The steady propagation of air bubbles through a Hele-Shaw channel with either a rectangular or partially occluded cross section is known to exhibit solution multiplicity for steadily propagating bubbles, along with complicated transient behaviour where the bubble may visit several edge states or even change topology several times, before typically reaching its final propagation mode. Many of these phenomena can be observed both in experimental realisations and in numerical simulations based on simple Darcy models of flow and bubble propagation in a Hele-Shaw cell. In this paper, we investigate the corresponding problem for the propagation of a viscous drop (with viscosity \(\nu \) relative to the surrounding fluid) using a Darcy model. We explore the effect of drop viscosity on the steady solution structure for drops in rectangular channels or with imposed height variations. Under the Darcy model in a uniform channel, steady solutions for bubbles map directly on to those for drops with any internal viscosity \(\nu \ne 1\). Hence, the solution multiplicity predicted for bubbles also occurs for drops, although for \(\nu >1\), the interface shape is reversed with inflection points appearing at the rear rather than the front of the drop. The equivalence between bubbles and drops breaks down for transient behaviour, at the introduction of any height variation, for multiple bodies of different viscosity ratios and for more detailed models which produce a more complicated flow in the interior of the drop. We show that the introduction of topography variations affects bubbles and drops differently, with very viscous drops preferentially moving towards more constricted regions of the channel. Both bubbles and drops can undergo transient behaviour which involves breakup into two almost equal bodies, which then symmetry break before either recombining or separating indefinitely.

众所周知，气泡通过具有矩形或部分封闭横截面的 Hele-Shaw 通道稳定传播，显示出稳定传播气泡的解多样性，以及复杂的瞬态行为，其中气泡可能会访问多个边缘状态甚至改变拓扑几个次，通常在达到其最终传播模式之前。许多这些现象都可以在实验实现和基于 Hele-Shaw 单元中流动和气泡传播的简单达西模型的数值模拟中观察到。在本文中，我们研究了粘性液滴传播的相应问题（具有粘性\(\nu \)相对于周围流体）使用达西模型。我们探讨了液滴粘度对矩形通道中液滴或强加高度变化的稳定溶液结构的影响。在均匀通道中的达西模型下，气泡的稳定解直接映射到具有任何内部粘度\(\nu \ne 1\) 的液滴。因此，对于气泡预测的解多样性也发生在液滴中，尽管对于\(\nu >1\)，界面形状相反，拐点出现在水滴的后部而不是前部。在引入任何高度变化时，对于具有不同粘度比的多个物体以及在液滴内部产生更复杂流动的更详细模型，气泡和液滴之间的等效性会因瞬态行为而失效。我们表明，地形变化的引入对气泡和液滴的影响不同，非常粘稠的液滴优先向通道中更狭窄的区域移动。气泡和液滴都可以经历瞬态行为，包括分裂成两个几乎相等的物体，然后在重新组合或无限期分离之前对称性破裂。