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The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

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Abstract

A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. The problem is motivated by modeling energy systems in rooms that possess an outlet where the fluid can freely flow, known as an “open boundary”. Numerical and experimental investigations available from the literature on heated cavities with open boundaries suggest that the heat transfer at the outlet depends on the temperature at the boundary, the velocity of the fluid, and the outside temperature. Aiming to include this feature in the model, we propose a novel non-smooth boundary condition which is deduced from physical assumptions. We show that this condition is compatible with two approaches at dealing with the “do-nothing” boundary condition for the fluid: (1) the “directional do-nothing” condition and (2) the “do-nothing” condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.

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References

  1. Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Arndt, D., Braack, M., Lube, G.: Finite Elements for the Navier–Stokes problem with outflow condition. Lecture Notes in Computational Science and Engineering, pp. 95–103. Springer, Berlin (2016)

    MATH  Google Scholar 

  3. Beneš, M.: A note on regularity and uniqueness of natural convection with effects of viscous dissipation in 3D open channels. Zeitschrift für angewandte Mathematik und Physik 65(5), 961–975 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beneš, M.: A note on the regularity of thermally coupled viscous flows with critical growth in nonsmooth domains. Math. Methods Appl. Sci. 36(10), 1290–1300 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beneš, M.: The “do nothing” problem for Boussinesq fluids. Appl. Math. Lett. 31, 25–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beneš, M., Kučera, P.: On the Navier–Stokes flows for heat-conducting fluids with mixed boundary conditions. J. Math. Anal. Appl. 389(2), 769–780 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beneš, M., Kučera, P.: Solutions to the Navier–Stokes equations with mixed boundary conditions in two-dimensional bounded domains. Mathematische Nachrichten 289(2–3), 194–212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beneš, M., Pažanin, I.: On existence, regularity and uniqueness of thermally coupled incompressible flows in a system of three dimensional pipes. Nonlinear Anal. Theory Methods Appl. 149, 56–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beneš, M., Tichý, J.: On coupled Navier–Stokes and energy equations in exterior-like domains. Comput. Math. Appl. 70(12), 2867–2882 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bertoglio, C., Caiazzo, A., Bazilevs, Y., Braack, M., Esmaily, M., Gravemeier, V., Marsden, A., Pironneau, O., Vignon-Clementel, I., Wall, W.: Benchmark problems for numerical treatment of backflow at open boundaries. Int. J. Numer. Methods Biomed. Eng. 34(2), 2–34 (2018)

    MathSciNet  Google Scholar 

  11. Boussinesq, J.: Théorie analytique de la Chaleur. Gauthier-Villars, Paris (1903)

    Google Scholar 

  12. Braack, M., Mucha, P.: Directional do-nothing condition for the Navier–Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bruneau, C.H., Fabrie, P.: New efficient boundary conditions for incompressible Navier–Stokes equations: a well posedness result. Math. Model. Numer. Anal. 30(7), 815–840 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Burns, J., He, X., Hu, W.: Feedback stabilization of a thermal fluid system with mixed boundary control. Comput. Math. Appl. 71(11), 2170–2191 (2016)

    Article  MathSciNet  Google Scholar 

  15. Chan, Y.L., Tien, C.L.: A numerical study of two-dimensional natural convection in square open cavities. Numer. Heat Transf. 8(1), 65–80 (1985)

    Article  MATH  Google Scholar 

  16. Chan, Y.L., Tien, C.L.: Laminar natural convection in shallow open cavities. J. Heat Transf. 108(2), 305–309 (1986)

    Article  Google Scholar 

  17. Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integral Equ. Oper. Theory 15(2), 227–261 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Doktor, P., Ženíšek, A.: The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions. Appl. Math. 51(5), 517–547 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feistauer, M., Neustupa, T.: On non-stationary viscous incompressible flow through a cascade of profiles. Math. Methods Appl. Sci. 29(16), 1907–1941 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer, New York (2011)

