# American Institute of Mathematical Sciences

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

* Corresponding author: Élder J. Villamizar-Roa

Received  January 2020 Published  May 2020

Fund Project: The first and third authors were supported by Vicerrectoría de Investigación y Extensión of the Universidad Industrial de Santander (UIS), proyecto Capital semilla-2412. The second author was supported by Vicerrectoría de Investigación y Extensión-UIS

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.

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.
Temperature and velocity field in Test 1
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: