Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T04:00:29.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Feasibility, efficiency and transportability of short-horizon optimal mixing protocols

Published online by Cambridge University Press:  01 February 2008

LUCA CORTELEZZI ,
ALESSANDRA ADROVER  and
MASSIMILIANO GIONA
LUCA CORTELEZZI*
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal, Canada
ALESSANDRA ADROVER
Affiliation:
Dipartimento di Ingegneria Chimica, Università di Roma ‘La Sapienza’, Rome, Italy
MASSIMILIANO GIONA
Affiliation:
Dipartimento di Ingegneria Chimica, Università di Roma ‘La Sapienza’, Rome, Italy
*
†Author to whom correspondence should be addressed: [email protected]; also affiliated with the Dipartimento di Fisica, Università di Udine, 33100 Udine, Italia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider, as a case study, the optimization of mixing protocols for a two-dimensional, piecewise steady, nonlinear flow, the sine flow, for both the advective–diffusive and purely advective cases. We use the mix-norm as the cost function to be minimized by the optimization procedure. We show that the cost function possesses a complex structure of local minima of nearly the same values and, consequently, that the problem possesses a large number of sub-optimal protocols with nearly the same mixing efficiency as the optimal protocol. We present a computationally efficient optimization procedure able to find a sub-optimal protocol through a sequence of short-time-horizon optimizations. We show that short-time-horizon optimal mixing protocols, although sub-optimal, are both feasible and efficient at mixing flows with and without diffusion. We also show that these optimized protocols can be derived, at lower computational cost, for purely advective flows and successfully transported to advective–diffusive flows with small molecular diffusivity. We characterize our results by discussing the asymptotic properties of the optimized protocols both in the pure advection and in the advection–diffusion cases. In particular, we quantify the mixing efficiency of the optimized protocols using the Lyapunov exponents and Poincaré sections for the pure advection case, and the eigenvalue–eigenfunction spectrum for the advection–diffusion case. Our results indicate that the optimization over very short-time horizons could in principle be used as an on-line procedure for enhancing mixing in laboratory experiments, and in future engineering applications.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 597 , 25 February 2008 , pp. 199 - 231
Copyright
Copyright © Cambridge University Press 2008

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.)

