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Thermoconvective instabilities of a non-uniform Joule-heated high Prandtl number liquid

Abstract : This numerical study is devoted to the thermoconvection occurring when a liquid is heated by the Joule dissipation. The model is based on the coupling between the Navier-Stokes equations written in the framework of the Boussinesq approximation, energy equation and electric potential conservation. A numerical solver has been developed using standard and discontinuous Galerkin finite element method. The problem is controlled by the Rayleigh number Ra in which the temperature scale depends on the 23 ème Congrès Français de Mécanique Lille, 28 au 1 er Septembre 2017 volumetric source term and by the Prandtl number Pr which is considered larger than one. The domain is a 2d rectangular cavity with a length equal to twice of the height. The electric field is provided with two vertical electrodes corresponding to a fraction of the vertical walls of the enclosure. A uniform temperature is applied in the upper horizontal wall. When the electrode length is equal to the height of the cavity, the convection appears above a critical Rayleigh number independent on the Prandtl number. The nature of the instability is analyzed by studying the motion intensity determined in term of a Péclet number close to the critical Rayleigh number. We perform a numerical study when the electrode lengths are equal to the 2/3 of the cavity height. In this case, the motion appears without threshold. Two regimes of motion are established: the " conductive " regime observed when Ra < 10 3 for which the Péclet number is proportional to the Rayleigh number and the " convective " regime in which the Péclet number scales as the square root of Ra whatever the Prandtl number. A thermoconvection instability is established in which the symmetric structure of flow breaks down to evolve toward an asymmetric structure. The critical Rayleigh number corresponding to the onset of this transition depends strongly on the Prandtl number when Pr is lesser than 3 while for larger values of Pr, the critical Rayleigh number becomes independent on Pr. By analyzing the flow obtained for Rayleigh numbers larger than the critical values, we pinpoint that the flow becomes periodic in time and the onset of periodic solution obeys to a Hopf bifurcation.
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Submitted on : Friday, November 10, 2017 - 4:34:02 PM
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Franck Pigeonneau. Thermoconvective instabilities of a non-uniform Joule-heated high Prandtl number liquid. 23ème Congrès Français de Mécanique [CFM2017], Association Française de Mécanique (AFM), Aug 2017, Lille, France. 18 p. ⟨hal-01632867⟩



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