Abstract Dynamical behaviors of compound droplets including those interacting with rigid walls in an axisymmetric channel appear in various industrial and natural processes. However, so far, no detailed investigation has been carried out for such interactions of compound droplets. Motivating from this missing gap, we here numerically study the finite deformation and breakup of an initially concentric compound droplet when it moves toward a rigid wall at the bottom of an axisymmetric vertical channel. The method used is a finite difference-based front-tracking method. The numerical results reveal that when the compound droplet is delivered toward the wall, it is deformed and can break up into smaller droplets. For the cases of finite deformation (i.e. non-breakup), while the outer droplet is radially stretched, the inner droplet first moves downward in the direction of the outer flow but then gets back. Thereby, a thin film is created between the outer and inner interfaces at the droplet top and thus prevents the outer droplet further deforming and breaking up. In contrast, if breakup happens, the outer droplet is further stretched, and most of the middle fluid moves outward toward the outer droplet edge to form a blob. Breakup can be available in one of three patterns: off-axis breakup, on-axis breakup, and inner breakup. The off-axis breakup mode only happens with the outer droplet while the inner breakup mode is only for the inner droplet. Various parameters are investigated to show the transition between a non-breakup mode to a mode of breakup. Such parameters contributing the transition include the Capillary number Ca (varied in the range of 0.01–2.5), the channel aspect ratio (varied in the range of 0.4–2.0), the ratio of the inner to outer droplet radii (varied in the range of 0.3–0.8), the droplet size relative to the channel size (varied in the range of 0.2–0.9), the interfacial tension ratio of the inner to outer interfaces (varied in the range of 0.1–4.0), and the viscosity ratio of the middle to outer fluids (varied in the range of 0.16–6.3). In contrast, some others, e.g. the Reynolds number, the viscosity ratio of the inner to the outer, do not induce any transition. From the numerical results, regime diagrams of breakup and non-breakup based on these parameters are proposed.

中文翻译：

---

复合液滴在轴对称通道中向刚性壁移动的数值研究

摘要 复合液滴的动力学行为，包括与轴对称通道中刚性壁相互作用的液滴，出现在各种工业和自然过程中。然而，到目前为止，还没有对复合液滴的这种相互作用进行详细的研究。从这个缺失的间隙出发，我们在这里数值研究了初始同心复合液滴在向轴对称垂直通道底部的刚性壁移动时的有限变形和破裂。使用的方法是基于有限差分的前向跟踪方法。数值结果表明，当复合液滴向壁输送时，它会变形并分解成更小的液滴。对于有限变形（即非破裂）的情况，而外层液滴被径向拉伸，内部液滴首先沿外部流动的方向向下移动，然后返回。因此，在液滴顶部的外部和内部界面之间形成薄膜，从而防止外部液滴进一步变形和破裂。相反，如果发生破裂，外部液滴会进一步拉伸，并且大部分中间流体向外移向外部液滴边缘以形成斑点。破碎可以采用以下三种模式之一：离轴破碎、轴上破碎和内部破碎。离轴破碎模式仅发生在外部液滴，而内部破碎模式仅发生于内部液滴。研究了各种参数以显示非破裂模式到破裂模式之间的转变。这些有助于转变的参数包括毛细管数 Ca（在 0.01-2 的范围内变化。5)、通道纵横比（在 0.4-2.0 范围内变化）、内外液滴半径之比（在 0.3-0.8 范围内变化）、液滴尺寸相对于通道尺寸（在0.2-0.9），内外界面的界面张力比（在 0.1-4.0 范围内变化），以及中外流体的粘度比（在 0.16-6.3 范围内变化）。相比之下，其他一些，例如雷诺数，内部与外部的粘度比，不会引起任何转变。根据数值结果，提出了基于这些参数的破碎和非破碎状态图。相对于通道尺寸的液滴尺寸（在 0.2-0.9 的范围内变化），内部与外部界面的界面张力比（在 0.1-4.0 的范围内变化），以及中外流体的粘度比（在 0.16-6.3 的范围内变化）。相比之下，其他一些，例如雷诺数，内部与外部的粘度比，不会引起任何转变。根据数值结果，提出了基于这些参数的破碎和非破碎状态图。相对于通道尺寸的液滴尺寸（在 0.2-0.9 的范围内变化），内部与外部界面的界面张力比（在 0.1-4.0 的范围内变化），以及中外流体的粘度比（在 0.16-6.3 的范围内变化）。相比之下，其他一些，例如雷诺数，内部与外部的粘度比，不会引起任何转变。根据数值结果，提出了基于这些参数的破碎和非破碎状态图。