References

REFERENCES

Arnold, V. I. & Avez, A. 1989 Ergodic Problems of Classical Mechanics. Addison-Wesley.Google Scholar
Alvarez, M. M., Arratia, P. E. & Muzzio, F. J. 2002 a Laminar mixing in eccentric stirred tank systems. Can. J. Chem. Engng 80, 546–557.CrossRefGoogle Scholar
Alvarez, M. M., Shinbrot, T., Zalc, J. & Muzzio, F. J. 2002 b Practical chaotic mixing. Chem. Engng Sci. 17, 3749–3753.CrossRefGoogle Scholar
Balasuriya, S. 2005 Optimal perturbation for enhanced chaotic transport. Physica D 202, 155–176.Google Scholar
Beigie, D., Leonard, A. & Wiggins, S. 1994 Invariant manifold templates for chaotic advection. Chaos Solitons Fractals 4, 749–868.CrossRefGoogle Scholar
Boyland, P. L., Aref, H. & Stremler, M. A. 2000 Topological fluid mechanics of stirring. J. Fluid Mech. 403, 277–304.CrossRefGoogle Scholar
Cerbelli, S., Vitacolonna, V., Adrover, A. & Giona, M. 2004 Eigenvalue–eigenfunction analysis of infinitely fast reactions and micromixing regimes in regular and chaotic bounded flows. Chem. Engng Sci. 59, 2125–2144.CrossRefGoogle Scholar
Devaney, R. 1989 An Introduction to Chaotic Dynamical Systems. 2nd edn. Perseus Books.Google Scholar
D'Alessandro, D., Dahleh, M. & Mezić, I. 1999 Control of mixing in fluid flow: A maximum entropy approach. IEEE Trans. Aut. Contr. 44, 1852–1863.CrossRefGoogle Scholar
Fountain, G. O., Khakhar, D. V. & Ottino, J. M. 1997 Visualization of three-dimensional chaos. Science 31, 683–686.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.CrossRefGoogle Scholar
Giona, M., Cerbelli, S. & Adrover, A. 2002 Geometry of reaction interfaces in chaotic flows. Phys. Rev. Lett. 88, 024501 I–IV.CrossRefGoogle ScholarPubMed
Giona, M., Cerbelli, S., & Vitacolonna, V. 2004 a Universality and imaginary potentials in advection–diffusion equations in closed flows. J. Fluid Mech. 513, 221–237.CrossRefGoogle Scholar
Giona, M., Adrover, A., Cerbelli, S. & Vitacolonna, V. 2004 b Spectral properties and transport mechanisms of partially chaotic bounded flows in the presence of diffusion. Phys. Rev. Lett. 92, 114101 I–IV.CrossRefGoogle ScholarPubMed
Harvey, A. D. III, Wood, S. P. & Leng, D. E. 1997 Experimental and computational study of multiple impeller flows. Chem. Engng Sci. 52, 1479–1491.CrossRefGoogle Scholar
Hobbs, D. M., & Muzzio, F. J. 1998 Optimization of a static mixer using dynamical systems techniques. Chem. Engng Sci. 53, 3199–3213.CrossRefGoogle Scholar
Hobbs, D. M., Alvarez, M. M. & Muzzio, F. J. 1997 Mixing in globally chaotic flows: A self-similar process. Fractals 5, 395–425.CrossRefGoogle Scholar
Katok, A. & Hasselblatt, B. 1995 Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press.CrossRefGoogle Scholar
Lamberto, D. J., Muzzio, F. J., Swanson, P. D. & Tonkovich, A. L 1996 Using time-dependent RPM to enhance mixing in stirred vessels. Chem. Engng Sci. 51, 733–741.CrossRefGoogle Scholar
Liu, W. & Haller, G. 2004 Inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos. Physica D 194, 297–319.Google Scholar
Liu, M., Muzzio, F. J. & Peskin, R. L. 1994 a Quantification of mixing in aperiodic chaotic flows. Chaos Solitons Fractals 4, 869–893.CrossRefGoogle Scholar
Liu, M., Peskin, R. L., Muzzio, F. J., & Leong, C. W. 1994 b Structure of the stretching field in chaotic cavity flows. AIChE J. 40, 1273–1286.CrossRefGoogle Scholar
Mathew, C., Mezić, I., & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 23–46.Google Scholar
Mathew, C., Mezić, I., Grivopoulos, S., Vaidya, U. & Petzold, L. 2007 Optimal control of mixing in Stokes fluid flows. J. Fluid Mech. 580, 261–281.CrossRefGoogle Scholar
Metcalfe, G., Rudman, M., Brydon, A., Graham, L. J. W. & Hamilton, R. 2005 Composing chaos: An experimental and numerical study of an open duct mixing flow. AIChE J. 52, 9–28.CrossRefGoogle Scholar
Noack, B. R., Mezić, I., Tadmor, G. & Banaszuk, A. 2004 Optimal mixing in recirculation zones. Phys. Fluids 16, 867–888.CrossRefGoogle Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press.Google Scholar
Ottino, J. M. & Wiggins, S. 2004 a Designing optimal micromixers. Science 305, 485–486.CrossRefGoogle ScholarPubMed
Ottino, J. M. & Wiggins, S. 2004 b Introduction: Mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362, 923–935.CrossRefGoogle ScholarPubMed
Rice, M., Hall, J., Papadakis, G. & Yanneskis, M. 2006 Investigation of laminar flow in a stirred vessel at low Reynolds numbers. Chem. Engng Sci. 61, 2762–2770.CrossRefGoogle Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytic study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Sharma, A. & Gupte, N. 1997 Control methods for problems of mixing and coherence in chaotic maps and flows. Pramana J. Phys. 488, 231–248.CrossRefGoogle Scholar
Stremler, M. A., Haselton, F. R. & Aref, H. 2004 Designing for chaos: Applications of chaotic advection at the microscale. Phil. Trans. R. Soc. Lond. A 362, 1019–1036.CrossRefGoogle ScholarPubMed
Tabeling, P. 2005 Introduction to Microfluidics. Oxford University Press.CrossRefGoogle Scholar
Tabeling, P., Chabert, M., Dodge, A., Jullien, C. & Okkels, F. 2004 Chaotic mixing in cross-channel micromixer. Phil. Trans. R. Soc. Lond. 362, 987–1000.CrossRefGoogle Scholar
Tay, F. E. H. 2002 Microfluidics and BioMEMS Applications. Kluwer.CrossRefGoogle Scholar
Thiffeault, J. L., Doering, C. R. & Gibbon, J. D. 2004 A bound on mixing efficiency for the advection–diffusion equation. J. Fluid Mech. 521, 105–114.CrossRefGoogle Scholar
Thiffeault, J.-L. & Finn, M. D. 2006 Topology, braids and mixing in fluids. Phil. Trans. R. Soc. Lond. A 364, 3251–3266.Google ScholarPubMed
Toussaint, V., Carriere, P., & Raynal, F. 1995 A numerical Eulerian approach to mixing by chaotic advection. Phys. Fluids 7, 2587–2600.CrossRefGoogle Scholar
Toussaint, V., Carriere, P., Scott, J. & Gence, J. N. 2000 Spectral decay of a passive scalar in chaotic mixing. Phys. Fluids 12, 2834–2844.CrossRefGoogle Scholar
Vikhansky, A. 2002 a Enhancement of laminar mixing by optimal control methods. Chem. Engng Sci. 57, 2719–2725.CrossRefGoogle Scholar
Vikhansky, A. 2002 b Control of stretching rate in time-periodic chaotic flows. Phys. Fluids 14, 2752–2756.CrossRefGoogle Scholar
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362, 937–970.CrossRefGoogle ScholarPubMed
Wojtkowski, M. 1981 A model problem with the coexistence of stochastic and integrable behavior. Commun. Math. Phys. 80, 453–464.CrossRefGoogle Scholar
Xia, H. M., Shu, C., Wan, S. Y. M. & Chen, Y. T. 2006 Influence of the Reynolds number on chaotic mixing in a spatially periodic micromixer and its characterization using dynamical system techniques. J. Micromech. Microengng 16, 53–61.CrossRefGoogle Scholar
Zalc, J. M., Szalai, E. S. & Muzzio, F. J. 2003 Mixing dynamics in the SMX static mixer as a function of injection location and flow ratio. Polym. Engng Sci. 43, 875–890.CrossRefGoogle Scholar