On the Stability of Rotating Drops
On the Stability of Rotating Drops
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关于旋转水滴的稳定性
DOI：
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发表时间：
2015
影响因子：
1.5
通讯作者：
G. McFadden
中科院分区：
文献类型：
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作者：
A. K. Nurse;S. Coriell;G. McFadden
我们通过一个变分原理来考虑旋转轴对称液滴的平衡和稳定性，该原理将平衡描述为包含表面能和转动能贡献的泛函的定态，并加上体积约束。液滴的线性稳定性是通过求解与能量泛函的二次变分相关的本征值问题来确定的。我们计算了扁形和长形以及环状的平衡，并跟踪了它们随自转速度的演变。稳定性的结果是在两种情况下得到的：(I)系统的规定转速(“驱动滴”)，或(Ii)规定角动量(“孤立滴”)。对于轴对称液滴族，轴对称或非轴对称扰动都可能发生不稳定性；后者对应于可能出现非轴对称形状的分叉点。我们采用了角度弧长公式，允许计算在球坐标中不是单值的平衡形状。我们能够说明从在旋转轴上有强烈压痕的球状液滴到不延伸到旋转轴的环状液滴的转变。具有大纵横比(大半径与小半径)的环状液滴在较高模数下容易发生方位不稳定性，这类似于圆柱界面的瑞利不稳定性。如果密度较低的液滴在密度较高的介质中旋转，就会出现长球形；这些液滴看起来是线性稳定的。这项工作是由最近对环状组织簇的研究推动的，这些环状组织簇被观察到在自组装后爬上圆锥形障碍[Nurse等人，应用力学杂志79(2012年)051013]。
We consider the equilibrium and stability of rotating axisymmetric fluid drops by appealing to a variational principle that characterizes the equilibria as stationary states of a functional containing surface energy and rotational energy contributions, augmented by a volume constraint. The linear stability of a drop is determined by solving the eigenvalue problem associated with the second variation of the energy functional. We compute equilibria corresponding to both oblate and prolate shapes, as well as toroidal shapes, and track their evolution with rotation rate. The stability results are obtained for two cases: (i) a prescribed rotational rate of the system (“driven drops”), or (ii) a prescribed angular momentum (“isolated drops”). For families of axisymmetric drops instabilities may occur for either axisymmetric or non-axisymmetric perturbations; the latter correspond to bifurcation points where non-axisymmetric shapes are possible. We employ an angle-arc length formulation of the problem which allows the computation of equilibrium shapes that are not single-valued in spherical coordinates. We are able to illustrate the transition from spheroidal drops with a strong indentation on the rotation axis to toroidal drops that do not extend to the rotation axis. Toroidal drops with a large aspect ratio (major radius to minor radius) are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. Prolate spheroidal shapes occur if a drop of lower density rotates within a denser medium; these drops appear to be linearly stable. This work is motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013].