Applied Mathematics & Optimization

, Volume 78, Issue 1, pp 201–217 | Cite as

Boundary Control for Optimal Mixing by Stokes Flows



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.


Optimal mixing Unsteady stokes flow Navier slip boundary conditions Bilinear system 

Mathematics Subject Classification

35Q93 37A25 49J20 49K20 76B75 76F25 



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.


  1. 1.
    Aamo, O.M., Krstić, M., Bewley, T.R.: Control of mixing by boundary feedback in 2D channel flow. Automatica 39, 1597–1606 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 3.
    Barbu, V., Lasiecka, I., Triggiani, R.: Tangential Boundary Stabilization of Navier-Stokes Equations, vol. 181. American Mathematical Society, Providence (2006)MATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chakravarthy, V.S., Ottino, J.M.: Mixing of two viscous fluids in rectangular cavity. Chem. Eng. Sci. 51(14), 3613–3622 (1996)CrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    DiPerna, J.R., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gubanov, O., Cortelezzi, L.: Towards the design of an optimal mixer. J. Fluid Mech. 651, 27–53 (2010)CrossRefMATHGoogle Scholar
  12. 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)MathSciNetCrossRefGoogle Scholar
  13. 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)CrossRefGoogle Scholar
  14. 14.
    Gouillart, E., Thiffeault, J.-L., Dauchot, O.: Rotation shields chaotic mixing regions from no-slip walls. Phys. Rev. Lett. 104(20), 204502 (2010)CrossRefGoogle Scholar
  15. 15.
    Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54, 081507 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lions, J.-L.: Quelques méthodes de résolution des problemes aux limites non linéaires, vol. 31. Dunod, Paris (1969)MATHGoogle Scholar
  18. 18.
    Lions, P.-L.: Mathematical Topics in Fluid Mechanics: Vol. 2: Compressible Models. Oxford University Press, New York (1998)MATHGoogle Scholar
  19. 19.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  20. 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lion, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefGoogle Scholar
  22. 22.
    Lin, Z., Thiffeault, J.-L., Doering, C.R.: Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Liu, W.: Mixing enhancement by optimal flow advection. SIAM J. Control Optim. 47(2), 624–638 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 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)MathSciNetCrossRefMATHGoogle Scholar
  25. 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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mathew, G., Mezić, I., Petzold, L.: A multiscale measure for mixing. Physica D 211(1), 23–46 (2005)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Navier, C.L.: Mémoire sur les lois du mouvement des fluids. Mém. Acad. R. Sci. 6, 389–440 (1823)Google Scholar
  28. 28.
    Ottino, J.M.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge, UK (1989)MATHGoogle Scholar
  29. 29.
    Shankar, P.N.: Slow Viscous Flows: Qualitative Feature and Quantitative Analysis Using Complex Eigenfunction Expansions. Imperial College Press, London (2007)CrossRefMATHGoogle Scholar
  30. 30.
    Sharma, A., Gupte, N.: Control methods for problems of mixing and coherence in chaotic maps and flows. Pramana 48, 231–248 (1997)CrossRefGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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. 33.
    Thiffeault, J.-L.: Using multiscale norms to quantify mixing and transport. Nonlinearity 25(2), R1–R44 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Thiffeault, J.-L., Gouillart, E., Dauchot, O.: Moving walls accelerate mixing. Phys. Rev. E 84(3), 036313 (2011)CrossRefGoogle Scholar
  35. 35.
    Vikhansky, A.: Enhancement of laminar mixing by optimal control methods. Chem. Eng. Sci. 57(14), 2719–2725 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for Mathematics and Its ApplicationsUniversity of MinnesotaMinneapolisUSA

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