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Magnetic stabilization of the buoyant convection between infinite horizontal walls with a horizontal temperature gradient. (English) Zbl 1063.76547

Summary: Buoyant convection induced between infinite horizontal walls by a horizontal temperature gradient is characterized by simple monodimensional parallel flows. In a layer of low-Prandtl-number fluid, these flows can involve two types of instabilities: two-dimensional stationary transverse instabilities and three-dimensional oscillatory longitudinal instabilities. The stabilization of such flows by a constant magnetic field (vertical, or horizontal with a direction transverse or longitudinal to the flow) is investigated in this paper through a linear stability analysis and energy considerations. The vertical magnetic field stabilizes the instabilities more quickly than the horizontal fields, but the stabilization is only obtained up to moderate values of Hartmann number \(Ha\) (before disappearance of the instabilities). Characteristic laws, given by the critical Grashof number \(Gr_c\) as a function of \(Ha\) (proportional to the intensity of the magnetic field), have been found for the initial stabilization at small \(Ha\). They are \(Gr_c \sim Gr_{c_0} \exp(Ha^2)\) for the two-dimensional instabilities and \(Gr_c - Gr_{c_0} \sim Ha^2\) for the three-dimensional instabilities (where \(Gr_{c_0}\) is the critical Grashof number at \(Ha=0\)), indicating that the three-dimensional instabilities, less stabilized, will prevail in a vertical magnetic field. It has been shown by an energy analysis that the strong stabilization of the two-dimensional instabilities is connected to the strong diminution of the destabilizing shear energy term when the velocity profiles are modified by the vertical magnetic field, and affected little by the Lorentz energy term. For the horizontal magnetic fields, the stabilization is very weak at small \(Ha\), but then reaches an asymptotic behaviour corresponding to \(Gr_c \sim Ha\). This asymptotic stabilization is connected to the decrease of the destabilizing shear energy term due to the increase of the marginal cell length in the horizontal magnetic field. In fact, this stabilization only concerns the two-dimensional modes in the longitudinal field and the three-dimensional modes in the transverse field.

MSC:

76E06 Convection in hydrodynamic stability
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
80A20 Heat and mass transfer, heat flow (MSC2010)
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