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Linear dynamics of a thick liquid layer subjected to an oblique temperature gradient

Published online by Cambridge University Press:  22 May 2024

Ganesh Dixit ,
Syed Furqan Bukhari  and
Ramkarn Patne Open the ORCID record for Ramkarn Patne [Opens in a new window]
Ganesh Dixit
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana 502285, India
Syed Furqan Bukhari
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana 502285, India
Ramkarn Patne*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana 502285, India
*
†Email address for correspondence: ramkarn@che.iith.ac.in
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Abstract

The present study aims to demonstrate the application of the oblique temperature gradient (OTG) to manipulate the buoyancy instabilities and resulting mixing. To accomplish this, we consider a model consisting of a liquid layer supported by a perfectly conducting bottom wall from below and the upper surface is exposed to ambient inert gas in a shallow container. The stability analysis employs the pseudospectral method and perturbation energy budget approach. The imposed OTG consists of a negative vertical temperature gradient (VTG), which leads to unstable density stratification, and a negative horizontal temperature gradient (HTG) component responsible for the Hadley circulation, which can be utilised to control instabilities. Without the HTG, a Rayleigh–Bénard mode of instability exists due to the imposed VTG termed as a VTG mode. The imposed HTG induces a linear VTG term and a quintic polynomial VTG term in the bottom wall-normal coordinate $y$. The induced linear VTG term reinforces the imposed VTG, while the induced quintic polynomial VTG term opposes the imposed VTG. For the vanishing Biot number ($Bi$), the induced quintic VTG term stabilises the VTG mode for the Prandtl number, $Pr=7$. However, for $Pr=0.01$, the Reynolds stresses associated with the Hadley circulation destabilise the VTG mode. For non-zero $Bi$, new streamwise and spanwise instability modes arise for $Pr=7$ due to the induced linear VTG term. Physical arguments show that the unstable density stratification near the gas–liquid interface caused by the synergistic effect of the imposed VTG and the induced linear VTG term lead to the existence of the predicted new modes. The present study thus shows that the strength of the HTG can be utilised to control the instability modes and critical parameters, thereby demonstrating the utility of the OTG in manipulating the buoyancy instabilities.

JFM classification

Convection: Buoyancy-driven instability Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 987 , 25 May 2024 , A32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

‡

G.D. and S.F.B. contributed equally.

