
- Previous Article
- JCD Home
- This Issue
-
Next Article
A functional analytic approach to validated numerics for eigenvalues of delay equations
On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis
Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia |
This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Bénard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of $ \mathbb{R}^3 $ and the convergence, in the norm of $ L^{2}(\Omega), $ to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.
References:
[1] |
H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Revue Gén. Sci. Pures Appl., 11 (1900), 1261–1271, 1309–1328. Google Scholar |
[2] |
M. J. Block,
Surface tension as the cause of Bénard cells and surface deformation of a liquid film, Nature, 178 (1965), 650-651.
doi: 10.1038/178650a0. |
[3] |
T. Chacón-Rebollo and F. Guillén-González,
An intrinsic analysis of the hydrostatic approximation of Navier-Stokes equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 841-846.
doi: 10.1016/S0764-4442(00)00266-4. |
[4] |
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of
Monographs on Physics Clarendon Press, Oxford, 1961. |
[5] |
P. Colinet, J. C. Legros and M. G. Velarde, Nonlinear Dynamics of Surface-Tension-Driven Instabilities, WILEY-VCH Verlag Berlin GmbH, Berlin, 2001.
doi: 10.1002/3527603115. |
[6] |
P. C. Dauby and G. Lebon,
Bénard-Marangoni instability in rigid rectangular containers, J. Fluid Mech., 329 (1996), 25-64.
doi: 10.1017/S0022112096008816. |
[7] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
On the stability problem for the Boussinesq equations in weak-$L^p$ spaces, Commun. Pure Appl. Anal., 9 (2010), 667-684.
doi: 10.3934/cpaa.2010.9.667. |
[8] |
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[9] |
F. Guillén-González and M. V. Redondo-Neble,
Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations, Appl. Numer. Math., 111 (2017), 219-245.
doi: 10.1016/j.apnum.2016.09.011. |
[10] |
J. G. Heywood and R. Rannacher,
Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.
doi: 10.1137/0727022. |
[11] |
S. Hoyas, H. Herrero and A. M. Mancho,
Thermal convection in a cylindrical annulus heated laterally, J. Phys. A, 35 (2002), 4067-4083.
doi: 10.1088/0305-4470/35/18/306. |
[12] |
S. Hoyas, Estudio Teórico y Numérico de un Problema de Convección de Bénard-Marangoni en un Anillo, Ph.D thesis, U. Complutense de Madrid, 2003. Google Scholar |
[13] |
E. L. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge Monographs on Mechanics
and Applied Mathematics, Cambridge University Press, New York, 1993. |
[14] |
A. M. Kvarving, T. Bjontegaard and E. M. Ronquist,
On pattern selection in three-dimensional Bénard-Marangoni flows, Commun. Comput. Phys., 11 (2012), 893-924.
doi: 10.4208/cicp.280610.060411a. |
[15] |
M. Lappa, Thermal Convection: Patterns, Evolution and Stability, John Wiley & Sons, Ltd., Chichester, 2010. |
[16] |
G. Lebon, D. Jou and J. Casas-Vázquez, Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-74252-4. |
[17] |
S. A. Lorca and J. L. Boldrini,
Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406.
doi: 10.1006/jdeq.1996.0016. |
[18] |
S. A. Lorca and J. L. Boldrini,
The initial value problem for the generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480.
doi: 10.1016/S0362-546X(97)00635-4. |
[19] |
C. Marangoni, Sull'espansione delle gocce di un liquido gallegianti sulla superficie di altro liquido. Pavia: Tipografia dei fratelli Fusi, Ann. Phys. Chem., 143 (1871), 337-354. Google Scholar |
[20] |
I. Mutabazi, J. E. Wesfreid and E. Guyon, Dynamics of Spatio-Temporal Cellular Structures. Henri Bénard Centenary Review, Springer, New York, 2006.
doi: 10.1007/b106790. |
[21] |
J. Necas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-10455-8. |
[22] |
D. A. Nield,
Surface tension and buoyancy effects in cellular convection, J. Fluid Mech., 19 (1964), 341-352.
doi: 10.1017/S0022112064000763. |
[23] |
R. Pardo, H. Herrero and S. Hoyas,
Theoretical study of a Bénard-Marangoni problem, Journal of Mathematical Analysis and Applications, 376 (2011), 231-246.
doi: 10.1016/j.jmaa.2010.10.064. |
[24] |
J. R. A. Pearson,
On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.
doi: 10.1017/S0022112058000616. |
[25] |
P. H. Rabinowitz,
Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Ration. Mech. Anal., 29 (1968), 32-57.
doi: 10.1007/BF00256457. |
[26] |
L. Rayleigh, On convection currents in horizontal layer of fluid when the higher temperature
is on the under side, Philos. Mag. Ser. (6), 32 (1916), 529–546.
doi: 10.1080/14786441608635602. |
[27] |
M. A. Rodríguez-Bellido, M. A. Rojas-Medar and E. J. Villamizar-Roa,
The Boussinesq system with mixed nonsmooth boundary data, C. R. Math. Acad. Sci. Paris, 343 (2006), 191-196.
doi: 10.1016/j.crma.2006.06.011. |
[28] |
D. A. Rueda-Gómez and E. J. Villamizar-Roa,
On the Rayleigh-Bénard-Marangoni system and a related optimal control problem, Computers and Mathematics with Applications, 74 (2017), 2969-2991.
doi: 10.1016/j.camwa.2017.07.038. |
[29] |
J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987),
65–96.
