Inﬂuence of an axial magnetic ﬁeld on the stability of convective ﬂows between non-isothermal concentric spheres V. Travnikov ⇑ , K. Eckert, S. Odenbach Technische Universität Dresden, Institute of Fluid Mechanics, Chair of Magnetoﬂuiddynamics, 01062 Dresden, Germany article info Article history: Received 11 April 2012 Received in revised form 6 July 2012 Accepted 18 July 2012 Available online xxxx Keywords: Natural convection Magnetoﬂuiddynamics Spectral methods Linear stability abstract This paper deals with the linear stability analysis of the convective ﬂow of an electrically conducting ﬂuid in a spherical gap in the presence of an axial magnetic ﬁeld which is parallel to the vector of gravitational acceleration. The inner shell is warmer then the outer one (T 1 > T 2 ). The numerical investigations are per- formed for the radii ratios g ¼ R 1 R 2 ¼ 0:4—0:8. The corresponding stability diagrams, i.e. the critical values of the Grashof number Gr c and the wave number m c , are presented in dependence on the Hartmann num- ber Ha. We show that the critical Grashof number decreases with increasing g in the case of absence of the magnetic ﬁeld. A steady axial magnetic ﬁeld stabilizes the ﬂow, i.e. Gr c increases with the Hartmann number Ha for each g investigated. The instability sets in either as a Hopf bifurcation or a steady pitchfork bifurcation in dependence on g and Ha. The stability analysis is accompanied by simulation of the three- dimensional states. We found that according to the bifurcation type there are two classes of 3D states: steady and oscillatory. Consequences of the symmetry with respect to ±/ direction are discussed. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Natural convection in a spherical gap is a classical problem in ﬂuid dynamics. Buoyancy-driven ﬂows in spherical annuli are used in many technologically important applications, e.g. spherical tanks used in storage systems for high-pressure ﬂuids, or nuclear reactors, etc. The analytical and experimental study of this problem began in the 1960s. Initial works [1,2] deal with the analytical simulation and the experimental investigation of the axisymmetric states for small Grashof numbers, Gr, and Pr = 0.7. These results could not be compared directly because the theoretically accessible Grashof numbers at that time (Gr  3000) were much smaller than those investigated experimentally (Gr  2  10 4  3.6  10 6 ). Hence only simple ﬂow patterns were obtained analytically, while much more complex ﬂows were observed experimentally. It has been established that the ﬂow depends not only on the Grashof or Ray- leigh number but also crucially on the geometry parameter, i.e. radii ratio g ¼ R 1 R 2 . Three different regimes were found experimentally: the steady monocellular (i) ‘‘crescent-eddy’’, (ii) ‘‘kidney-shaped- eddy’’ ﬂows for wide (g = 0.318) and moderate (g = 0.58) gaps, respectively, and (iii) unsteady bicellular ﬂows for the narrow gap (g = 0.84). An experimental study of the convective ﬂows is ana- lyzed in [3] for the narrower radii ratio g = 0.92 and larger Grashof numbers Gr = 1.5  10 7 for water and air as the working ﬂuids. The ﬂow patterns were categorized in terms of steady and unsteady. Systematic numerical analysis of the 2D ﬂows for Pr = 0.7, g = 0.8 and g = 0.83 was performed in [4], including the investigation of the transitions between steady and multiple unsteady states and the description of the hysteresis. 2D ﬂows were studied numeri- cally for a wide range of Prandtl numbers, Pr, in [5] using the ﬁnite volume method. For small Prandtl numbers such as Pr = 0.02 a good agreement was found between the ﬂow structure of [5] and our re- sults (cf. Section 4). The ﬁrst ﬂow stability analyses appeared after 1985. Almost all of these investigations were performed for narrow gaps (g > 0.9), for which the basic ﬂow can be calculated analytically [6,7]. Fur- thermore, strong restrictions were made. First, only axisymmetric perturbations (m = 0) were considered. Second, the principle of the exchange of stability was frequently assumed to be valid, i.e. the imaginary part of the eigenvalue was set to zero, implying that the instability sets in as a steady bifurcation [6]. The inﬂuence of the Prandtl number on the stability was investigated in [7], who showed that a transition Prandtl number Pr tr exists, the value of which depends on the radii ratio. While for Pr < Pr tr the instability sets in as a Hopf bifurcation and is axisymmetric, the bifurcation is steady and non-axisymmetric for Pr > Pr tr . At this point it is important to note that the present class of con- vective problems, where the ﬂow occurs in the presence of a con- stant gravitational acceleration due to the radial temperature gradient, differs fundamentally from problems where gravity is a function of the radial coordinate. Those problems are relevant in astrophysics and were investigated e.g. in [8–12]. In those cases 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.047 ⇑ Corresponding author. E-mail addresses: Vadim.Travnikov@tu-dresden.de (V. Travnikov), Kerstin. Eckert@tu-dresden.de (K. Eckert), Stefan.Odenbach@tu-dresden.de (S. Odenbach). International Journal of Heat and Mass Transfer xxx (2012) xxx–xxx Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Please cite this article in press as: V. Travnikov et al., Inﬂuence of an axial magnetic ﬁeld on the stability of convective ﬂows between non-isothermal con- centric spheres, Int. J. Heat Mass Transfer (2012), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.047