Abstract
The stability of natural convection in an internally heated vertical porous layer confined between two impermeable boundaries which are kept at different constant temperatures is investigated. The momentum transfer is modeled by adopting Darcy’s law including time-dependent velocity term contribution. The conduction stream function and temperature fields are significantly altered due to internal heating, and the linear instability is analyzed through a study of normal mode perturbations on the base flow. The neutral stability curves and the critical Darcy–Rayleigh number for the onset of instability are evaluated by solving the stability eigenvalue problem numerically. It has been established that a uniform volumetric heat source and the Prandtl–Darcy number reinforce together in initiating the instability of the base flow under certain conditions despite their isolation presence evidences stability for all infinitesimal perturbations. Although the internal heat source strength is to hasten the onset of instability, and the Prandtl–Darcy number is found to induct both destabilizing and stabilizing impact on the base flow.
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References
Barletta, A., Celli, M.: Instability of parallel buoyant flow in a vertical porous layer with an internal heat source. Int. J. Heat Mass Transfer 111, 1063–1070 (2017)
Barletta, A., de Alves, L.S.B.: On Gill’s stability problem for non-Newtonian Darcy’s flow. Int. J. Heat Mass Transfer 79, 759–768 (2014)
Barletta, A., Rees, D.A.S., Pulvirenti, B.: Buoyant flow instability induced by a uniform internal heat source in a vertical annular porous layer. Int. J. Heat Mass Transfer 194, 122935 (2022)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004)
Gasser, R.D., Kazimi, M.S.: Onset of convection in a porous medium with internal heat generation. J. Heat Transfer 98, 49–54 (1976)
George, J.H., Gunn, R.D., Straughan, B.: Patterned ground formation and penetrative convection in porous media. Geophys. Astrophys. Fluid Dyn. 46, 135–158 (1989)
Gill, A.E.: A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545–547 (1969)
He, X.S., Georgiadis, J.G.: Natural convection in porous media: effect of weak dispersion on bifurcation. J. Fluid Mech. 216, 285–298 (1990)
Khalili, A., Shivakumara, I.S.: Onset of convection in a porous layer with net through-flow and internal heat generation. Phys. Fluids 10, 315–317 (1998)
Lazzari, S., Celli, M., Barletta, A.: Stability of a buoyant Oldroyd-B flow saturating a vertical porous layer with open boundaries. Fluids 6, 375 (2021)
McKenzie, D.P., Roberts, J.M., Weiss, N.O.: Convection in the earth’s mantle: towards a numerical simulation. J. Fluid Mech. 62, 465–538 (1974)
Naveen, S.B., Shankar, B.M., Shivakumara, I.S.: Finite Darcy–Prandtl number and maximum density effects on Gill’s stability problem. J. Heat Transfer 142, 102601 (2020)
Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)
Nield, D.A., Kuznetsov, A.V.: Onset of convection with internal heating in a weakly heterogeneous porous medium. Transp. Porous Med. 98, 543–552 (2013)
Nouri-Borujerdi, A., Noghrehabadi, A.R., Rees, D.A.S.: Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model. Transp. Porous Med. 69, 343–357 (2007)
Nouri-Borujerdi, A., Noghrehabadi, A.R., Rees, D.A.S.: Influence of Darcy number on the onset of convection in a porous layer with a uniform heat source. Int. J. Therm. Sci. 47, 1020–1025 (2008)
Rees, D.A.S.: The stability of Prandtl–Darcy convection in a vertical porous layer. Int. J. Heat Mass Transfer 31, 1529–1534 (1988)
Rhee, S.J., Dhir, V.K.: Catton, I: natural convection heat transfer in beds of inductively heated particles. J. Heat Transfer 100, 78–85 (1978)
Shankar, B.M., Shivakumara, I.S.: On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid. Theor. Comput. Fluid Dyn. 31, 221–231 (2017)
Shankar, B.M., Shivakumara, I.S.: Gill’s stability problem may be unstable with horizontal heterogeneity in permeability. J. Fluid Mech. 943, A20 (2022)
Shankar, B.M., Kumar, J., Shivakumara, I.S.: Stability of natural convection in a vertical layer of Brinkman porous medium. Acta Mech. 228, 1–19 (2017)
Shankar, B.M., Shivakumara, I.S., Kumar, J.: Benchmark solution for the hydrodynamic stability of plane porous-Couette flow. Phys. Fluids 32, 104104 (2020)
Shankar, B.M., Kumar, J., Shivakumara, I.S.: Stability of double-diffusive natural convection in a vertical fluid layer. Phys. Fluids 33, 094113 (2021a)
Shankar, B.M., Shivakumara, I.S., Naveen, S.B.: Density maximum and finite Darcy–Prandtl number outlooks on Gill’s stability problem subject to a lack of thermal equilibrium. Phys. Fluids 33, 124108 (2021b)
Straughan, B.: Convection with local thermal non-equilibrium and microfluidic effects, Advances in Mechanics and Mathematics, vol. 32. Springer, Cham (2015)
Straughan, B., Walker, D.W.: Anisotropic porous penetrative convection. Proc. R. Soc. London Ser. A 452, 97–115 (1996)
Vadasz, P.: Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J. Fluid Mech. 376, 351–375 (1998)
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We are indebted to two anonymous referees who made substantial observations which have led to improvements in the original manuscript.
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B.M.S. gratefully acknowledges the financial support received from the PES University, India through Grant PESUIRF/Math/2020/11.
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Nagamani, K.V., Shankar, B.M. & Shivakumara, I.S. The Prandtl–Darcy Convection in a Vertical Porous Layer may be Unstable with Internal Heating. Transp Porous Med 148, 417–431 (2023). https://doi.org/10.1007/s11242-023-01954-0
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DOI: https://doi.org/10.1007/s11242-023-01954-0