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Vibrational convection in a heterogeneous binary mixture. Part 2. Frozen waves

Published online by Cambridge University Press:  14 May 2019

Anatoliy Vorobev*
Affiliation:
Department of Engineering and Physical Sciences, University of Southampton, Southampton SO17 1BJ, UK
Tatyana Lyubimova
Affiliation:
Institute of Continuous Media Mechanics, Ural Branch RAS, Perm 614013, Russia Perm State University, Perm 614990, Russia
*
Email address for correspondence: A.Vorobev@soton.ac.uk

Abstract

The action of high-frequency vibrations on a heterogeneous binary mixture that fills in a closed container is numerically modelled to validate the theoretical model obtained in the first part of the work, and to investigate the role of interfacial stresses in the evolution of miscible boundaries. Only weightlessness conditions are considered. A recent experimental study reports the threshold ignition of the frozen waves at a miscible interface even under weightlessness conditions, which cannot be explained on the basis of the classical approach that represents a binary mixture as a single-phase fluid with an impurity. This effect, however, can be well explained on the basis of the phase-field equations that were derived in the first part of our work. In particular, we found that when the vibrational forcing is sufficiently strong (the vibrational forcing is primarily determined by the amplitude of the vibrational velocity), above a certain threshold value, then the interface becomes shaped into a ‘frozen’ (time independent to the naked eye) structure of several pillars (the frozen waves) with axes perpendicular to the directions of the vibrations. The threshold level of the vibrations is determined by the interfacial stresses that need to be associated with miscible interfaces. The time needed for setting up the frozen pattern is relatively small, determined by hydrodynamic processes, however this time grows exponentially near the threshold. The frozen pattern remains stable either indefinitely long (if liquids are partially miscible) or until the interface becomes invisible due to diffusive smearing (if liquids are miscible in all proportions). A further increase of the vibrational forcing alters the number of the pillars, which happens discretely when the intensity of the vibrations surpasses a sequence of further critical levels. Correlation of the results with the previous experimental and theoretical studies validates the new approach making it a useful tool for tracing thermo- and hydrodynamic changes in heterogeneous mixtures.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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References

