Non-linear approximations for natural convection in a horizontal annulus

Duka, Bejo; Ferrario, C.; Passerini, Arianna; Piva, S.

doi:10.1016/j.ijnonlinmec.2007.05.006

Cited by 13 publications

(21 citation statements)

References 7 publications

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“…Such an approach is referred to as the nonlinear Stokes problem (cf. Duka et al 2007). Standard indicial notation is employed throughout, with the symbol D representing the Laplace operator and kZ(0, 0, 1).…”

Section: Formation Of the Problemmentioning

confidence: 99%

Global stability for thermal convection in a fluid overlying a highly porous material

Hill

Straughan

2008

Proc. R. Soc. A.

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This paper investigates the instability thresholds and global nonlinear stability bounds for thermal convection in a fluid overlying a highly porous material. A two-layer approach is adopted, where the Darcy-Brinkman equation is employed to describe the fluid flow in the porous medium. An excellent agreement is found between the linear instability and unconditional nonlinear stability thresholds, demonstrating that the linear theory accurately emulates the physics of the onset of convection.

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Section: Formation Of the Problemmentioning

confidence: 99%

Global stability for thermal convection in a fluid overlying a highly porous material

Hill

Straughan

2008

Proc. R. Soc. A.

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“…In fact, the velocity field and temperature are usually found as the solution of a coupled system of partial differential equations, the Boussinesq system, representing an incompressible approximation (recently re-discussed in [15]) of the full Navier-Stokes-Fourier system. In the present paper, the solutions to Boussinesq's system of equations are replaced by solutions to a further approximation, already proposed in [5,16] and reconsidered here. Such a system, named nonlinear Stokes problem, is chosen in place of the fully linearized one, proposed in [17] and theoretically analyzed in [4], because the fully linearized model approximates the Nusselt number too roughly.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…Thus, we study τ as relevant temperature (i.e., deviation from pure conduction) and, consequently, the boundary conditions for (3)(4)(5) are fully homogeneous:…”

Section: Mathematical Modelmentioning

confidence: 99%

“…In [19] it is shown that for sufficiently small Ra a solution can always be found for system (3)(4)(5). However, a simpler system would be easier to handle.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…All these properties occur even for very small Rayleigh numbers, so small that one would observe the fluid almost at rest if A were also small. This would happen because, as shown in [4,5], the upper bound for the solutions (the one usually related to classical energy stability analysis as in [13,Sect. 54]) is of order Ra/b, where the parameter b := log(R o /R i ) = log(1 + 2/A) is large for small A. Oppositely, in the region defined by Ra < 16 but A > 8.5, the upper bound for the solutions shows a non-negligible order of magnitude, since b vanishes as A tends to infinity.…”

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confidence: 99%

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Natural convection in horizontal annuli: a lower bound for the energy

2007

Self Cite

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We study natural convection in the gap between two infinitely\ud long horizontal coaxial cylindrical surfaces, each of which is\ud maintained at constant temperature. If the inverse relative gap\ud width $\cA$ is large,\ud relevant steady convective flow and considerable heat transfer\ud are observed, even for extremely small Rayleigh numbers $\Ra$. A lower\ud bound for the norm of the velocity and of the temperature is\ud rigorously found by studying an approximation problem, which is a\ud good model in the parameter range where steady stable flow\ud occurs. The lower bound depends on $\cA$ only, and is an\ud increasing function of $\cA$. This means that the bound can be\ud arbitrarily increased via the geometry only - no matter how small\ud the temperature difference is, and independently of the Prandtl\ud number $\Pr$

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Natural convection between two horizontal coaxial cylinders

Passerini

Růžička

Thäter³

2009

Z Angew Math Mech

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The objective of our paper is to start the theoretical investigation of the most commonly used mathematical model for natural convection in a horizontal annulus. First, one shows existence of solutions for the unsteady problem. Then, for any Prandtl number and inverse relative gap width, existence, and stability of a steady solution is proved, provided that the Rayleigh number is sufficiently small. The same is also proved for any Rayleigh number provided the inverse relative gap width is sufficiently small. Furthermore, exponential decay to the steady symmetric solution and uniqueness hold true under stronger restrictions to the Rayleigh number

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Non-linear approximations for natural convection in a horizontal annulus

Cited by 13 publications

References 7 publications

Global stability for thermal convection in a fluid overlying a highly porous material

Global stability for thermal convection in a fluid overlying a highly porous material

Natural convection in horizontal annuli: a lower bound for the energy

Natural convection between two horizontal coaxial cylinders

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