The effects of boundary roughness on the MHD duct flow with slip hydrodynamic condition
Authors:
Igor Pažanin and Marcone Corrêa Pereira
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35B25, 35B40, 76W05
DOI:
https://doi.org/10.1090/qam/1686
Published electronically:
January 31, 2024
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Abstract: In this paper we present the analytical study of the magnetohydrodynamic (MHD) flow through a rectangular duct driven by the pressure gradient and under the action of the transverse magnetic field. Motivated by various MHD applications in which hydrodynamic slip naturally occur, we prescribe the slipping boundary condition on the upper boundary which contains irregularities as well. Depending on the period of the boundary roughness, we derive three different limit problems by using rigorous analysis in the appropriate functional setting. This approach also enables us to determine the relative contribution of the MHD effect and the slip itself in the governing coupled system satisfied by the velocity and induced magnetic field.
References
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- Eduard Marušić-Paloka, Igor Pažanin, and Marko Radulović, MHD flow through a perturbed channel filled with a porous medium, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2441–2471. MR 4489571, DOI 10.1007/s40840-022-01356-3
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- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
- N. V. Priezjev, A. A Darhuber, and S. M. Trojan, Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamic simulations, Phys. Rew. E 71 (2005), 041608.
- M. Rivero and S. Cuevas, Analysis of the slip condition in magnetohydrodynamic (mhd) pumps, Sens. Actuators B: Chem. 166 (2012), 884–892.
- Münevver Sezgin, Magnetohydrodynamic flow in a rectangular duct, Internat. J. Numer. Methods Fluids 7 (1987), no. 7, 697–718. MR 899414, DOI 10.1002/fld.1650070703
- B. Singh and J. Lal, Finite element method of MHD channel flow with arbitrary wall conductivity, J. Math. Phys. Sci. 18 (1984), 501–516.
- S. Smolentsev, MHD duct flows under hydrodynamic slip condition, Theor. Comput. Fluid Dyn. 23 (2009), 557–570.
- A. Yakhot, M. Arad, and G. Ben-Dor, Numerical investigation of a laminar pulsating flow in a rectangular duct, Int. J. Numer. Methods Fluids 29 (1999), no. (8), 935–950.
- L. Yang and J. Mao, B. Xiong, Numerical simulation of liquid metal MHD flows in a conducting rectangular duct with triangular strips, Fusion Eng. Des. 163 (2021), 112152.
References
- Gleiciane S. Aragão, Antônio L. Pereira, and Marcone C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Methods Appl. Sci. 35 (2012), no. 9, 1110–1116. MR 2931215, DOI 10.1002/mma.2525
- José M. Arrieta and Simone M. Bruschi, Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation, Math. Models Methods Appl. Sci. 17 (2007), no. 10, 1555–1585. MR 2359916, DOI 10.1142/S0218202507002388
- José M. Arrieta and Simone M. Bruschi, Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 2, 327–351. MR 2660861, DOI 10.3934/dcdsb.2010.14.327
- José M. Arrieta, Alexandre N. Carvalho, Marcone C. Pereira, and Ricardo P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal. 74 (2011), no. 15, 5111–5132. MR 2810693, DOI 10.1016/j.na.2011.05.006
- José M. Arrieta and Marcone C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, J. Math. Anal. Appl. 404 (2013), no. 1, 86–104. MR 3061383, DOI 10.1016/j.jmaa.2013.02.061
- Cemre Aydin and Munevver Tezer-Sezgin, DRBEM solution of the Cauchy MHD duct flow with a slipping perturbed boundary, Eng. Anal. Bound. Elem. 93 (2018), 94–104. MR 3809242, DOI 10.1016/j.enganabound.2018.04.007
- Michael J. Bluck and Michael J. Wolfendale, An analytical solution to electromagnetically coupled duct flow in MHD, J. Fluid Mech. 771 (2015), 595–623. MR 3359624, DOI 10.1017/jfm.2015.202
- C. Bozkaya and M. Tezer-Sezgin, Fundamental solution for coupled magnetohydrodynamic flow equations, J. Comput. Appl. Math. 203 (2007), no. 1, 125–144. MR 2313825, DOI 10.1016/j.cam.2006.03.013
- Juan Casado-Díaz, Enrique Fernández-Cara, and Jacques Simon, Why viscous fluids adhere to rugose walls: a mathematical explanation, J. Differential Equations 189 (2003), no. 2, 526–537. MR 1964478, DOI 10.1016/S0022-0396(02)00115-8
- J. Casado-Díaz, M. Luna-Laynez, and F. J. Suárez-Grau, Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall, Math. Models Methods Appl. Sci. 20 (2010), no. 1, 121–156. MR 2606246, DOI 10.1142/S0218202510004179
- J.-H. J. Cho, B. M. Law, and F. Ricutord, Probing nanoscale dipole-dipole interactions by electric force microscopy, Phys. Rev. Lett. 92 (2004), 166101.
