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Solutal Marangoni instability in layered two-phase flows

Published online by Cambridge University Press:  14 March 2016

Jason R. Picardo ,
T. G. Radhakrishna  and
S. Pushpavanam
Jason R. Picardo
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, TN 600036, India
T. G. Radhakrishna
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, TN 600036, India
S. Pushpavanam*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, TN 600036, India
*
†Email address for correspondence: spush@iitm.ac.in
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Abstract

In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided that a concentration gradient is maintained across the fluids. One long-wave and two short-wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long-wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long- and short-wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large-interfacial-tension limit), the flow can become unstable to short-wave disturbances. On increasing the Reynolds number, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.

JFM classification

Instability: Instability Convection: Marangoni convection Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 793 , 25 April 2016 , pp. 280 - 315
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Copyright
© 2016 Cambridge University Press 

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References

Aljbour, S., Yamada, H. & Tagawa, T. 2010 Sequential reaction-separation in a microchannel reactor for liquid–liquid phase transfer catalysis. Top. Catal. 53, 694–699.Google Scholar
Assmann, N., Ładosz, A. & Rudolf von Rohr, P. 2013 Continuous micro liquid–liquid extraction. Chem. Engng Technol. 36 (6), 921–936.CrossRefGoogle Scholar
Blyth, M. G., Luo, H. & Pozrikidis, C. 2007 Surfactant-driven instability in two-fluid pipe and channel flows. Proc. Appl. Maths Mech. 7, 1100601–1100602.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004a Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis. J. Engng Maths 50 (2–3), 329–341.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004b Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250.Google Scholar
Boomkamp, P. A. M., Boersma, B. J., Miesen, R. H. M. & Beijnon, G. V. 1997 A Chebyshev collocation method for solving two-phase flow stability problems. J. Comput. Phys. 132 (2), 191–200.Google Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.Google Scholar
Chomaz, J. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Dutta, N. N. & Patil, G. S. 1993 Effect of phase transfer catalysts on the interfacial tension of water/toluene system. Can. J. Chem. Engng 71, 802–804.Google Scholar
Evans, A. W. 1937 The effect of uni-univalent electrolytes upon the interfacial tension between normal-hexane and water. Trans. Faraday Soc. 33, 794–800.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14 (7), L45–L48.Google Scholar
Frenkel, A. L. & Halpern, D. 2005 Effect of inertia on the insoluble-surfactant instability of a shear flow. Phys. Rev. E 71 (1), 16302.Google Scholar
Fries, D. Maria, Voitl, T. & von Rohr, P. Rudolf 2008 Liquid extraction of vanillin in rectangular microreactors. Chem. Engng Technol. 31 (8), 1182–1187.Google Scholar
Goussis, D. A. & Kelly, R. E. 1990 On the thermocapillary instabilities in a liquid layer heated from below. Intl J. Heat Mass Transfer 33 (10), 2237–2245.Google Scholar
Gumerman, R. J. & Homsy, G. M. 1974 Convective instabilities in concurrent two phase flow: part I. Linear stability. AIChE J. 20 (5), 981–988.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A. L. 2003 Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J. Fluid Mech. 485, 191–220.Google Scholar
Harkins, W. D. & Humphery, E. C. 1916 The surface tension at the interface between two liquids, and the effect of acids, salts and bases upon the interfacial tension. J. Am. Chem. Soc. 38 (2), 242–246.Google Scholar
Hennenberg, M., Sorensen, T. S. & Sanfeld, A. 1977 Deformational instability of a plane interface with transfer of matter. Part 1. Non-oscillatory critical states with a linear concentration profile. J Chem. Soc. Faraday Trans. II 73, 48–66.CrossRefGoogle Scholar
