We present convective instability analyses of the boundary-layer flows originated over cones (with half-angle $\psi \le 90^{\circ }$) rotating in an otherwise still conducting fluid. A uniform magnetic field (with magnetic strength parameter, $m\ge 0$) is acting normally on the surface of each cone. In the non-magnetic case of cones, comparison of present results with existing experimental and theoretical studies lead *to propose that onset of instabilities may be attributed to crossflow (type I) modes and streamline curvature (type II) modes for broad cones (with $\psi \ge 40^{\circ }$) and more slender cones (with $\psi \le 15^{\circ }$), respectively. For slender cones with half-angle $\psi$ in the vicinity of 25° the presented results validate the hypothesis of centrifugal (type III) modes ( discovered by Garrett et al. (2014)). In the magnetic case, increasing magnetic strength $m\in (0,11]$ considerably delays the onset of instabilities (due to both type I and type II modes) over each rotating cone with fixed $\psi$. The stabilising influence of magnetism is in agreement with the meagrely available theoretical studies for magnetic rotating disk case $\psi =90^{\circ }$. In addition, our results lead to suggest that streamline curvature mode over each cone with fixed half-angle $\psi$ become sensitive with increasing m. A minimum $m_0\in [0,11]$ exists for each fixed half-angle ${\psi }_0\le 90^{\circ }$ such that whenever $m\ge m_0$ the onset of instabilities (in the sense of minimum critical Reynolds numbers) over every cone with half-angle ($\psi \le {\psi }_0$) are commenced by type II mode instead of type I mode whilst the aforementioned conjecture of stabilising effect of magnetism is not violated in the considered range of parameters. Under non-stationary vortices assumption with magnetism, we show that crossflow modes travelling at around 75% of the cone surface are likely to be selected in applications where smooth and frictionless surfaces are used. This finding is in complete agreement with the non-magnetic case of rotating cone in existing studies.

我们对源自圆锥体（具有半角）的边界层流进行对流不稳定性分析 $\psi \le 90^{\circ }$）在其他情况下仍在导电的流体中旋转。均匀的磁场（具有磁强度参数，$米\ge 0$）正常作用在每个锥体的表面上。在圆锥体的非磁性情况下，将当前结果与现有的实验和理论研究进行比较，可以得出*不稳定现象的起因可以归因于宽圆锥体（具有）的横流（I型）模式和流线曲率（II型）模式。$\psi \ge 40^{\circ }$）和更细长的锥体（带有 $\psi \le \mathrm{1个}{5}^{\circ }$）， 分别。适用于半角细长锥$\psi$在25°附近，本文给出的结果验证了离心（III型）模式的假设（由Garrett等人（2014年）发现）。在磁性情况下，增加磁场强度$米\in （0，11]$ 固定旋转的每个旋转锥大大延迟了不稳定的发生（由于I型和II型模式） $\psi$。磁性的稳定作用与旋转磁盘盒的理论研究很少$\psi =90^{\circ }$。另外，我们的结果表明，在固定半角的情况下，每个锥体的流线曲率模式$\psi$随着m的增加变得敏感。最低要求${米}_0\in [0，11]$ 每个固定的半角都存在 ${\psi }_0\le 90^{\circ }$ 这样每当 $米\ge {米}_0$ 每个半角锥的不稳定性的发生（在最小临界雷诺数的意义上）（$\psi \le {\psi }_0$）由II型模式代替I型模式开始，而在所考虑的参数范围内没有违反上述磁场稳定作用的推测。在具有磁性的非平稳涡旋假设下，我们表明，在使用光滑无摩擦表面的应用中，可能会选择在锥面约75％处传播的错流模式。这一发现与现有研究中的非磁性旋转锥情况完全吻合。