References

Bau, H.H., Zhong, J.H. & Yi, M.Q. 2001 A minute magneto hydrodynamic (MHD) mixer. Sensors Actuators B 79, 207–215.CrossRefGoogle Scholar
Braunsfurth, M.G., Skeldon, A.C., Juel, A., Mullin, T. & Riley, D.S. 1997 Free convection in liquid gallium. J. Fluid Mech. 342, 295.CrossRefGoogle Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Drazin, P.G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Ezersky, A.B., Garcimartin, A., Mancini, H.L. & Perez-Garcia, C. 1993 Spatiotemporal structure of hydrothermal waves in Marangoni convection. Phys. Rev. E 48, 4414.CrossRefGoogle ScholarPubMed
Gill, A.E. 1974 A theory of thermal oscillations in liquid metals. J. Fluid Mech. 64, 577.CrossRefGoogle Scholar
Hart, J.E. 1972 Stability of thin non-rotating Hadley circulations. J. Atmos. Sci. 29, 687.2.0.CO;2>CrossRefGoogle Scholar
Hart, J.E. 1983 A note of stability of low-Prandtl-number Hadley circulations. J. Fluid Mech. 132, 271.CrossRefGoogle Scholar
Hof, B., Juel, A., Zhao, L., Henry, D., Ben Hadid, H. & Mullin, T. 2004 On the onset of oscillatory convection in molten gallium. J. Fluid Mech. 515, 391.CrossRefGoogle Scholar
Hu, K.X., He, M. & Chen, Q.S. 2016 Instability of thermocapillary liquid layers for Oldroyd-B fluid. Phys. Fluids 28, 033105.CrossRefGoogle Scholar
Hu, K.X., He, M., Chen, Q.S. & Liu, R. 2017 Linear stability of thermocapillary liquid layers of a shear-thinning fluid. Phys. Fluids 29, 073101.CrossRefGoogle Scholar
Hurle, D.T.J., Jakeman, E. & Johnson, C.P. 1974 Convective temperature oscillations in molten gallium. J. Fluid Mech. 64, 565–576.CrossRefGoogle Scholar
Juel, A., Mullin, J., Ben Hadid, H. & Henry, D. 2001 Three-dimensional free convection in molten gallium. J. Fluid Mech. 436, 267.CrossRefGoogle Scholar
Kistler, S.F. & Schweizer, P.M. 1997 Liquid Film Coating: Scientific Principles and Their Technological Implications. Springer.CrossRefGoogle Scholar
Kowal, K.N., Davis, S.H. & Voorhees, P.W. 2018 Thermocapillary instabilities in a horizontal liquid layer under partial basal slip. J. Fluid Mech. 855, 839–859.CrossRefGoogle Scholar
Kuo, H.P. & Korpela, S.A. 1988 Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side. Phys. Fluids 33, 31.Google Scholar
Lappa, M. 2010 Thermal Convection: Patterns, Evolution and Stability. Wiley.Google Scholar
Laure, P. & Roux, B. 1989 Linear and non-linear analysis of the Hadley circulation. J. Cryst. Growth 97, 226–234.CrossRefGoogle Scholar
Levenspiel, O. 1999 Chemical Reaction Engineering. John Wiley & Sons.Google Scholar
Li, M., Xu, S. & Kumacheva, E. 2000 Convection in polymeric fluids subjected to vertical temperature gradients. Macromolecules 33, 4972–4978.CrossRefGoogle Scholar
Mercier, J.F. & Normand, C. 1996 Buoyant-thermocapillary instabilities of differentially heated liquid layers. Phys. Fluids 8, 1433.CrossRefGoogle Scholar
Mizev, A.I. & Schwabe, D. 2009 Convective instabilities in liquid layers with free upper surface under the action of an inclined temperature gradient. Phys. Fluids 21, 112102.CrossRefGoogle Scholar
Nepomnyashchy, A.A. & Simanovskii, I.B. 2009 Dynamics of ultra-thin two-layer films under the action of inclined temperature gradients. J. Fluid Mech. 631, 165–197.CrossRefGoogle Scholar
Nield, D.A. 1994 Convection induced by an inclined temperature gradient in a shallow horizontal layer. Intl J. Heat Fluid Flow 15, 157.CrossRefGoogle Scholar
Oddy, M.H., Santiago, J.G. & Mikkelson, J.C. 2001 Electrokinetic instability micromixing. Anal. Chem. 73, 5822–5832.CrossRefGoogle ScholarPubMed
Ortiz-Pérez, A.S. & Dávalos-Orozco, L.A. 2011 Convection in a horizontal fluid layer under an inclined temperature gradient. Phys. Fluids 23 (8), 084107.CrossRefGoogle Scholar
Ortiz-Pérez, A.S. & Dávalos-Orozco, L.A. 2014 Convection in a horizontal fluid layer under an inclined temperature gradient for Prandtl numbers $Pr >1$. Intl J. Heat Mass Transfer 68, 444–455.CrossRefGoogle Scholar
Ospennikov, N.A. & Schwabe, D. 2004 Thermocapillary flow without return flow-linear flow. Exp. Fluids 36, 938–945.CrossRefGoogle Scholar
Patne, R. 2024 Effect of inhaled air temperature on mucus layer in the proximal airways. J. Fluid Mech. 978, A15.CrossRefGoogle Scholar
Patne, R., Agnon, Y. & Oron, A. 2020 a Marangoni instability in the linear Jeffreys fluid with a deformable surface. Phys. Rev. Fluids 5, 084005.CrossRefGoogle Scholar
Patne, R., Agnon, Y. & Oron, A. 2020 b Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient: effect of a prescribed normal temperature gradient at the substrate. Phys. Fluids 32, 112109.CrossRefGoogle Scholar
Patne, R., Agnon, Y. & Oron, A. 2021 a Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient. J. Fluid Mech. 906, A12.CrossRefGoogle Scholar
Patne, R., Agnon, Y. & Oron, A. 2021 b Thermocapillary instability in a viscoelastic liquid layer under an imposed oblique temperature gradient. Phys. Fluids 33, 012107.CrossRefGoogle Scholar
Patne, R. & Chandarana, J. 2023 Spatio-temporal dynamics of a two-layer pressure-driven flow subjected to a wall-normal temperature gradient. J. Fluid Mech. 957, A11.CrossRefGoogle Scholar
Patne, R. & Oron, A. 2022 Buoyancy instabilities in a liquid layer subjected to an oblique temperature gradient. J. Fluid Mech. 937, A11.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shklyaev, O.E. & Nepomnyashchy, A.A. 2004 Thermocapillary flows under an inclined temperature gradient. J. Fluid Mech. 504, 99–132.CrossRefGoogle Scholar
Smith, M.K. & Davis, S.H. 1983 a Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119–144.CrossRefGoogle Scholar
Smith, M.K. & Davis, S.H. 1983 b Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-wave instabilities. J. Fluid Mech. 132, 145–162.CrossRefGoogle Scholar
Sweet, D., Jakeman, E. & Hurle, D.T.J. 1977 Free convection in the presence of both vertical and horizontal temperature gradients. Phys. Fluids 20, 1412.CrossRefGoogle Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Walton, I.C. 1985 The effect of a shear flow on convection near a two-dimesional hot-patch. Q. J. Mech. Appl. Maths 38, 561.CrossRefGoogle Scholar
Wang, T.-M. & Korpela, S.A. 1989 Convection rolls in a shallow cavity heated from the side. Phys. Fluids A 1, 947.CrossRefGoogle Scholar
Weber, J.E. 1973 On thermal convection between non-uniformly heated planes. Intl J. Heat Mass Transfer 16, 961.CrossRefGoogle Scholar
Weber, J.E. 1978 On the stability of thermally driven shear flow heated from below. J. Fluid Mech. 87, 65.CrossRefGoogle Scholar
Yi, M.Q., Qian, S.Z. & Bau, H.H. 2002 A magnetohydrodynamic chaotic stirrer. J. Fluid Mech. 468, 153–157.CrossRefGoogle Scholar