Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-17T02:13:07.688Z Has data issue: false hasContentIssue false

Reaction-induced Kelvin–Helmholtz instability in a layered channel flow

Published online by Cambridge University Press:  19 January 2023

Surya Narayan Maharana
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
Kirti Chandra Sahu
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi 502284, Sangareddy, Telangana, India
Manoranjan Mishra*
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
*
Email address for correspondence: manoranjan@iitrpr.ac.in

Abstract

We show that a vertical viscosity stratification at a localized region caused by a chemical reaction yields an inconspicuous shear layer. A chemo-hydrodynamic Kelvin–Helmholtz instability or cat-eye-shaped morphology develops at one reaction front, while the other front diffuses steadily over time. Through linear stability and nonlinear simulations, the existence of such instabilities is established if the log-mobility ratio exceeds a critical value. We find unique scalings between the stable and unstable zones that demonstrate how the influence of variations in solute diffusion on instability can be eliminated. The observed unstable patterns agree with existing experimental results.

Type
JFM Papers
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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

Caulfield, C.P. 2021 Layering, instabilities, and mixing in turbulent stratified flows. Annu. Rev. Fluid Mech. 53 (1), 113145.CrossRefGoogle Scholar
De Wit, A. 2020 Chemo-hydrodynamic patterns and instabilities. Annu. Rev. Fluid Mech. 52 (1), 531555.CrossRefGoogle Scholar
D'olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2009 Convective/absolute instability in miscible core-annular flow. Part 1: experiments. J. Fluid Mech. 618, 305322.CrossRefGoogle Scholar
Drazin, P.G. & Reid, W.H. 1985 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gálfi, L. & Rácz, Z. 1988 Properties of the reaction front in an $A+B \rightarrow C$ type reaction–diffusion process. Phys. Rev. A 38, 31513154.CrossRefGoogle Scholar
Gallaire, F. & Brun, P.-T. 2017 Fluid dynamic instabilities: theory and application to pattern forming in complex media. Phil. Trans. R. Soc. A 375 (2093), 20160155.CrossRefGoogle ScholarPubMed
Gérard, T. & De Wit, A. 2009 Miscible viscous fingering induced by a simple $A+B \rightarrow C$ chemical reaction. Phys. Rev. E 79, 016308.CrossRefGoogle Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 11771192.CrossRefGoogle Scholar
Govindarajan, R., L'vov, V.S. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrasts. Phys. Rev. Lett. 87, 174501.CrossRefGoogle ScholarPubMed
Govindarajan, R. & Sahu, K.C. 2014 Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46 (1), 331353.CrossRefGoogle Scholar
Hejazi, S.H., Trevelyan, P.M.J., Azaiez, J. & De Wit, A. 2010 Viscous fingering of a miscible reactive $A+B \rightarrow C$ interface: a linear stability analysis. J. Fluid Mech. 652, 501528.CrossRefGoogle Scholar
Horner-Devine, A.R., Hetland, R.D. & MacDonald, D.G. 2015 Mixing and transport in coastal river plumes. Annu. Rev. Fluid Mech. 47 (1), 569594.CrossRefGoogle Scholar
Hu, X. & Cubaud, T. 2018 Viscous wave breaking and ligament formation in microfluidic systems. Phys. Rev. Lett. 121, 044502.CrossRefGoogle ScholarPubMed
Jayaprakash, V., Costalonga, M., Dhulipala, S. & Varanasi, K.K. 2020 Enhancing the injectability of high concentration drug formulations using core annular flows. Adv. Healthc. Mater. 9 (18), 2001022.CrossRefGoogle ScholarPubMed
Joseph, D.D., Bai, R., Chen, K.P. & Renardy, Y.Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29 (1), 6590.CrossRefGoogle Scholar
Landel, J.R. & Wilson, D.I. 2021 The fluid mechanics of cleaning and decontamination of surfaces. Annu. Rev. Fluid Mech. 53 (1), 147171.CrossRefGoogle Scholar
Maharana, S.N. & Mishra, M. 2021 Reaction induced interfacial instability of miscible fluids in a channel. J. Fluid Mech. 925, A3.CrossRefGoogle Scholar
Maharana, S.N. & Mishra, M. 2022 Effects of low and high viscous product on Kelvin–Helmholtz instability triggered by $A+B\rightarrow C$ type reaction. Phys. Fluids 34 (1), 012104.CrossRefGoogle Scholar
Nagatsu, Y., Matsuda, K., Kato, Y. & Tada, Y. 2007 Experimental study on miscible viscous fingering involving viscosity changes induced by variations in chemical species concentrations due to chemical reactions. J. Fluid Mech. 571, 475493.CrossRefGoogle Scholar
Nepf, H.M. 2012 Flow and transport in regions with aquatic vegetation. Annu. Rev. Fluid Mech. 44 (1), 123142.CrossRefGoogle Scholar
Podgorski, T., Sostarecz, M.C., Zorman, S. & Belmonte, A. 2007 Fingering instabilities of a reactive micellar interface. Phys. Rev. E 76, 016202.CrossRefGoogle ScholarPubMed
Rayleigh, Lord 1879 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 1 (1), 5772.CrossRefGoogle Scholar
Regner, M., Henningsson, M., Wiklund, J., Östergren, K. & Trägårdh, C. 2007 Predicting the displacement of yoghurt by water in a pipe using CFD. Chem. Engng Technol. 30 (7), 844853.CrossRefGoogle Scholar
Riolfo, L.A., Nagatsu, Y., Iwata, S., Maes, R., Trevelyan, P.M.J. & De Wit, A. 2012 Experimental evidence of reaction-driven miscible viscous fingering. Phys. Rev. E 85, 015304.CrossRefGoogle ScholarPubMed
Roediger, E., Kraft, R.P., Nulsen, P., Churazov, E., Forman, W., Brüggen, M. & Kokotanekova, R. 2013 Viscous Kelvin–Helmholtz instabilities in highly ionized plasmas. Mon. Not. R. Astron. Soc. 436 (2), 17211740.CrossRefGoogle Scholar
Rongy, L., Trevelyan, P.M.J. & De Wit, A. 2008 Dynamics of $A+B \rightarrow C$ reaction fronts in the presence of buoyancy-driven convection. Phys. Rev. Lett. 101, 084503.CrossRefGoogle Scholar
Sahu, K.C., Ding, H., Valluri, P. & Matar, O.K. 2009 Linear stability analysis and numerical simulation of miscible two-layer channel flow. Phys. Fluids 21 (4), 042104.CrossRefGoogle Scholar
Sahu, K.C. & Govindarajan, R. 2016 Linear stability analysis and direct numerical simulation of two-layer channel flow. J. Fluid Mech. 798, 889909.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flows with viscosity stratification. J. Fluid Mech. 592, 2349.CrossRefGoogle Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323348.CrossRefGoogle Scholar
Sharma, V., Pramanik, S., Chen, C.-Y. & Mishra, M. 2019 A numerical study on reaction-induced radial fingering instability. J. Fluid Mech. 862, 624638.CrossRefGoogle Scholar
Talon, L. & Meiburg, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities: instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484506.CrossRefGoogle Scholar
Tan, C.T. & Homsy, G.M. 1987 Stability of miscible displacements in porous media: radial source flow. Phys. Fluids 30 (5), 12391245.CrossRefGoogle Scholar
Winant, C.D. & Browand, F.K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63 (2), 237255.CrossRefGoogle Scholar