When a viscous liquid is released on an inclined solvophilic substrate, it spreads downwards by gravity and tends to coat the substrate. Here we report the gravitational axisymmetric spreading of a viscous liquid with a fixed volume on inclined geometries that cause converging flows, for example, on the inside of a downward-pointing hollow cone (funnel) and the lower part of a sphere (bowl). For the limit that the effect of surface tension is negligible, we analytically determine the thickness profile as well as the time-dependent position of the front of the spreading liquid. In the typical scenario that a liquid spreads on a geometry such as an inclined plate, the thickness of the front of the spreading liquid monotonically decreases in time throughout the spreading and a fingering instability occurs while the liquid film thins. However, we show that on an inclined geometry, where convergence of the flow occurs, the thickness of the spreading front first decreases in time and then increases. We also predict a critical volume $V_c$ such that a fingering instability occurs if the volume of the spreading liquid is less than $V_c$. Experiments of axisymmetric spreading on a downward-pointing cone (funnel) are then performed and the measured position of the front and the critical liquid volume of the fingering instability agree with our theoretical predictions. This study highlights the effect of a geometry that focuses and thickens a thin film.

中文翻译：

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沿导致汇流的几何结构排水和扩散：指向下方的圆锥体和碗状半球上的粘性重力流

当粘性液体在倾斜的亲溶剂性基材上释放时，它会在重力作用下向下扩散并趋于覆盖基材。在这里，我们报告了在倾斜几何形状上具有固定体积的粘性液体的重力轴对称分布，这些几何形状会导致会聚流动，例如在指向下方的空心圆锥（漏斗）的内部以及球体（碗）的下部。为了使表面张力的影响可以忽略不计，我们分析性地确定了铺展液的厚度轮廓以及随时间变化的位置。在液体在诸如斜板的几何形状上扩散的典型情况下，在整个扩散过程中，扩散液体的前部的厚度在时间上单调减小，并且当液膜变薄时会出现指状不稳定性。然而，我们表明，在倾斜的几何形状上，在发生流动收敛的情况下，扩展前沿的厚度首先随时间减小，然后增大。我们还预测临界量${\mathrm{伏特}}_C$ 如果撒布液的体积小于 ${\mathrm{伏特}}_C$。然后在指向下的圆锥体（漏斗）上进行轴对称扩展的实验，所测得的前部位置和指状不稳定性的临界液体体积与我们的理论预测相符。这项研究突出了聚焦和加厚薄膜的几何形状的效果。