Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall te
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O R I G I NA L A RT I C L E
M. Dostalík · V. Pruša ˚
· K. R. Rajagopal
Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature
Received: 22 December 2019 / Accepted: 3 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Navier–Stokes–Fourier fluid · Finite amplitude stability · Thermodynamically open system · Non-equilibrium steady state
1 Introduction The everyday experience is that if a fluid is put into a vessel, and if it is not allowed to substantially interact with the outside environment, then it eventually comes to the rest state. Moreover, the rest state is attained irrespective of the initial state of the fluid. The question is whether this behaviour can be deduced using the corresponding governing equations such as the Navier–Stokes–Fourier equations for an incompressible fluid. In the case of a vessel that is completely isolated from the outside environment, see Fig. 1a, the answer to this question is straightforward and positive. One can exploit basic ideas from continuum thermodynamics, Communicated by Andreas Öchsner. Vít Pr˚uša thanks the Czech Science Foundation, Grant Number 20-11027X, for its support. Mark Dostalík has been supported by Charles University Research Program No. UNCE/SCI/023 and GAUK 1652119. M. Dostalík · V. Pr˚uša (B) Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague, CZ 186 75, Czech Republic E-mail: [email protected] M. Dostalík E-mail: [email protected] R. Rajagopal Department of Mechanical Engineering 3123 TAMU, Texas A&M University, College Station, TX 77843-3123, United States of America E-mail: [email protected]
M. Dostalík et al.
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Fig. 1 Long time behaviour under various types of temperature boundary conditions.
see [4,13,14] and early considerations by [7], and prove that both the velocity and temperature field eventually reach the spatially homogeneous equilibrium rest state. The same holds also for simple open systems such as a vessel immersed in a thermal bath, that is for the mechanically isolated vessel with walls kept at constant spat
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