Flow-field development during finger splitting at an exothermic chemical reaction front

L. Šebestíková, J. D’Hernoncourt, M. J. B. Hauser, S. C. Müller, and A. De Wit

Phys. Rev. E 75, 026309 – Published 22 February 2007

Abstract

Fingertip splitting may be observed at chemical reaction fronts subject to buoyancy-induced Rayleigh-Taylor fingering, as investigated in ascending fronts of the iodate-arsenous acid reaction in vertical Hele-Shaw cells. We study the properties of the flow-field evolution during a tip-splitting event both experimentally and theoretically. Experimental particle-image velocimetry techniques show that the flow field associated to a finger displays a quadrupole of vortices. The evolution of the flow field and the reorganization of the vortices after a tip-splitting event are followed experimentally in detail. Numerical integration of a model reaction-diffusion-convection system for an exothermic reaction taking into account possible heat losses through the walls of the reactor shows that the nonlinear properties of the flow field are different whether the walls are insulating or conducting. In insulating systems, the flow field inside one finger features only one pair of vortices. A quadrupole of flow vortices arranged around a saddle-node structure similar to the one observed experimentally is obtained in the presence of heat losses suggesting that heat effects, even if of very small amplitude, are important in understanding the nonlinear properties of fingering of exothermic chemical fronts.

Received 12 September 2006

DOI:https://doi.org/10.1103/PhysRevE.75.026309

Authors & Affiliations

L. Šebestíková^1,*,†, J. D’Hernoncourt^2,*,‡, M. J. B. Hauser^1,§, S. C. Müller¹, and A. De Wit^2,∥

¹Institut für Experimentelle Physik, Abteilung Biophysik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg Germany
²Nonlinear Physical Chemistry Unit and Center for Nonlinear Phenomena and Complex Systems, CP 231, Université Libre de Bruxelles, 1050 Brussels, Belgium

^*These authors contributed equally to this work.
^†Electronic address: lenka.sebestikova@physik.uni-magdeburg.de
^‡Electronic address: jdhernon@ulb.ac.be
^§Electronic address: marcus.hauser@physik.uni-magdeburg.de
^∥Electronic address: adewit@ulb.ac.be

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Issue

Vol. 75, Iss. 2 — February 2007

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Images

Figure 1
Fingertip splitting and the adjacent flow field. (a) At $86 s$ a fully developed finger is accompanied by two convection rolls at the front. The flow streams at the very bottom suggest the existence of a second pair of vortices in a deeper position of the liquid layer. (b) At $t \sim 151 s$ the finger starts to split while the macroscopic flow-field pattern is quite reminiscent of the situation prior to finger splitting. (c) At $t \sim 236 s$ the splitting is clearly seen. The macroscopic flow field is being rearranged. (d) Finally, at $t = 296 s$ the new fingers and the rearranged macroscopic flow field is fully developed. The bold arrows indicate the general direction of the flow field.Reuse & Permissions
Figure 2
Details of the finger splitting and adjacent flow-field development. (a) A fully developed finger is associated with two convection rolls located at the reaction front $(t = 109 s)$ . (b) At $t \sim 151 s$ the finger splits. (c) At $t \sim 174 s$ the splitting becomes more pronounced. The macroscopic flow-field pattern is quite reminiscent to the situation prior to finger splitting. (d) At $t = 223 s$ , the macroscopic flow field is being rearranged, forming a new macroscopic convection roll. The black lines at the reaction front are segments of a circle and are intended as a guide for the eyes, to help to visualize the finger-splitting process. The arrows represent vectors of the hydrodynamic flow field; they show the direction and magnitude of the flow.Reuse & Permissions
Figure 3
Density plots of the single-finger asymptotic dynamics of the fingering of a chemical front in an insulated system for $α = 0$ showing from left to right at a time $t = 1300$ the concentration $C$ , the temperature $T$ , the stream function $ψ$ , and the isovelocity lines.Reuse & Permissions
Figure 4
Density plots of the tip-splitting dynamics of the fingering of a chemical front in a system with conductive walls for $α = 0.01$ showing from left to right the concentration $C$ , the temperature $T$ , the stream function $ψ$ , and the isovelocity lines at time $t = 1700$ .Reuse & Permissions
Figure 5
Evolution of the concentration (upper line) and of the velocity field (lower line) showing the onset of tip splittings for $α = 0.01$ from $t = 1300$ to $t = 1700$ .Reuse & Permissions
Figure 6
Reconstructed numerical velocity field in the case of conducting walls $(α = 0.005)$ at time $t = 1800$ . The general quadrupole structure as well as the saddle-node point are clearly visible in the center of each finger. Little vortices favoring tip splittings can be seen inside the reaction zone.Reuse & Permissions

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