Triply resonant penetrative convection

Straughan, Brian

doi:10.1098/rspa.2012.0211

Cited by 16 publications

(26 citation statements)

References 41 publications

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“…(In ongoing three-dimensional computations on a triply penetrative convection problem, cf. Straughan [32], we definitely find the presence of subcritical instabilities, especially when oscillatory convection occurs.) As H → ∞, our computations indicate that the curves a, b and c asymptote to E. If one uses the analysis of Rees [21] to estimate H for a high porosity material such as a metallic foam, then with the solid CuO or Al 2 O 3 and the fluid water, we find with = 0.9, k f = 0.6, k s = 30 W m K −1 , then H ≈ 1.23C, where C is a geometrical factor depending on the structure of the porous medium.…”

Section: Numerical Results and Conclusionmentioning

confidence: 97%

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

Straughan

2013

Proc. R. Soc. A.

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There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

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Section: Numerical Results and Conclusionmentioning

confidence: 97%

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

Straughan

2013

Proc. R. Soc. A.

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“…In [427] investigated a situation of thermal convection in a fluid layer where resonance may possibly occur simultaneously in three distinct sublayers. To achieve this he assumes the density-temperature relation in the buoyancy term is quadratic and he allows the heat source/sink to vary linearly with vertical height z.…”

Section: Cellular Instability Structurementioning

confidence: 99%

“…This has some resemblance to the experimental situation of [217], although it is different, and appropriate experiments would be appreciated. [427] derives linear instability results as well as global nonlinear stability ones and he finds that there is a very interesting parameter range where the Rayleigh number for instability rises very rapidly. The basic steady state configuration of interest is one where there are three potential sub-layers which may give rise to instability which can then penetrate into the other layer(s).…”

Section: Cellular Instability Structurementioning

confidence: 99%

“…while the balance of energy equation is Denote by Δ T 1 = T m − T 0 , then [427] shows the function Q which yields the scenario of figure 14.16 is given by…”

Section: The Modelmentioning

confidence: 99%

“…To study stability of the basic steady state solution [427] introduces perturbations u i , θ , π by v i =v i + u i , and non-dimensionalizes with length, velocity and temperature scales h, ν/h = U,U ν/κgαh together with the Prandtl and Rayleigh numbers defined by…”

Section: The Modelmentioning

confidence: 99%

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Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

113

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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Internally Heated Convection and Rayleigh-Bénard Convection

Goluskin¹

2016

SpringerBriefs in Applied Sciences and Technology

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PrefaceThe purpose of this SpringerBrief is to review heat transfer in layers of convective fluid. Six different configurations are considered-three that are versions of Rayleigh-Bénard (RB) convection, which is driven by differential heating at the boundaries, and three that are driven by uniform internal heating. The essential features of all six models are derived mathematically. The experimental literature is reviewed in depth for the models of internally heated (IH) convection, which are much less studied than their RB counterparts. Experiments on RB convection are treated in less depth, as they have been thoroughly reviewed elsewhere.Along with placing the various convective models within a conceptual framework that brings out their similarities, we give some minor results not published elsewhere. For instance, a few of the linear instability and energy stability thresholds given in tables 2.2 and 2.3 either have not been reported before or have been reported with less precision. One of the bounds proven in §2.3 is also new, as are the visualizations of simulations included as figure 1.2.Chapter 1 provides background and then defines the six configurations under study, the governing equations of our models, and their basic features. Chapter 2 presents results that can be derived mathematically from the governing equations: linear and nonlinear stability thresholds of static states, along with proven bounds on mean temperatures and heat fluxes. For the IH cases only, chapter 3 gives a quantitative survey of heat transport in both laboratory experiments and numerical simulations, followed by suggestions for future work.

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Triply resonant penetrative convection

Cited by 16 publications

References 41 publications

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Internally Heated Convection and Rayleigh-Bénard Convection

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