Hotokezaka, H., Tokeshi, M., Harada, M., Kitamori, T. & Ikeda, Y. 2005 System for high-level radioactive waste using microchannel chip–extraction behavior of metal ions from aqueous phase to organic phase in microchannel. Prog. Nucl. Energy 47 (1), 439–447.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Javed, K. H., Thornton, J. D. & Anderson, T. J. 1989 Surface phenomena and mass transfer rates in liquid–liquid systems: part 2. AIChE J. 35 (7), 1125–1136.Google Scholar
Johns, L. E. & Narayanan, R. 2002 Interfacial Instability. Springer.Google Scholar
Kovalchuk, N. M. & Vollhardt, D. 2006 Marangoni instability and spontaneous non-linear oscillations produced at liquid interfaces by surfactant transfer. Adv. Colloid Interface Sci. 120, 1–31.Google Scholar
Leal, G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.Google Scholar
Lin, S. P. & Chen, J. N. 1998 Role played by the interfacial shear in the instability mechanism of a viscous liquid jet surrounded by a viscous gas in a pipe. J. Fluid Mech. 376, 37–51.Google Scholar
Picardo, J. R., Radhakrishna, T. G., Anil, B. V., Sundari, R. & Pushpavanam, S. 2015 Modelling extraction in microchannels with stratified flow: channel geometry, flow configuration and Marangoni stresses. Indian Chem. Engr. 57 (3–4), 322–358.Google Scholar
Samanta, A. 2013 Effect of surfactant on two-layer channel flow. J. Fluid Mech. 735, 519–552.CrossRefGoogle Scholar
Schwarzenberger, K., Köllner, T., Linde, H., Boeck, T., Odenbach, S. & Eckert, K. 2014 Pattern formation and mass transfer under stationary solutal Marangoni instability. Adv. Colloid Interface Sci. 206, 344–371.Google Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J. Fluid Mech. 19 (3), 321–340.CrossRefGoogle Scholar
Šinkovec, E., Pohar, A. & Krajnc, M. 2013 Phase transfer catalyzed esterification: modeling and experimental studies in a microreactor under parallel flow conditions. Microfluid. Nanofluid. 14, 489–498.Google Scholar
Smith, K. A. 1966 On convective instability induced by surface-tension gradients. J. Fluid Mech. 24 (2), 401–414.CrossRefGoogle Scholar
Sternling, C. V. & Scriven, L. E. 1959 Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AIChE J. 5 (4), 514–523.Google Scholar
Sun, Z. F. & Fahmy, M. 2006 Onset of Rayleigh–Benard–Marangoni convection in gas–liquid mass transfer with two-phase flow: theory. Ind. Engng Chem. Res. 45 (9), 3293–3302.Google Scholar
Tadmouri, R., Kovalchuk, N. M., Pimienta, V., Vollhardt, D. & Micheau, J. 2010 Transfer of oxyethylated alcohols through water/heptane interface: transition from non-oscillatory to oscillatory behaviour. Colloids Surf. A 354, 134–142.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.Google Scholar
Wang, P. & Anderko, A. 2013 Modeling interfacial tension in liquid–liquid systems containing electrolytes. Ind. Engng Chem. Res. 52, 6822–6840.Google Scholar
Wei, H. 2005 On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173–200.Google Scholar
Wei, H. 2006 Shear-flow and thermocapillary interfacial instabilities in a two-layer viscous flow. Phys. Fluids 18, 064109.Google Scholar
Wei, H. 2007 Role of base flows on surfactant-driven interfacial instabilities. Phys. Rev. E 75, 036306.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1988 Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids 31 (11), 3225–3238.Google Scholar
Yih, C. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (2), 337–352.Google Scholar
You, X., Zhang, L. & Zheng, J. 2014 Marangoni instability of immiscible liquid–liquid stratified flow with a planar interface in the presence of interfacial mass transfer. J. Taiwan Inst. Chem. E 45 (3), 772–779.Google Scholar
Zaisha, M., Ping, L., Guangji, Z. & Chao, Y. 2008 Numerical simulation of the Marangoni effect with interphase mass transfer between two planar liquid layers. Chin. J. Chem. Engng 16 (2), 161–170.Google Scholar
Znidarsic-Plazl, P. & Plazl, I. 2007 Steroid extraction in a microchannel system – mathematical modelling and experiments. Lab. on a Chip 7, 883–889.Google Scholar