Hostname: page-component-5f745c7db-96s6r Total loading time: 0 Render date: 2025-01-06T20:54:46.808Z Has data issue: true hasContentIssue false
  • English
  • Français

Gill's stability problem may be unstable with horizontal heterogeneity in permeability

Published online by Cambridge University Press:  09 June 2022

B.M. Shankar Open the ORCID record for B.M. Shankar [Opens in a new window]  and
I.S. Shivakumara Open the ORCID record for I.S. Shivakumara [Opens in a new window]
B.M. Shankar*
Affiliation:
Department of Mathematics, PES University, Bangalore 560 085, India
I.S. Shivakumara
Affiliation:
Department of Mathematics, Bangalore University, Bangalore 560 056, India
*
† Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of thermal buoyant flow in a fluid-saturated vertical porous slab is studied under the assumption of weak and strong horizontal heterogeneities of the permeability. The two end vertical isothermal boundaries are impermeable and some paradigmatic cases of linear, quadratic and exponential heterogeneity models are deliberated. The stability/instability of the basic flow is examined by carrying out a numerical solution of the governing equations for the disturbances as Gill's proof (A.E. Gill, J. Fluid Mech, vol. 35, 1969, pp. 545–547) of linear stability is found to be ineffective. The possibilities of base flow becoming unstable due to heterogeneity in permeability are recognized, in contrast to Gill's stability problem. The neutral stability curves are presented and the critical Darcy–Rayleigh number for the onset of convective instability is computed for different values of the variable permeability constant. The similarities and differences between different heterogeneity models on the stability of fluid flow are clearly discerned.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 943 , 25 July 2022 , A20
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