    Book  MATH  Google Scholar 

  21. Gresho, P.M.: Some current CFD issues relevant to the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 87, 201–252 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Heywood, J.E., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hintermüller, M., Rautenberg, C., Mohammadi, M., Kanitsar, M.: Optimal sensor placement: a robust approach. SIAM J. Control Optim. 55(6), 3609–3639 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Maz’ya, V., Rossmann, J.: Mixed boundary value problems for the stationary Navier–Stokes system in polyhedral domains. Arch. Ration. Mech. Anal. 194(2), 669–712 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hintermüller, M., Rautenberg, C.N.: On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces. J. Math. Anal. Appl. 426(1), 585–593 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hintermüller, M., Rautenberg, C.N., Rösel, S.: Density of convex intersections and applications. Proc. A. 473(2205), 20160919, 28 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Hu, W., Kukavica, I., Ziane, M.: Persistence of regularity for the viscous Boussinesq equations with zero diffusivity. Asymptot. Anal. 91(2), 111–124 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hu, W., Wang, Y., Wu, J., Xiao, B., Yuan, J.: Partially dissipative 2D Boussinesq equations with Navier type boundary conditions. Physica D Nonlinear Phenom. (2017)

  29. Kračmar, S., Neustupa, J.: Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities. ZAMM J. Appl. Math. Mech. 74(6), 637–639 (1994)

    MATH  Google Scholar 

  30. Kračmar, S., Neustupa, J.: A weak solvability of a steady variational inequality of the Navier–Stokes type with mixed boundary conditions. Nonlinear Anal. 27, 4169–4180 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kračmar, S., Neustupa, J.: Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Mathematische Nachrichten 291, 1801–1814 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kundu, P., Cohen, I.: Fluid Mechanics, 2nd edn. Academic Press, Cambridge (2001)

    Google Scholar 

  33. Neustupa, T.: A steady flow through a plane cascade of profiles with an arbitrarily large inflow\_The mathematical model, existence of a weak solution. Appl. Math. Comput. 272, 687–691 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Neustupa, T.: The weak solvability of the steady problem modelling the flow of a viscous incompressible heat-conductive fluid through the profile cascade. Int. J. Numer. Methods Heat Fluid Flow 27, 1451–1466 (2017)

    Article  Google Scholar 

  35. Nečas, J.: Direct Methods in the Theory of Elliptic Differential Equations. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  36. Nguyen, J., Raymond, J.: Boundary stabilization of the Navier–Stokes equations in the case of mixed boundary conditions. SIAM J. Control Optim. 53(5), 3006–3039 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pérez, C.E., Thomas, J.M., Blancher, S., Creff, R.: The steady Navier–Stokes/energy system with temperature-dependent viscosity–part 1: analysis of the continuous problem. Int. J. Numer. Methods Fluids 56(1), 63–89 (2008)

    Article  MATH  Google Scholar 

  38. Pérez, C.E., Thomas, J.M., Blancher, S., Creff, R.: The steady Navier–Stokes/energy system with temperature-dependent viscosity—part 2: the discrete problem and numerical experiments. Int. J. Numer. Methods Fluids 56(1), 91–114 (2008)

    Article  MATH  Google Scholar 

  39. Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Commun. Partial Differ. Equ. 22(5–6), 869–899 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B. Springer, New York (1990). Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron

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Acknowledgements

The authors would like to thank the two anonymous referees for the helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Andrea N. Ceretani & Carlos N. Rautenberg

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Carlos N. Rautenberg

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  1. Andrea N. Ceretani
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  2. Carlos N. Rautenberg
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Correspondence to Carlos N. Rautenberg.

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CNR has been supported via the framework of Matheon by the Einstein Foundation Berlin within the ECMath Project SE15/SE19 and acknowledges the support of the DFG through the DFG-SPP 1962: Priority Programme “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” within Project 11.

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Ceretani, A.N., Rautenberg, C.N. The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow. Z. Angew. Math. Phys. 70, 14 (2019). https://doi.org/10.1007/s00033-018-1058-y

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  • DOI: https://doi.org/10.1007/s00033-018-1058-y

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