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. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. HoyasH. 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.  Google Scholar

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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

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E. L. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1993.  Google Scholar

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A. M. KvarvingT. 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.  Google Scholar

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M. Lappa, Thermal Convection: Patterns, Evolution and Stability, John Wiley & Sons, Ltd., Chichester, 2010.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

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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.  Google Scholar

[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.  Google Scholar

[22]

D. A. Nield, Surface tension and buoyancy effects in cellular convection, J. Fluid Mech., 19 (1964), 341-352.  doi: 10.1017/S0022112064000763.  Google Scholar

[23]

R. PardoH. 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.  Google Scholar

[24]

J. R. A. Pearson, On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.  doi: 10.1017/S0022112058000616.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. A. Rodríguez-BellidoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[4]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. HoyasH. 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.  Google Scholar

[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.  Google Scholar

[14]

A. M. KvarvingT. 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.  Google Scholar

[15]

M. Lappa, Thermal Convection: Patterns, Evolution and Stability, John Wiley & Sons, Ltd., Chichester, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. A. Nield, Surface tension and buoyancy effects in cellular convection, J. Fluid Mech., 19 (1964), 341-352.  doi: 10.1017/S0022112064000763.  Google Scholar

[23]

R. PardoH. 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.  Google Scholar

[24]

J. R. A. Pearson, On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.  doi: 10.1017/S0022112058000616.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. A. Rodríguez-BellidoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

Figure 1.  Temperature and velocity field in Test 1
Figure 2.  Temperature and velocity field in Test 2
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