Barletta, A. 2015 A proof that convection in a porous vertical slab may be unstable. J. Fluid Mech. 770, 273–288.CrossRefGoogle Scholar
Barletta, A. 2016 Instability of stationary two-dimensional mixed convection across a vertical porous layer. Phys. Fluids 28, 014101.CrossRefGoogle Scholar
Barletta, A. & Celli, M. 2017 Instability of parallel buoyant flow in a vertical porous layer with an internal heat source. Intl J. Heat Mass Transfer 111, 1063–1070.CrossRefGoogle Scholar
Barletta, A. & Celli, M. 2021 Anisotropy and the onset of the thermoconvective instability in a vertical porous layer. J. Heat Transfer 143, 102601.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Kuznetsov, A. 2012 Heterogeneity and onset of instability in Darcy's flow with a prescribed horizontal temperature gradient. J. Heat Transfer 134, 042602.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Ouarzazi, M.N. 2017 Unstable buoyant flow in a vertical porous layer with convective boundary conditions. Intl J. Therm. Sci. 120, 427–436.CrossRefGoogle Scholar
Barletta, A. & De B. Alves, L.S. 2014 On Gill's stability problem for non-Newtonian Darcy's flow. Intl J. Heat Mass Transfer 79, 759–768.CrossRefGoogle Scholar
Barletta, A. & Nield, D.A. 2012 On Hadley flow in a porous layer with vertical heterogeneity. J. Fluid Mech. 710, 304–323.CrossRefGoogle Scholar
Benenati, R.F. & Brosilow, C.B. 1962 Void fraction distribution in beds of spheres. AIChE J. 8, 359–361.CrossRefGoogle Scholar
Benham, G.P., Bickle, M.J. & Neufeld, J.A. 2021 Upscaling multiphase viscous-to-capillary transitions in heterogeneous porous media. J. Fluid Mech. 911, A59.CrossRefGoogle Scholar
Braester, C. & Vadasz, P. 1993 The effect of a weak heterogeneity of a porous medium on natural convection. J. Fluid Mech. 254, 345–362.CrossRefGoogle Scholar
Celli, M., Barletta, A. & Rees, D.A.S. 2017 Local thermal non-equilibrium analysis of the instability in a vertical porous slab with permeable sidewalls. Transp. Porous Med. 119, 539–553.CrossRefGoogle Scholar
Damene, D., Alloui, Z., Alloui, I. & Vasseur, P. 2021 Variable permeability effects on natural convection in a vertical porous layer with uniform heat flux from the side. Transp. Porous Med. 137, 287–306.CrossRefGoogle Scholar
Fajraoui, N., Fahs, M., Younes, A. & Sudret, B. 2017 Analyzing natural convection in porous enclosure with polynomial chaos expansions: effect of thermal dispersion, anisotropic permeability and heterogeneity. Intl J. Heat Mass Transfer 115, 205–224.CrossRefGoogle Scholar
Flavin, J.N. & Rionero, S. 1999 Nonlinear stability for a thermofluid in a vertical porous slab. Contin. Mech. Thermodyn. 11, 173–179.CrossRefGoogle Scholar
Gill, A.E. 1969 A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545–547.CrossRefGoogle Scholar
Naveen, S.B., Shankar, B.M. & Shivakumara, I.S. 2020 Finite Darcy-Prandtl number and maximum density effects on Gill's stability problem. J. Heat Transfer 142, 102601.CrossRefGoogle Scholar
Nguyen, V.T., Graf, T. & Guevara Morel, C.R. 2016 Free thermal convection in heterogeneous porous media. Geothermics 64, 152–162.CrossRefGoogle Scholar
Nield, D.A. 2008 General heterogeneity effects on the onset of convection in a porous medium. In Emerging Topics in Heat and Mass Transfer in Porous Media (ed. P. Vadasz), pp. 63–84. Springer.CrossRefGoogle Scholar
Nield, D.A. & Bejan, A. 2017 Convection in Porous Media, 5th edn. Springer International Publishing.CrossRefGoogle Scholar
Nield, D.A. & Kuznetsov, A.V. 2007 The effect of combined vertical and horizontal heterogeneity on the onset of convection in a bidisperse porous medium. Intl J. Heat Mass Transfer 50, 3329–3339.CrossRefGoogle Scholar
Rees, D.A.S. 1988 The stability of Prandtl–Darcy convection in a vertical porous layer. Intl J. Heat Mass Transfer 31, 1529–1534.CrossRefGoogle Scholar
Rees, D.A.S. 2011 The effect of local thermal nonequilibrium on the stability of convection in a vertical porous channel. Transp. Porous Med. 87, 459–464.CrossRefGoogle Scholar
Rees, D.A.S. & Pop, I. 2000 Vertical free convection in a porous medium with variable permeability effects. Intl J. Heat Mass Transfer 43, 2565–2571.CrossRefGoogle Scholar
Rionero, S. 2011 Onset of convection in porous materials with vertically stratified porosity. Acta Mech. 222, 261–272.CrossRefGoogle Scholar
Roblee, L.H.S., Baird, R.M. & Tierney, J.W. 1958 Radial porosity variations in packed beds. AIChE J. 4, 460–464.CrossRefGoogle Scholar
Salibindla, A.K.R., Subedi, R., Shen, V.C., Masuk, A.U.M. & Ni, R. 2018 Dissolution-driven convection in a heterogeneous porous medium. J. Fluid Mech. 857, 61–79.CrossRefGoogle Scholar
Schwartz, C.E. & Smith, J.M. 1953 Flow distribution in packed beds. Ind. Engng Chem. 45, 1209–1218.CrossRefGoogle Scholar
Scott, N.L. & Straughan, B. 2013 A nonlinear stability analysis of convection in a porous vertical channel including local thermal nonequilibrium. J. Math. Fluid Mech. 15, 171–178.CrossRefGoogle Scholar
Shafabakhsh, P., Fahs, M., Ataie-Ashtiani, B. & Simmons, C.T. 2019 Unstable density-driven flow in fractured porous media: the fractured elder problem. Fluids 4, 168.CrossRefGoogle Scholar
Shankar, B.M., Kumar, J. & Shivakumara, I.S. 2017 Stability of natural convection in a vertical layer of Brinkman porous medium. Acta Mech. 228, 1–19.CrossRefGoogle Scholar
Shankar, B.M., Kumar, J. & Shivakumara, I.S. 2020 Benchmark solution for the hydrodynamic stability of plane porous-Couette flow. Phys. Fluids 32, 104101.CrossRefGoogle Scholar
Shankar, B.M., Kumar, J. & Shivakumara, I.S. 2021 Numerical investigation of the stability of mixed convection in a differentially heated vertical porous slab. Appl. Maths Comput. 389, 125486.CrossRefGoogle Scholar
Shankar, B.M., Naveen, S.B. & Shivakumara, I.S. 2022 Stability of double-diffusive natural convection in a vertical porous layer. Transp. Porous Med. 141, 87–105.CrossRefGoogle Scholar
Shankar, B.M. & Shivakumara, I.S. 2017 On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid. Theor. Comput. Fluid Dyn. 31, 221–231.CrossRefGoogle Scholar
Shankar, B.M. & Shivakumara, I.S. 2020 Stability of porous-Poiseuille flow with uniform vertical throughflow: high accurate solution. Phys. Fluids 32, 044101.CrossRefGoogle Scholar
Shankar, B.M. & Shivakumara, I.S. 2021 Stability of Poiseuille flow in an anisotropic porous layer with oblique principal axes: more accurate solution. Z. Angew. Math. Mech. 101, e201900264.CrossRefGoogle Scholar
Shankar, B.M., Shivakumara, I.S. & Naveen, S.B. 2021 Density maximum and finite Darcy-Prandtl number outlooks on Gill's stability problem subject to a lack of thermal equilibrium. Phys. Fluids 33, 124108.CrossRefGoogle Scholar
Straughan, B. 1988 A nonlinear analysis of convection in a porous vertical slab. Geophys. Astrophys. Fluid Dyn. 42, 269–275.CrossRefGoogle Scholar
Wolanski, E.J. 1973 Convection in a vertical porous slab. Phys. Fluids 16, 2014–2016.CrossRefGoogle Scholar