- E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations 138 (1997), no. 1, 86–132. MR 1458457, DOI 10.1006/jdeq.1997.3256
- Lazăr Dragoş, Magnetofluid dynamics, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1975. Translated from the Romanian by Vasile Zoiţa; Translation edited by John Hammel. MR 391687
- Hande Fendoğlu, Canan Bozkaya, and Münevver Tezer-Sezgin, MHD flow in a rectangular duct with a perturbed boundary, Comput. Math. Appl. 77 (2019), no. 2, 374–388. MR 3913600, DOI 10.1016/j.camwa.2018.09.040
- J. Hartmann, Hg-dynamics I: theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 15 (1937), 1–28.
- J. C. R. Hunt, Magnetohydrodynamic flow in rectangular ducts, J. Fluid Mech. 21 (1965), 577–590. MR 175474, DOI 10.1017/S0022112065000344
- J. C. R. Hunt and K. Stewartson, Magnetohydrodynamic flow in rectangular ducts. II, J. Fluid Mech. 23 (1965), 563–581. MR 191252, DOI 10.1017/S0022112065001544
- Gregory A. Chechkin, Avner Friedman, and Andrey L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231 (1999), no. 1, 213–234. MR 1676697, DOI 10.1006/jmaa.1998.6226
- U. S. Mahabaleshwar, I. Pažanin, M. Radulović, and F. J. Suarez-Grau, Effects of small boundary perturbation on the MHD duct flow, Theor. Appl. Mech. 44 (2017), 83–101.
- Eduard Marušić-Paloka, Effects of small boundary perturbation on flow of viscous fluid, ZAMM Z. Angew. Math. Mech. 96 (2016), no. 9, 1103–1118. MR 3550599, DOI 10.1002/zamm.201500195
- Eduard Marušić-Paloka and Igor Pažanin, On the Darcy-Brinkman flow through a channel with slightly perturbed boundary, Transp. Porous Media 117 (2017), no. 1, 27–44. MR 3615489, DOI 10.1007/s11242-016-0818-4
- Eduard Marušić-Paloka and Igor Pažanin, Reaction of the fluid flow on time-dependent boundary perturbation, Commun. Pure Appl. Anal. 18 (2019), no. 3, 1227–1246. MR 3917704, DOI 10.3934/cpaa.2019059
- Eduard Marušić-Paloka, Igor Pažanin, and Marko Radulović, MHD flow through a perturbed channel filled with a porous medium, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2441–2471. MR 4489571, DOI 10.1007/s40840-022-01356-3
- E. Marušić-Paloka, I. Pažanin, and M. Radulović, Analytical solution for the magnetohydrodynamic duct flow with slip condition on the perturbed boundary, submitted (2023).
- Ariadne Nogueira, Jean Carlos Nakasato, and Marcone Corrêa Pereira, Concentrated reaction terms on the boundary of rough domains for a quasilinear equation, Appl. Math. Lett. 102 (2020), 106120, 7. MR 4032774, DOI 10.1016/j.aml.2019.106120
- Jean Carlos Nakasato, Igor Pažanin, and Marcone Corrêa Pereira, On the non-isothermal, non-Newtonian Hele-Shaw flows in a domain with rough boundary, J. Math. Anal. Appl. 524 (2023), no. 1, Paper No. 127062, 21. MR 4545175, DOI 10.1016/j.jmaa.2023.127062
- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
- N. V. Priezjev, A. A Darhuber, and S. M. Trojan, Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamic simulations, Phys. Rew. E 71 (2005), 041608.
- M. Rivero and S. Cuevas, Analysis of the slip condition in magnetohydrodynamic (mhd) pumps, Sens. Actuators B: Chem. 166 (2012), 884–892.
- Münevver Sezgin, Magnetohydrodynamic flow in a rectangular duct, Internat. J. Numer. Methods Fluids 7 (1987), no. 7, 697–718. MR 899414, DOI 10.1002/fld.1650070703
- B. Singh and J. Lal, Finite element method of MHD channel flow with arbitrary wall conductivity, J. Math. Phys. Sci. 18 (1984), 501–516.
- S. Smolentsev, MHD duct flows under hydrodynamic slip condition, Theor. Comput. Fluid Dyn. 23 (2009), 557–570.
- A. Yakhot, M. Arad, and G. Ben-Dor, Numerical investigation of a laminar pulsating flow in a rectangular duct, Int. J. Numer. Methods Fluids 29 (1999), no. (8), 935–950.
- L. Yang and J. Mao, B. Xiong, Numerical simulation of liquid metal MHD flows in a conducting rectangular duct with triangular strips, Fusion Eng. Des. 163 (2021), 112152.
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Additional Information
Igor Pažanin
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
ORCID:
0000-0003-3384-5184
Email:
pazanin@math.hr
Marcone Corrêa Pereira
Affiliation:
Department of Applied Mathematics, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, Brazil
MR Author ID:
710820
Email:
marcone@ime.usp.br
Keywords:
MHD duct flow,
rough boundary,
hydrodynamic slip,
analytical results
Received by editor(s):
October 17, 2023
Received by editor(s) in revised form:
December 29, 2023
Published electronically:
January 31, 2024
Additional Notes:
The authors were supported by the Croatian Science Foundation under the project Multiscale problems in fluid mechanics - MultiFM (IP-2019-04-1140). Also, the second author was partially supported by CNPq 308950/2020-8, FAPESP 2020/04813-0 and 2020/14075-6 (Brazil).
Igor Pažanin is the corresponding author
Article copyright:
© Copyright 2024
Brown University