The present study is devoted to the problem of the onset of instability in time dependent convective ﬂows of an electrically conducting ﬂuid, subject to an externally applied magnetic ﬁeld and to gravity forces. Linear stability theory is applied to investigate convective ﬂows conﬁned by two rigid walls and opposed by the magnetic ﬁeld. These magnetohydrodynamic channel ﬂows are studied for non-zero magnetic Prandtl numbers as well as in the inductionless approximation. The following conﬁgurations are considered: the Rayleigh-Benard problem, the horizontal layer with longitudinal temperature gradient and the vertical channel with internal heating sources. The numerical results are obtained giving characteristic laws by critical values of parameters, beyond which the ﬂows become unstable. The comparison is made between these problems for small, but non-zero Prm, with the inductionless approximation, in order to determine the validity of that approximation. The analysis of perturbations shows that the instabilities critically depend on the electrical and thermal boundary conditions and on the Prandtl (Pr), magnetic Prandtl (Prm), and Hartmann (Ha) numbers. The instabilities are driven by different mechanisms and set in either as two- or three-dimensional modes, stationary or oscillatory, depending on these parameters and on the orientation of magnetic ﬁeld and gravity. New instability structures are observed in the horizontal layer heated for the side for small ranges of Prm. The MATLAB code created for the purpose of this thesis allows the study of convective ﬂows in the presence of high magnetic ﬁelds up to Ha = 10 5 and the asymptotic relations are successfully reached for the critical values of parameters in most cases.
|Date of Award
|Sergei Molokov (Supervisor)