Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-31T09:07:55.018Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal control of mixing in Stokes fluid flows

Published online by Cambridge University Press:  21 May 2007

GEORGE MATHEW ,
IGOR MEZIĆ ,
SYMEON GRIVOPOULOS ,
UMESH VAIDYA  and
LINDA PETZOLD
GEORGE MATHEW
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
IGOR MEZIĆ
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
SYMEON GRIVOPOULOS
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
UMESH VAIDYA
Affiliation:
Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA
LINDA PETZOLD
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the problem of microfluidic mixing, optimal control of advective mixing in Stokes fluid flows is considered. The velocity field is assumed to be induced by a finite set of spatially distributed force fields that can be modulated arbitrarily with time, and a passive material is advected by the flow. To quantify the degree of mixedness of a density field, we use a Sobolev space norm of negative index. We frame a finite-time optimal control problem for which we aim to find the modulation that achieves the best mixing for a fixed value of the action (the time integral of the kinetic energy of the fluid body) per unit mass. We derive the first-order necessary conditions for optimality that can be expressed as a two-point boundary value problem (TPBVP) and discuss some elementary properties that the optimal controls must satisfy. A conjugate gradient descent method is used to solve the optimal control problem and we present numerical results for two problems involving arrays of vortices. A comparison of the mixing performance shows that optimal aperiodic inputs give better results than sinusoidal inputs with the same energy.

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 580 , 10 June 2007 , pp. 261 - 281
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Arnold, V. I. & Avez, A. 1968 Ergodic Problems of Classical Mechanics. Benjamin.Google Scholar
Bajer, K. & Moffatt, H. K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363.CrossRefGoogle Scholar
Ball, J., Marsden, J. E. & Slemrod, M. 1982 Controllability for distributed bilinear systems. SIAM. J. Control Optim. 20, 575.Google Scholar
Balogh, A., Aamo, O. M. & Kristic, M. 2005 Optimal mixing enhancement in 3-d pipe flow. IEEE Trans. Control Syst. Tech. 13, 27–41.Google Scholar
Banks, S. P. 1987 Optimal control of distributed bilinear systems. Syst. Control Lett. 9, 183–189.CrossRefGoogle Scholar
Boyland, P. L., Aref, H. & Stremler, M. A. 2000 Topological fluid mechanics of stirring. J. Fluid Mech. 403, 277–304.Google Scholar
Chrisohoides, A. & Sotiropoulos, F. 2003 Experimental visualization of Lagrangian coherent structures in aperiodic flows. Phys. Fluids 15 (3), L25.Google Scholar
D'Alessandro, D., Dahleh, M. & Mezić, I. 1999 Control of Mixing in Fluid Flow: A Maximum Entropy Approach. IEEE Trans. Automatic Control 44, 1852–1864.Google Scholar
Fountain, G. O., Khakhar, D. V., Mezić, I. & Ottino, J. M. 2000 Chaotic mixing in a bounded three-dimensional flow. J. Fluid Mech. 417, 265–301.Google Scholar
Franjione, J. G. & Ottino, J. M. 1992 Symmetry concepts for geometric analysis of mixing flows. Phil. Trans. R. Soc. Lond. A 338, 301–323.Google Scholar
Frigo, M. & Johnson, S. G. 1998 FFTW: An adaptive software architecture for the FFT. Proc. ICASSP 3, 1381–1384.Google Scholar
Gelfand, I. M. & Fomin, S. V. 1963 Calculus of Variations. Prentice-Hall.Google Scholar
Grigoriev, R. O. 2005 Chaotic mixing in thermocapillary-driven microdroplets. Phys. Fluids 17, 033601.Google Scholar
Haller, G. & Poje, A. C. 1998 Finite time transport in aperiodic flows. Physica D 119, 352–380.Google Scholar
Jurdjevic, V. & Quinn, J. P. 1978 Controllability and stability. J. Diffl. Equat. 28, 381–389.Google Scholar
Kirk, D. E. 1970 Optimal Control Theory – An Introduction. Prentice-Hall.Google Scholar
Lasota, A. & Mackey, M. 1994 Chaos, Fractals and Noise. Applied Mathematical Sciences, vol. 97. Springer.Google Scholar
Liu, W. & Haller, G. 2004 Strange eigenmodes and decay of variance in the mixing of diffusive tracers. Physica D 188, 1–39.Google Scholar
Luenberger, D. G. 1984 Linear and Nonlinear Programming, 2nd edn. Addison-Wesley.Google Scholar
Malhotra, N. & Wiggins, S. 1998 Geometric structures, Lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby Wave Flow. J. Nonlinear Sci. 8, 401–456.Google Scholar
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 23–46.Google Scholar
Mezić, I. & Wiggins, S. 1994 On the integrability and perturbation of three dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press.Google Scholar
Petersen, K. 1983 Ergodic Theory. Cambridge University Press.Google Scholar
Pierrehumbert, R. T. 1994 Tracer microstructure in the large-eddy dominated regime. Chaos, Solitons Fractals 4, 1091–1110.CrossRefGoogle Scholar
Poje, A., Haller, G. & Mezić, I. 1999 The geometry and statistics of mixing in aperiodic flows. Phys. Fluids 11, 2963.CrossRefGoogle Scholar
Rothstein, D., Henry, E. & Gollub, J. P. 1999 Persistent patterns in transient chaotic fluid mixing. Nature 401, 770–772.CrossRefGoogle Scholar
Sharma, A. & Gupte, N. 1997 Control methods for problems of mixing and coherence in chaotic maps and flows. Pramana – J. Phys. 48, 231–248.Google Scholar
Slemrod, M. 1978 Stabilization of bilinear control systems with applications to non-conservative problems in elasticity. SIAM J. Control 16, 131–141.Google Scholar
Stone, H. A., Nadim, A. & Strogatz, S. H. 1991 Chaotic streamlines inside drops immersed in steady Stokes flows. J. Fluid Mech. 232, 629–646.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.Google Scholar
Stone, Z. B. & Stone, H. A. 2005 Imaging and quantifying mixing in a model droplet micromixer. Phys. Fluids 17, 063103.Google Scholar
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezić, I. & Stone, H. A. 2002 Chaotic mixer for microchannels. Science 295, 647–651.CrossRefGoogle ScholarPubMed
Thiffeault, J.-L. & Finn, M. D. 2006 Topology, Braids, and Mixing in Fluids. Phil. Trans. R. Soc. Lond. A 364, 3251–3266.Google Scholar
Wang, Y., Haller, G., Banaszuk, A. & Tadmor, G. 2003 Closed-loop Lagrangian separation control in a bluff body shear flow model. Phys. Fluids 15, 2251–2266.Google Scholar
Wiggins, S. 1992 Chaotic Transport in Dynamical Systems. Springer.Google Scholar