A Stokes-like system for natural convection in a horizontal annulus

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

doi:10.1016/j.nonrwa.2006.11.008

Cited by 8 publications

(14 citation statements)

References 8 publications

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“…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. In fact, it is proved that its solutions would produce the same Nu as for pure conduction.…”

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%

“…In (14), and can be either scalar-, vector-or tensor-valued functions and the dot between and denotes multiplication for scalar functions or the corresponding scalar product in case of vectors or tensors, respectively. Next, the norm of the difference between the gradients of solutions to problems (3)(4)(5) and (10)(11)(12), namely, the quantity ∇(v − w) 2 + ∇(τ − τ 0 ) 2 , is taken as a measure of the absolute error. Finally, an upper bound for the absolute error is found; such a bound essentially depends on Pr −1 (Ra/b) 2 .…”

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

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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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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%

Section: Mathematical Modelmentioning

confidence: 99%

mentioning

confidence: 99%

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

2007

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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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Natural convection in horizontal annuli: evaluation of the error for two approximations

2011

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We consider convection inside annuli, driven by a uniform temperature gap on the boundaries and gravitation as outer force. It takes place for any Rayleigh number while steady convective motions are observed only for small ones (but any Prandtl number and gap width). We provide estimates for the relative error of two popular approximations to the full Navier–Stokes–Fourier equations. For this we propose a new method. In particular we have to derive a lower bound for the norm of the velocity and the temperature both for steady nonlinear coupled and decoupled approximations in two space dimensions

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A Stokes-like system for natural convection in a horizontal annulus

Cited by 8 publications

References 8 publications

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

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

Natural convection between two horizontal coaxial cylinders

Natural convection in horizontal annuli: evaluation of the error for two approximations

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