Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T06:11:58.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Buoyancy instabilities in a liquid layer subjected to an oblique temperature gradient

Published online by Cambridge University Press:  22 February 2022

Ramkarn Patne Open the ORCID record for Ramkarn Patne [Opens in a new window]  and
Alexander Oron Open the ORCID record for Alexander Oron [Opens in a new window]
Ramkarn Patne*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology (ISM) Dhanbad, Jharkhand 826004, India Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana 502285, India
Alexander Oron
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the temporal and spatio-temporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an oblique temperature gradient (OTG) with horizontal and vertical components, denoted as HTG and VTG, respectively. General linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers $Pr>1$ via inducing an extra vertical temperature gradient opposing the VTG through energy convection. For $Pr<1$, a new mode of instability arises as a result of a velocity jump in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatio-temporal dynamics of the temperature perturbations is derived. Spatio-temporal stability analysis reveals the existence of a convectively unstable long-wave regime due to the HTG. Weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to Hopf in the presence of the OTG. Numerical investigation of the spatio-temporal dynamics of the temperature disturbances in the layer in the weakly nonlinear regime reveals the emergence of travelling wave regimes propagating against the direction of the HTG and whose phase speed depends on $Pr$. In the case of a small but non-zero Biot number, the wavelength of these travelling waves is larger than that of the fastest-growing mode obtained from GLSA.

JFM classification

Convection: Buoyancy-driven instability Flow Control: Instability control Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A11
Check for updates
Copyright
© The Author(s), 2022. 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

Boyd, J.P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Braunsfurth, M.G., Skeldon, A.C., Juel, A., Mullin, T. & Riley, D.S. 1997 Free convection in liquid gallium. J. Fluid Mech. 342, 295–314.CrossRefGoogle Scholar
Briggs, R.J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Chapman, C.J. & Proctor, M.R.E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759–782.CrossRefGoogle Scholar
Cheng, P.J., Chen, C.K. & Lai, H.Y. 2001 Nonlinear stability analysis of thin viscoelastic film flow traveling down along a vertical cylinder. Nonlinear Dyn. 24, 305–332.CrossRefGoogle Scholar
Drazin, P.G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gertsberg, V.L. & Sivashinsky, G.I. 1981 Large cells in nonlinear Rayleigh–Bénard convection. Prog. Theor. Phys. 66, 1219.CrossRefGoogle Scholar
Gill, A.E. 1974 A theory of thermal oscillations in liquid metals. J. Fluid Mech. 64, 577–588.CrossRefGoogle Scholar
Hart, J.E. 1972 Stability of thin non-rotating Hadley circulations. J. Atmos. Sci. 29, 687–697.2.0.CO;2>CrossRefGoogle Scholar
Hart, J.E. 1983 A note of stability of low-Prandtl-number Hadley circulations. J. Fluid Mech. 132, 271–181.CrossRefGoogle Scholar
Hinch, E.J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.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, 391–413.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
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 173–537.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, 267–281.CrossRefGoogle Scholar
Kistler, S.F. & Schweizer, P.M. 1997 Liquid Film Coating: Scientific Principles and Their Technological Implications. Springer.CrossRefGoogle Scholar
Knobloch, E. 1990 Pattern selection in long-wavelength convection. Physica D 41, 450–479.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
Kupfer, K., Bers, A. & Ram, A.K. 1987 The cusp map in complex-frequency plane for absolute instabilities. Phys. Fluids 30, 3075–3082.CrossRefGoogle Scholar
Lappa, M. 2010 Thermal Convection: Patterns, Evolution and Stability. Wiley.Google Scholar
Levenspiel, O. 1999 Chemical Reaction Engineering. John Wiley & Sons.Google Scholar
Mercier, J.F. & Normand, C. 1996 Buoyant-thermocapillary instabilities of differentially heated liquid layers. Phys. Fluids 8, 1433.CrossRefGoogle Scholar
Nepomnyashchy, A.A. & Simanovskii, I.B. 2004 Influence of thermocapillary effect and interfacial heat release on convective oscillations in a two-layer system. Phys. Fluids 16, 1127.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
Oron, A. 2000 Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films. Phys. Fluids 12, 1633–1645.CrossRefGoogle Scholar
Oron, A. & Bankoff, S.G. 1999 Dewetting of a heated surface by an evaporating liquid film under conjoining/disjoining pressures. J. Colloid Interface Sci. 218, 152–166.CrossRefGoogle ScholarPubMed
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636.CrossRefGoogle Scholar
Oron, A. & Nepomnyashchy, A.A. 2004 Long-wavelength thermocapillary instability with the Soret effect. Phys. Rev. E 69, 016313.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
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
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, 561–574.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, 65–84.CrossRefGoogle Scholar