Abstract
Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of oscillatory convection, which is a previously unobserved phenomenon in radiation-induced convection. The numerical results for the linear instability analysis suggest a critical value γ c of γ, a measure for the strength of the internal heat source, for which oscillatory convection is inhibited when γ > γ c . Linear instability analyses on the effect of varying the ratio of the salt concentrations at the upper and lower boundaries conclude that the ratio has a significant effect on the stability boundary. A nonlinear analysis using an energy approach confirms that the linear theory describes the stability boundary most accurately when γ is such that the linear theory predicts the onset of mostly stationary convection. Nevertheless, the agreement between the linear and nonlinear stability thresholds deteriorates for larger values of the solute Rayleigh number for any value of γ.
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References
Huppert H.E., Turner J.S.: Double-diffusive convection. J. Fluids Mech. 106, 299–329 (1981)
Nield D.A., Bejan A.: Convection in Porous Media. Springer, New York (2006)
Giestas M., Pina H., Joyce A.: The influence of radiation absorption on solar pond stability. Int. J. Heat Mass Transf. 18, 3873–3885 (1996)
Tabor H.: Non-convection solar ponds. Philos. Trans. R. Soc. Lond. A 295, 423–433 (1980)
Tabor H.: Solar ponds. Solar Energy 27, 181–194 (1981)
Zangrando F.: On the hydrodynamics of salt-gradient solar ponds. Solar Energy 46, 323–341 (1991)
Kudish A.I., Wolf D.: A compact shallow solar pond hot water heater. Solar Energy 21, 317–322 (1978)
Krishnamurti R.: Convection induced by selective absorption of radiation: a laboratory model of conditional instability. Dyn. Atmos. Oceans 27, 367–382 (1997)
Straughan B.: Global stability for convection induced by absorption of radiation. Dynam. Atmos. Oceans 35, 351–361 (2002)
Hill A.A.: Convection due to the selective absorption of radiation in a porous medium. Contin. Mech. Thermodyn. 15, 275–285 (2003)
Hill A.A.: Convection induced by the selective absorption of radiation for the Brinkman model. Contin. Mech. Thermodyn. 16, 43–52 (2004)
Hill A.A.: Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium. Contin. Mech. Thermodyn. 16, 305–318 (2004)
Hill A.A.: Penetrative convection induced by the absorption of radiation with a nonlinear internal heat source. Dyn. Atmos. Oceans 38, 57–67 (2004)
Hill A.A.: Double-diffusive convection in a porous medium with a concentration based internal heat source. Proc. R. Soc. A 461, 561–574 (2005)
Chang M.H.: Stability of convection induced by selective absorption of radiation in a fluid overlying a porous layer. Phys. Fluids 16, 3690–3698 (2004)
Straughan B.: The Energy Method, Stability, and Nonlinear Convection Applied. Mathematical Sciences, vol. 91, 2nd edn. Springer, New York (2004)
Dongarra J.J., Straughan B., Walker D.W.: Chebyshev-τ−QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22, 399–434 (1996)
Straughan B., Walker D.W.: Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127, 128–141 (1996)
Hill A.A.: Global stability for penetrative double-diffusive convection in a porous medium. Acta Mechanica 200, 1–10 (2008)
Hill A.A., Malashetty M.S.: An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients. Proc. R. Soc. A 468, 323–336 (2012)
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Communicated by Andreas Öchsner.
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Wicks, T.J., Hill, A.A. Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer. Continuum Mech. Thermodyn. 24, 229–237 (2012). https://doi.org/10.1007/s00161-012-0234-0
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DOI: https://doi.org/10.1007/s00161-012-0234-0