doi: 10.1007/BF01762360. |
[30] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |
show all references
References:
[1] |
H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Revue Gén. Sci. Pures Appl., 11 (1900), 1261–1271, 1309–1328. Google Scholar |
[2] |
M. J. Block,
Surface tension as the cause of Bénard cells and surface deformation of a liquid film, Nature, 178 (1965), 650-651.
doi: 10.1038/178650a0. |
[3] |
T. Chacón-Rebollo and F. Guillén-González,
An intrinsic analysis of the hydrostatic approximation of Navier-Stokes equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 841-846.
doi: 10.1016/S0764-4442(00)00266-4. |
[4] |
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of
Monographs on Physics Clarendon Press, Oxford, 1961. |
[5] |
P. Colinet, J. C. Legros and M. G. Velarde, Nonlinear Dynamics of Surface-Tension-Driven Instabilities, WILEY-VCH Verlag Berlin GmbH, Berlin, 2001.
doi: 10.1002/3527603115. |
[6] |
P. C. Dauby and G. Lebon,
Bénard-Marangoni instability in rigid rectangular containers, J. Fluid Mech., 329 (1996), 25-64.
doi: 10.1017/S0022112096008816. |
[7] |
L. C. F. Ferreira and E. J. Villamizar-Roa,
On the stability problem for the Boussinesq equations in weak-$L^p$ spaces, Commun. Pure Appl. Anal., 9 (2010), 667-684.
doi: 10.3934/cpaa.2010.9.667. |
[8] |
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[9] |
F. Guillén-González and M. V. Redondo-Neble,
Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations, Appl. Numer. Math., 111 (2017), 219-245.
doi: 10.1016/j.apnum.2016.09.011. |
[10] |
J. G. Heywood and R. Rannacher,
Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.
doi: 10.1137/0727022. |
[11] |
S. Hoyas, H. Herrero and A. M. Mancho,
Thermal convection in a cylindrical annulus heated laterally, J. Phys. A, 35 (2002), 4067-4083.
doi: 10.1088/0305-4470/35/18/306. |
[12] |
S. Hoyas, Estudio Teórico y Numérico de un Problema de Convección de Bénard-Marangoni en un Anillo, Ph.D thesis, U. Complutense de Madrid, 2003. Google Scholar |
[13] |
E. L. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge Monographs on Mechanics
and Applied Mathematics, Cambridge University Press, New York, 1993. |
[14] |
A. M. Kvarving, T. Bjontegaard and E. M. Ronquist,
On pattern selection in three-dimensional Bénard-Marangoni flows, Commun. Comput. Phys., 11 (2012), 893-924.
doi: 10.4208/cicp.280610.060411a. |
[15] |
M. Lappa, Thermal Convection: Patterns, Evolution and Stability, John Wiley & Sons, Ltd., Chichester, 2010. |
[16] |
G. Lebon, D. Jou and J. Casas-Vázquez, Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-74252-4. |
[17] |
S. A. Lorca and J. L. Boldrini,
Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406.
doi: 10.1006/jdeq.1996.0016. |
[18] |
S. A. Lorca and J. L. Boldrini,
The initial value problem for the generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480.
doi: 10.1016/S0362-546X(97)00635-4. |
[19] |
C. Marangoni, Sull'espansione delle gocce di un liquido gallegianti sulla superficie di altro liquido. Pavia: Tipografia dei fratelli Fusi, Ann. Phys. Chem., 143 (1871), 337-354. Google Scholar |
[20] |
I. Mutabazi, J. E. Wesfreid and E. Guyon, Dynamics of Spatio-Temporal Cellular Structures. Henri Bénard Centenary Review, Springer, New York, 2006.
doi: 10.1007/b106790. |
[21] |
J. Necas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-10455-8. |
[22] |
D. A. Nield,
Surface tension and buoyancy effects in cellular convection, J. Fluid Mech., 19 (1964), 341-352.
doi: 10.1017/S0022112064000763. |
[23] |
R. Pardo, H. Herrero and S. Hoyas,
Theoretical study of a Bénard-Marangoni problem, Journal of Mathematical Analysis and Applications, 376 (2011), 231-246.
doi: 10.1016/j.jmaa.2010.10.064. |
[24] |
J. R. A. Pearson,
On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.
doi: 10.1017/S0022112058000616. |
[25] |
P. H. Rabinowitz,
Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Ration. Mech. Anal., 29 (1968), 32-57.
doi: 10.1007/BF00256457. |
[26] |
L. Rayleigh, On convection currents in horizontal layer of fluid when the higher temperature
is on the under side, Philos. Mag. Ser. (6), 32 (1916), 529–546.
doi: 10.1080/14786441608635602. |
[27] |
M. A. Rodríguez-Bellido, M. A. Rojas-Medar and E. J. Villamizar-Roa,
The Boussinesq system with mixed nonsmooth boundary data, C. R. Math. Acad. Sci. Paris, 343 (2006), 191-196.
doi: 10.1016/j.crma.2006.06.011. |
[28] |
D. A. Rueda-Gómez and E. J. Villamizar-Roa,
On the Rayleigh-Bénard-Marangoni system and a related optimal control problem, Computers and Mathematics with Applications, 74 (2017), 2969-2991.
doi: 10.1016/j.camwa.2017.07.038. |
[29] |
J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987),
65–96.
doi: 10.1007/BF01762360. |
[30] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |


[1] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
[2] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[3] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[4] |
Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 |
[5] |
Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020367 |
[6] |
Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 |
[7] |
Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230 |
[8] |
Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150 |
[9] |
Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 |
[10] |
Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087 |
[11] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
[12] |
Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021002 |
[13] |
Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 |
[14] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[15] |
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 |
[16] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[17] |
Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123 |
[18] |
Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 |
[19] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[20] |
Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]