JournalsrlmVol. 28, No. 1pp. 21–47

Dynamic of thermo-MHD flows via a new approach

  • Salvatore Rionero

    Università di Napoli Federico II, Italy and Accademia Nazionale dei Lincei, Roma, Italy
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Abstract

Via a new approach, the dynamic of thermo-MHD flows in horizontal layers heated from below, in the Boussinesq scheme, is investigated. Denoting by m0m_0, EE, PrP_r, PmP_m and RˉC\bar R_C the thermal conduction solution, the L2L^2-energy of the nonlinear perturbations to m0m_0, the Prandtl number, the magnetic Prandtl number and the steady convection onset critical value of the Rayleigh number R2R^2, first of all we obtain, for any initial data, the ultimately boundedness of EE, via the existence of L2L^2-absorbing sets. Successively we introduce a new approach for the E-longtime behaviour associated to the R2R^2-increasing. This new approach is based on an L2L^2-energy linearization principle and on a new way of analyzing and using the linear stability. As concerns the linearization principle, denoting by E\mathcal E the L2L^2-energy of the linear perturbations to m0m_0, it is shown that: "(dEdt)(t=0)<0\left(\frac{d\mathcal E}{dt}\right)_{(t=0)}<0 for any initial data, implies dEdt<0\frac{dE}{dt}<0 tR+\forall t\in\mathbb R^+“.

In order to obtain (dEdt)(t=0)<0\left(\frac{d\mathcal E}{dt}\right)_{(t=0)}<0 for any initial data, denoting by Ln\texttt L_n the linear operator governing the nth-Fourier component of perturbations, we introduce the characteristic values Ijn,(j=1,2,3)\texttt I_{jn},\,(j=1,2,3) of Ln\texttt L_n via the Ln\texttt L_n-entries and obtain the equation λ3I1nλ2+I2nλI3n=0\lambda^3-\texttt I_{1n}\lambda^2+\texttt I_{2n}\lambda-\texttt I_{3n}=0, governing the Ln\texttt L_n-eigenvalues. To this equation we apply the Hurwitz's Criterion guaranteeing that all the eigenvalues have negative real part. As matter of fact, the Hurwitz's Criterion, applied for each nNn\in\mathbb N, allows to obtain conditions necessary and sufficient for being (dEdt)(t=0)<0\left(\frac{d\mathcal E}{dt}\right)_{(t=0)}<0 for any initial data. Following this new approach, we show that the unconditional nonlinear asymptotic stability of m0m_0, with respect to the L2L^2-energy norm, is guaranteed by the linear stability and obtain – among other things – two conditions, in a very simple closed forms, guaranteeing the onset of oscillatory convection (overstability laws).

All the results, first obtained for the free-free boundary case, are successively generalized to the rigid-rigid, rigid-free and free-rigid boundary cases.

Cite this article

Salvatore Rionero, Dynamic of thermo-MHD flows via a new approach. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 21–47

DOI 10.4171/RLM/750