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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
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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.

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

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