Global stability in the B�nard problem for a mixture with superimposed plane parallel shear flows

Lombardo, Sara; Mulone, G.; Rionero, Salvatore

doi:10.1002/1099-1476(20001110)23:16<1447::aid-mma173>3.0.co;2-l

Cited by 19 publications

(3 citation statements)

References 17 publications

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“…A nonlinear energy stability analysis is really beneficial when it can be used to provide stability thresholds close to those of linear theory and valid for all initial conditions,as emphasized by Straughan (p. 157 of [19]). Many recent studies have focussed upon this problem (see Payne and Straughan [17], Budu [20], Lombardo et al [21], Straughan [22]). Flavin and Rionero [12][13][14] utilized the 'natural' transformation to deal with when the thermal conductivity is a nonlinear function of temperature, at least with regard to a particular class of nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

Harfash

Meften

2020

Phys. Scr.

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Models of double-diffusive convection in a fluid layer with the viscosity having linear dependence on the temperature are analyzed here. The density is assumed to have a quadratic dependence on temperature and a linear dependence on the species concentration. Conditional and unconditional nonlinear energy stability criteria for two different models are derived. A comparison of linear and nonlinear theories suggests that there are significant regions where the basic state is subject to a deep subcritical instability when the value of solute Rayleigh number is high.

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Section: Introductionmentioning

confidence: 99%

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

Harfash

Meften

2020

Phys. Scr.

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“…The stability of (6) has been considered by several authors (also when rotation about the vertical axis is incorporated) [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Precisely, denoting by (i) are periodic in the x and y directions respectively of periods 2π a x and 2π a y , (ii) on the periodicity cell Ω = [0, 2π a x ] × [0, 2π a y ] × [0, 1] (in order to guarantee uniqueness) u and v have zero mean value, (iii) belong to L 2 (Ω), ∀t ∈ R + , the results on nonlinear energy stability [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], as far as we know, can be summarized as follows: there exists a bounded positive number R * ∞ such that…”

Section: Introductionmentioning

confidence: 99%

A new approach to nonlinear L2-stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle

Rionero

2007

Journal of Mathematical Analysis and Applications

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A new approach to nonlinear L 2 -stability for double diffusive convection in porous media is given. An auxiliary system Σ of PDEs and two functionals V , W are introduced. Denoting by L and N the linear and nonlinear operators involved in Σ, it is shown that Σ-solutions are linearly linked to the dynamic perturbations, and that V and W depend directly on L-eigenvalues, while (along Σ) dV dt and dW dt not only depend directly on L-eigenvalues but also are independent of N . The nonlinear L 2 -stability (instability) of the rest state is reduced to the stability (instability) of the zero solution of a linear system of ODEs. Necessary and sufficient conditions for general, global L 2 -stability (i.e. absence of regions of subcritical instabilities for any Rayleigh number) are obtained, and these are extended to cover the presence of a uniform rotation about the vertical axis.

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“…Several recent articles have explicitly addressed this question in a variety of contexts (cf. Flavin & Rionero 1998, 1999aLombardo et al . 2000;Payne & Straughan 2000;Straughan 2001).…”

Section: Introductionmentioning

confidence: 99%

Sharp global nonlinear stability for temperature-dependent viscosity convection

Straughan

2002

Proc. R. Soc. Lond. A

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With regard to the dependence of viscosity on temperature, an unconditional nonlinear energy-stability analysis for thermal convection according to Navier-Stokes theory has not yet been developed. We here analyse three models of fluid behaviour proposed by O. A. Ladyzhenskaya. We show that by using these theories we can develop an unconditional analysis directly. Two of the models lead directly to an unconditional development employing L 2 theory, while the third necessitates the introduction of a generalized energy also involving an L 3 term. The nonlinear stability boundaries in all three cases are sharp when compared with the instability thresholds of linear theory.

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Global stability in the B�nard problem for a mixture with superimposed plane parallel shear flows

Cited by 19 publications

References 17 publications

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

A new approach to nonlinear L2-stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle

Sharp global nonlinear stability for temperature-dependent viscosity convection

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