Figure 1.
Temperature and velocity field in Test 1
- Previous Article
- JCD Home
- This Issue
-
Next Article
A functional analytic approach to validated numerics for eigenvalues of delay equations
June
2020, 7(1): 159-181.
doi: 10.3934/jcd.2020006
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.
Keywords:
Rayleigh-Bénard-Marangoni system,
global weak solutions,
finite elements,
convergence rates,
error estimates.
Mathematics Subject Classification:
Primary: 35Q35, 76D03, 76D05, 35D30; Secondary: 65L60, 65L70.
Citation:
Jhean E. Pérez-López, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis. Journal of Computational Dynamics,
2020, 7
(1)
: 159-181.
doi: 10.3934/jcd.2020006
![]()
References:
| [1] |
H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Revue Gén. Sci. Pures Appl., 11 (1900), 1261–1271, 1309–1328. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
Figure 2.
Temperature and velocity field in Test 2
| [1] |
Toshiyuki Ogawa. Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry. Communications on Pure and Applied Analysis, 2006, 5 (2) : 383-393. doi: 10.3934/cpaa.2006.5.383 |
| [2] |
B. A. Wagner, Andrea L. Bertozzi, L. E. Howle. Positive feedback control of Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619 |
| [3] |
Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure and Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591 |
| [4] |
Tingyuan Deng. Three-dimensional sphere $S^3$-attractors in Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 577-591. doi: 10.3934/dcdsb.2010.13.577 |
| [5] |
Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53 |
| [6] |
Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591 |
| [7] |
O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449 |
| [8] |
Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759 |
| [9] |
Ruiying Wei, Yin Li, Zheng-an Yao. Global existence and convergence rates of solutions for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2949-2967. doi: 10.3934/dcdsb.2020047 |
| [10] |
Quyen Tran, Harbir Antil, Hugo Díaz. Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022003 |
| [11] |
Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 |
| [12] |
Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681 |
| [13] |
Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091 |
| [14] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 |
| [15] |
Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051 |
| [16] |
Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751 |
| [17] |
Hengrong Du, Changyou Wang. Global weak solutions to the stochastic Ericksen–Leslie system in dimension two. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2175-2197. doi: 10.3934/dcds.2021187 |
| [18] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
| [19] |
Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 |
| [20] |
Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]