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Boundary Control for Optimal Mixing by Stokes Flows

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Abstract

We discuss the optimal boundary control problem for mixing an inhomogeneous distribution of a passive scalar field in an unsteady Stokes flow. The problem is motivated by mixing the fluids within a cavity or vessel at low Reynolds numbers by moving the walls or stirring at the boundary. It is natural to consider the velocity field which is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time optimal control problem of a bilinear system. In the current work, we consider a general open bounded and connected domain \(\Omega \subset \mathbb {R}^{d}, d=2,3\). We employ the Sobolev norm for the dual space \((H^{1}(\Omega ))'\) of \(H^{1}( \Omega )\) to quantify mixing of the scalar field in terms of the property of weak convergence. A rigorous proof of the existence of an optimal control is presented and the first-order necessary conditions for optimality are derived.

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References

  1. Aamo, O.M., Krstić, M., Bewley, T.R.: Control of mixing by boundary feedback in 2D channel flow. Automatica 39, 1597–1606 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balogh, A., Aamo, O.M., Krstić, M.: Optimal mixing enhancement in 3-d pipe flow. IEEE Trans. Control Syst. Technol. 13, 27–41 (2005)

    Article  Google Scholar 

  3. Barbu, V., Lasiecka, I., Triggiani, R.: Tangential Boundary Stabilization of Navier-Stokes Equations, vol. 181. American Mathematical Society, Providence (2006)

    MATH  Google Scholar 

  4. Barbu, V., Lasiecka, I., Triggiani, R.: Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers. Nonlinear Anal. 64(12), 2704–2746 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chakravarthy, V.S., Ottino, J.M.: Mixing of two viscous fluids in rectangular cavity. Chem. Eng. Sci. 51(14), 3613–3622 (1996)

    Article  Google Scholar 

  6. Clopeau, T., Robert, R., Mikelic, A.: On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coron, J.-M.: On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM Control Optim. Calc. Var. 1, 35–75 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. DiPerna, J.R., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Filho, M.C.L., Lopes, H.J.N., Planas, G.: On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foures, D.P.G., Caulfield, C.P., Schmid, P.J.: Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number. J. Fluid Mech. 748, 241–277 (2014)

    Article  MathSciNet  Google Scholar 

  11. Gubanov, O., Cortelezzi, L.: Towards the design of an optimal mixer. J. Fluid Mech. 651, 27–53 (2010)

    Article  MATH  Google Scholar 

  12. Gouillart, E., Dauchot, O., Dubrulle, B., Roux, S., Thiffeault, J.L.: Slow decay of concentration variance due to no-slip walls in chaotic mixing. Phys. Rev. E 78(2), 026211 (2008)

    Article  MathSciNet  Google Scholar 

  13. Gouillart, E., Kuncio, N., Dauchot, O., Dubrulle, B., Roux, S., Thiffeault, J.L.: Walls inhibit chaotic mixing. Phys. Rev. Lett. 99(11), 114501 (2007)

    Article  Google Scholar 

  14. Gouillart, E., Thiffeault, J.-L., Dauchot, O.: Rotation shields chaotic mixing regions from no-slip walls. Phys. Rev. Lett. 104(20), 204502 (2010)

    Article  Google Scholar 

  15. Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54, 081507 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kelliher, J.P.: Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lions, J.-L.: Quelques méthodes de résolution des problemes aux limites non linéaires, vol. 31. Dunod, Paris (1969)

    MATH  Google Scholar 

  18. Lions, P.-L.: Mathematical Topics in Fluid Mechanics: Vol. 2: Compressible Models. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  19. Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  20. Lasiecka, I., Tuffaha, A.: Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction. Syst. Control Lett. 58, 499–509 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lion, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  Google Scholar 

  22. Lin, Z., Thiffeault, J.-L., Doering, C.R.: Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, W.: Mixing enhancement by optimal flow advection. SIAM J. Control Optim. 47(2), 624–638 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lunasin, E., Lin, Z., Novikov, A., Mazzucato, A., Doering, C.R.: Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows. J. Math. Phys. 53, 115611 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mathew, G., Mezić, I., Grivopoulos, S., Vaidya, U., Petzold, L.: Optimal control of mixing in Stokes fluid flows. J. Fluid Mech. 580, 261–281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mathew, G., Mezić, I., Petzold, L.: A multiscale measure for mixing. Physica D 211(1), 23–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Navier, C.L.: Mémoire sur les lois du mouvement des fluids. Mém. Acad. R. Sci. 6, 389–440 (1823)

    Google Scholar 

  28. Ottino, J.M.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge, UK (1989)

    MATH  Google Scholar 

  29. Shankar, P.N.: Slow Viscous Flows: Qualitative Feature and Quantitative Analysis Using Complex Eigenfunction Expansions. Imperial College Press, London (2007)

    Book  MATH  Google Scholar 

  30. Sharma, A., Gupte, N.: Control methods for problems of mixing and coherence in chaotic maps and flows. Pramana 48, 231–248 (1997)

    Article  Google Scholar 

  31. Stremler, M.A., Cola, B.A.: A maximum entropy approach to optimal mixing in a pulsed source-sink flow. Phys. Fluids 18, 011701 (2006)

    Article  Google Scholar 

  32. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2. North-Holland, New York (1997)

    Google Scholar 

  33. Thiffeault, J.-L.: Using multiscale norms to quantify mixing and transport. Nonlinearity 25(2), R1–R44 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Thiffeault, J.-L., Gouillart, E., Dauchot, O.: Moving walls accelerate mixing. Phys. Rev. E 84(3), 036313 (2011)

    Article  Google Scholar 

  35. Vikhansky, A.: Enhancement of laminar mixing by optimal control methods. Chem. Eng. Sci. 57(14), 2719–2725 (2002)

    Article  Google Scholar 

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Acknowledgements

The author would like to thank Irena Lasiecka and Igor Kukavica for their valuable questions and suggestions to help improve the first version of the paper.

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  1. Institute for Mathematics and Its Applications, University of Minnesota, 436 Lind Hall, 207 Church Street SE, Minneapolis, MN, 55455-0436, USA

    Weiwei Hu

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  1. Weiwei Hu
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Hu, W. Boundary Control for Optimal Mixing by Stokes Flows. Appl Math Optim 78, 201–217 (2018). https://doi.org/10.1007/s00245-017-9404-6

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