The effect of the Prandtl number on magnetoconvection in a horizontal fluid layer

摘要

Linear stability of the buoyant convective flow subject to a uniform magnetic field is investigated. The flow configuration consists of a differentially heated extended horizontal layer of an electrically conducting fluid, placed in a constant vertical magnetic field. Critical values of parameters, marking the onset of the instability, are obtained for four combinations of thermal and electrical boundary conditions (perfectly insulating/conducting) and for wide ranges of the Prandtl, Pr, and Hartmann, Ha, numbers. The analysis of the most dangerous flow perturbations shows that the Hartmann number is not the only relevant parameter, and in general the instability strongly depends on the shape of the basic velocity profile, electrical and thermal conductivity of the walls, and critically on the type of an electrically conducting fluid considered. The linear stability analysis provides an insight into the basic mechanisms that govern the flow, and allows to identify the physical nature of the instabilities at constant values of the Prandtl number. Firstly, a dynamic instability develops due to the inflection point in the basic velocity profile. Secondly, the Rayleigh-Benard mechanism is identified as a source of instability in the regions of unstable thermal stratification near thermally conducting boundaries. Thirdly, we discuss the instability of the Hartmann boundary layers in the velocity profile modified by the magnetic field. The main interest here is in the variation of the Prandtl number depending on the type of an electrically conducting fluid (liquid metals, semiconductors or various kinds of electrolytes).