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.
Thermal convection induced simultaneously by horizontal temperature gradient and vibration in a rectangular cavity filled with molten silicon is investigated numerically and theoretically. The time averaged equations of convection are solved in the high-frequency vibration approximation. The Chebyshev spectral collocation method and a Newton-type method based on the Frechet derivative are used in the numerical solution of the streamfunction formulation of the incompressible Navier-Stokes equations. Validation by comparison with previous works has been performed. Different values of the Grashof number Gr and vibrational Grashof number Gr v and all the possible orientations of the vibrations are considered. Numerical results show that depending on the vibration direction, the flow can be amplified or damped, with even the possibility of flow inversion which can occur between critical vibration angles 1 and 2. A general theoretical expression is derived relating these critical angles and the ratio of vibrational to buoyant convection parameters, Gr v /Gr. A very good agreement between the theoretical and numerical results is obtained.
Studies of convection in the horizontal Bridgman configuration were performed to investigate the flow structures and the nature of the convective regimes in a rectangular cavity filled with an electrically conducting liquid metal when it is subjected to a constant vertical magnetic field. Under some assumptions analytical solutions were obtained for the central region and for the turning flow region. The validity of the solutions was checked by comparison with the solutions obtained by direct numerical simulations. The main effects of the magnetic field are first to decrease the strength of the convective flow and then to cause a progressive modification of the flow structure followed by the appearance of Hartmann layers in the vicinity of the rigid walls. When the Hartmann number is large enough, Ha > 10, the decrease in the velocity asymptotically approaches a power-law dependence on Hartmann number. All these features are dependent on the dynamic boundary conditions, e.g. confined cavity or cavity with a free upper surface, and on the type of driving force, e.g. buoyancy and/or thermocapillary forces. From this study we generate scaling laws that govern the influence of applied magnetic fields on convection. Thus, the influence of various flow parameters are isolated, and succinct relationships for the influence of magnetic field on convection are obtained. A linear stability analysis was carried out in the case of an infinite horizontal layer with upper free surface. The results show essentially that the vertical magnetic field stabilizes the flow by increasing the values of the critical Grashof number at which the system becomes unstable and modifies the nature of the instability. In fact, the range of Prandtl number over which transverse oscillatory modes prevail shrinks progressively as the Hartmann number is increased from zero to 5. Therefore, longitudinal oscillatory modes become the preferred modes over a large range of Prandtl number.
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