Ahmadlouydarab, M. & Feng, J. J. 2014 Motion and coalescence of sessile drops driven by substrate wetting gradient and external flow. J. Fluid Mech. 746, 214235.Google Scholar
Bezdenezhnyh, N. A., Briskman, V. A., Lapin, A. Y., Lyubimov, D. V., Lyubimova, T. P., Tcherepanov, A. A. & Zakharov, I. A. 1991 The influence of high frequency tangential vibrations on the stability of the fluid interface in microgravity. Intl J. Microgravity Res. Appl. 4, 9697.Google Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 20782095.Google Scholar
Gandikota, G., Chatain, D., Amiroudine, S., Lyubimova, T. & Beysens, D. 2014a Frozen-wave instability in near-critical hydrogen subjected to horizontal vibration under various gravity fields. Phys. Rev. E 89, 012309.Google Scholar
Gandikota, G., Chatain, D., Lyubimova, T. & Beysens, D. 2014b Dynamic equilibrium under vibrations of h 2 liquid–vapor interface at various gravity levels. Phys. Rev. E 89, 063003.Google Scholar
Gaponenko, Y., Torregrosa, M. M., Yasnou, V., Mialdun, A. & Shevtsova, V. 2015a Interfacial pattern selection in miscible liquids under vibration. Soft Matt. 11, 82218224.Google Scholar
Gaponenko, Y. A. & Shevtsova, V. 2010 Effects of vibrations on dynamics of miscible liquids. Acta Astron. 66, 174182.Google Scholar
Gaponenko, Y. A., Torregrosa, M., Yasnou, V., Mialdun, A. & Shevtsova, V. 2015b Dynamics of the interface between miscible liquids subjected to horizontal vibration. J. Fluid Mech. 784, 342372.Google Scholar
Gaponenko, Y. A., Volpert, V. A., Zen’kovskaya, S. M. & Pojman, D. A. 2006 Effect of high-frequency vibration on convection in miscible fluids. J. Fluid Mech. 47, 190198.Google Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96127.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 5788.Google Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids. Springer.Google Scholar
Landau, L. D. & Lifshitz, E. M. 2010 Statistical Physics, Course of Theoretical Physics, vol. 5. Elsevier, Butterworth Heinemann.Google Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (2), 26172654.Google Scholar
Lyubimov, D. V. & Cherepanov, A. A. 1986 Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn. 6, 813.Google Scholar
Lyubimov, D. V., Ivantsov, A. O., Lyubimova, T. P. & Khilko, G. L. 2016 Numerical modeling of frozen wave instability in fluids with high viscosity contrast. Fluid Dyn. Res. 48, 61415.Google Scholar
Lyubimov, D. V., Khilko, G. L., Ivantsov, A. O. & Lyubimova, T. P. 2017 Viscosity effect on the longwave instability of a fluid interface subjected to horizontal vibrations. J. Fluid Mech. 814, 2441.Google Scholar
Lyubimov, D. V., Lyubimova, T. P. & Cherepanov, A. A. 2003 Dynamics of Interfaces Subject to Vibrations. FizMatLit.Google Scholar
Lyubimova, T., Ivantsov, A., Garrabos, Y., Lecoutre, C., Gandikota, G. & Beysens, D. 2017 Band instability in near-critical fluids subjected to vibration under weightlessness. Phys. Rev. E 95, 013105.Google Scholar
Lyubimova, T., Vorobev, A. & Prokopev, S. 2019 Rayleigh–Taylor instability of a miscible interface in a confined domain. Phys. Fluids 31, 014104.Google Scholar
Pojman, J. A., Whitmore, C., Liveri, M. L. T., Lombardo, R., Marszalek, J., Parker, R. & Zoltowski, B. 2006 Evidence for the existence of an effective interfacial tension between miscible fluids: isobutyric acidwater and 1-butanolwater in a spinning-drop tensiometer. Langmuir 22, 25692577.Google Scholar
Prokopev, S., Vorobev, A. & Lyubimova, T. 2019 Phase-field modeling of an immiscible liquid–liquid displacement in a capillary. Phys. Rev. E 99, 033113.Google Scholar
Shevtsova, V., Gaponenko, Y., Yasnou, V., Mialdun, A. & Nepomnyashchy, A. 2015 Wall-generated pattern on a periodically excited miscible liquid/liquid interface. Langmuir 31, 55505553.Google Scholar
Shevtsova, V., Gaponenko, Y. A., Yasnou, V., Mialdun, A. & Nepomnyashchy, A. 2016 Two-scale wave patterns on a periodically excited miscible liquid–liquid interface. J. Fluid Mech. 795, 409422.Google Scholar
Stevar, M. S. P. & Vorobev, A. 2012 Shapes and dynamics of miscible liquid/liquid interfaces in horizontal capillary tubes. J. Colloid Interface Sci. 383, 184197.Google Scholar
Talib, E., Jalikop, S. V. & Juel, A. 2007 The influence of viscosity on the frozen wave instability: theory and experiment. J. Fluid Mech. 584, 4568.Google Scholar
Vorobev, A. 2014 Dissolution dynamics of miscible liquid/liquid interfaces. Curr. Opin. Colloid Interface Sci. 19, 300308.Google Scholar
Vorobev, A. & Boghi, A. 2016 Phase-field modelling of a miscible system in spinning droplet tensiometer. J. Colloid Interface Sci. 482, 193204.Google Scholar
Vorobev, A. & Khlebnikova, E. 2018 Modelling of the rise and adsorption of a fluid inclusion. Intl J. Heat Mass Transfer 125, 801814.Google Scholar
Wolf, G. H. 1961 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Phys. 227, 291300.Google Scholar
Wolf, G. H. 1970 Dynamic stabilization of the interchange instability of a liquid–gas interface. Phys. Rev. Lett. 24, 444446.Google Scholar
Wolf, G. H. 2018 Dynamic stabilization of the Rayleigh–Taylor instability of miscible liquids and the related ‘frozen waves’. Phys. Fluids 30, 021701.Google Scholar
Xie, R. & Vorobev, A. 2016 On the phase-field modelling of a miscible liquid/liquid boundary. J. Colloid Interface Sci. 464, 4858.Google Scholar
Yoshikawa, H. N. & Wesfreid, J. E. 2011a Oscillatory Kelvin–Helmholtz instability. Part 1. A viscous theory. J. Fluid Mech. 675, 223248.Google Scholar
Yoshikawa, H. N. & Wesfreid, J. E. 2011b Oscillatory Kelvin–Helmholtz instability. Part 2. An experiment in fluids with a large viscosity contrast. J. Fluid Mech. 675, 249267.Google Scholar
Zoltowski, B., Chekanov, Y., Masere, J., Pojman, J. A. & Volpert, V. 2007 Evidence for the existence of an effective interfacial tension between miscible fluids. 2. Dodecyl acrylatepoly(dodecyl acrylate) in a spinning drop tensiometer. Langmuir 23, 55225531.Google Scholar