论文标题
矩形细胞中金属垫不稳定性阈值的分形
Fractality of metal pad instability threshold in rectangular cells
论文作者
论文摘要
我们分析了由铝还原细胞的理想化模型中的界面波的线性稳定性，该模型由两个稳定的分层液体层组成，这些液体在共线外部磁场中携带垂直电流。如果电流和磁场的乘积根据细胞设计超过一定的临界阈值，则重力波模式的电磁耦合会导致自我扩增的旋转界面波，称为金属垫不稳定。 Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $α=\sqrt{m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds.临界纵横比的分形分布形成了与宽度比具有非零稳定性阈值的绝对不连续的密度集，可以通过线性稳定性问题的准确数值解确认。尽管考虑到粘性摩擦时的分形消失，但不稳定阈值逐渐平滑，其主要结构（由主要的临界宽高比与$ M $和$ n $相对应的主要临界纵横比的主导，但预付得很好，可提供相对较大的无二型粘性玻璃摩擦系数$γ\ sim 0.1 $ 0.1 $ 0.1 $。具有小的粘性摩擦，最稳定的是具有$α^{2} \ 2.13 $的单元，其稳定性阈值对应于电磁相互作用参数$β\约4.7 $。
We analyse linear stability of interfacial waves in an idealised model of an aluminium reduction cell consisting of two stably stratified liquid layers which carry a vertical electric current in a collinear external magnetic field. If the product of electric current and magnetic field exceeds a certain critical threshold depending on the cell design, the electromagnetic coupling of gravity wave modes can give rise to a self-amplifying rotating interfacial wave which is known as the metal pad instability. Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $α=\sqrt{m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds. This fractal distribution of critical aspect ratios, which form an absolutely discontinuous dense set of points interspersed with aspect ratios with non-zero stability thresholds, is confirmed by accurate numerical solution of the linear stability problem. Although the fractality vanishes when viscous friction is taken into account, the instability threshold is smoothed out gradually and its principal structure, which is dominated by the major critical aspect ratios corresponding to moderate values of $m$ and $n$, is well-preserved up to relatively large dimensionless viscous friction coefficients $γ\sim 0.1$. With a small viscous friction, the most stable are cells with $α^{2}\approx2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $β\approx 4.7$.
