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Continuum Mechanics and Thermodynamics

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15.09.2020 | Original Article

Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

verfasst von: M. Dostalík, V. Průša, K. R. Rajagopal

Erschienen in: Continuum Mechanics and Thermodynamics | Ausgabe 2/2021

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Abstract

A fluid occupying a mechanically isolated vessel with walls kept at spatially non-uniform temperature is in the long run expected to reach the spatially inhomogeneous steady state. Irrespective of the initial conditions the velocity field is expected to vanish, and the temperature field is expected to be fully determined by the steady heat equation. This simple observation is however difficult to prove using the corresponding governing equations. The main difficulties are the presence of the dissipative heating term in the evolution equation for temperature and the lack of control on the heat fluxes through the boundary. Using thermodynamical-based arguments, it is shown that these difficulties in the proof can be overcome, and it is proved that the velocity and temperature perturbations to the steady state actually vanish as the time goes to infinity.

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Further information on the product \({\widetilde{\mathbb {D}}} : {\widetilde{\mathbb {D}}}\) can be formally obtained by the multiplication of the evolution Eq. (2.12b) by the Laplacian of the velocity perturbation, see [19] for a similar manipulation, and also the discussion of slightly compressible convection in [31, 33]. This approach would however work only for constant viscosity, and it would allow one to prove stability only for a restricted set of initial perturbations (small perturbations). We, however, aim at unconditional result—the size of initial perturbation must not be limited. This is what we expect intuitively from our physical system.

Recall that the material time derivative in (3.13b) is taken with respect to the perturbed velocity field, that is \( \frac{\mathrm {d}{\eta }}{\mathrm {d}{t}}(\widehat{\varvec{W}} + \widetilde{\varvec{W}}) = \frac{\partial {\eta }}{\partial {t}}(\widehat{\varvec{W}} + \widetilde{\varvec{W}}) + \left( \widehat{\varvec{v}} + {\widetilde{\varvec{v}}}\right) \bullet \nabla \eta (\widehat{\varvec{W}} + \widetilde{\varvec{W}}) \), while the material time derivative in (3.13a) is taken with respect to the reference steady velocity field, that is \( \frac{\mathrm {d}{\eta }}{\mathrm {d}{t}}(\widehat{\varvec{W}}) = \frac{\partial {\eta }}{\partial {t}}(\widehat{\varvec{W}}) + \widehat{\varvec{v}} \bullet \nabla \eta (\widehat{\varvec{W}}) \) .

Auxiliary tools from the theory of function spaces are for convenience summarised in “Appendix C”.

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Titel: Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature
verfasst von: M. Dostalík
V. Průša
K. R. Rajagopal
Publikationsdatum: 15.09.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Continuum Mechanics and Thermodynamics / Ausgabe 2/2021
Print ISSN: 0935-1175
Elektronische ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-020-00925-w

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