1999.
G. Panasenko and R. Stavre. Three dimensional asymptotic analysis of an axisymmetric flow in a thin tube with thin stiff elastic wall. J. Math. Fluid Mech., 22(20), 2020. https://doi.org/10.1007/s00021-020-0484-8
G.P. Panasenko and R. Stavre. Viscous fluid-thin elastic plate interaction: asymptotic analysis with respect to the rigidity and density of the plate. Appl. Math. Optim., 81:141–194, 2020. https://doi.org/10.1007/s00245-018-9480-2
I. Pažanin and M. Radulović. Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe. Appl. Anal., 99(12):2045–2092, 2020. https://doi.org/10.1080/00036811.2018.1553036
T. Richter. Fluid-structure interactions, Models, analysis and finite elements, volume 118. Springer, 2017.
J. Simon. Compact sets in the space Lp(0,t;b). Ann. Mat. Pura Appl., 146(1):65–96, 1986. https://doi.org/10.1007/BF01762360
R. Stavre. Optimization and numerical approximation for micropolar fluids. Num. Funct. Anal. Optimiz., 24(3–4):223–241, 2003. https://doi.org/10.1081/NFA-120022919
R. Stavre. A boundary control problem for the blood flow in venous insufficiency. the general case. Nonlin. Anal. Real World Appl., 29:98–116, 2016. https://doi.org/10.1016/j.nonrwa.2015.11.003
R. Stavre. Optimization of the blood pressure with the control in coefficients. Evol. Equ. Control Theory, 9(1):131–151, 2020. ttps://doi.org/10.3934/eect.2020019
R. Temam. Navier-Stokes equations; theory & numerical analysis. Amsterdam, North-Holland, 1984.
D. Wang, J. Wu and Z. Ye. Global regularity of the three-dimensional fractional micropolar equations. J. Math. Fluid Mech., 22(28), 2020. https://doi.org/10.1007/s00021-020-0490-x
L. Xue. Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math. Methods Appl. Sci., 34(14):1760–1777, 2011. https://doi.org/10.1002/mma.1491
A. Zaman, N. Ali and O. Anwar Beg. Numerical simulation of the unsteady micropolar hemodynamics in a tapered catheterized artery with a combination of stenosis and aneurysm. Medical & Biological Engineering & Computing, 54:1423 –1436, 2016. https://doi.org/10.1007/s11517-015-1415-3
Y. Zócalo, D. Bia, E.I. Cabrera-Fischer, S. Wray, C. Galli and R.L. Armentano. Structural and functional properties of venous wall: relationship between elastin, collagen, and smooth muscle components and viscoelastic properties. International Scholarly Research Notices, 2013(1):906031, 2013. https://doi.org/10.1155/2013/906031