Journal of Nonlinear Science

, Volume 28, Issue 6, pp 2153–2186 | Cite as

A Shell Model for Optimal Mixing

  • Christopher J. Miles
  • Charles R. DoeringEmail author


What is the maximum mixing efficiency of an incompressible flow? To address this question we introduce a shell model—a reduced model mimicking the kinematics of advection and diffusion—to study the evolution of an initially inhomogeneous tracer concentration carried by a given incompressible fluid on a periodic spatial domain. We pose the mixing task as an optimization problem: Find the divergence-free velocity field (the control variable) that produces a well-mixed tracer concentration field (the state variable). We consider two alternative objectives: local-in-time optimization (maximize the instantaneous mixing rate) and global-in-time optimization (maximize mixing at a prescribed final time). Throughout, we use a shell-model analog of the \(H^{-1}\) mix-norm to measure mixing. In addition, lower bounds on the mix-norm are obtained and rule out perfect mixing in finite time in particular cases.


Mixing Advection–diffusion equation Passive scalar advection Dynamical systems Optimal control theory Fluid dynamics 



This work was supported in part by NSF Awards PHY-1205219 and DMS-1515161. One of us (CRD) is additionally grateful for fellowship funding from the John Simon Guggenheim Memorial Foundation. We thank the reviewers for helpful feedback and thoughtful suggestions.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Christopher J. Miles
    • 1
    • 2
    • 3
  • Charles R. Doering
    • 1
    • 2
    • 3
    Email author
  1. 1.Department of PhysicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Center for the Study of Complex SystemsUniversity of MichiganAnn ArborUSA

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