[论文解读] Stokes flows in a sessile hemispherical drop due to evaporation and surface tension gradient

П. В. Лебедев-Степанов|arXiv (Cornell University)|Feb 13, 2026

Nanomaterials and Printing Technologies被引用 0

一句话总结

分析型Stokes流解推导出一个带固定接触线的小蒸发性半球形自洽滴中的流动，展示在无滑与全滑基底条件下，Deegan 外向流（咖啡环流）和表面张力梯度引发的Marangoni流动。

ABSTRACT

Viscous hydrodynamic flow in a small, slowly evaporating, sessile hemispherical droplet with a pinned contact line is considered. Analytical solutions are obtained for the Deegan outward flow, which is responsible for the coffee ring effect, as well as the Marangoni flow excited by a surface tension gradient. It is assumed that the surface tension gradient may be caused by anisotropic cooling of droplet surface or other factors, such as nonuniform illumination of an optically active surfactant. Two main types of boundary conditions, no-slip and full-slip, are considered in describing the flow-substrate interaction. It is shown that under the no-slip condition, there is a rigid relationship between the evaporation rate and the surface tension gradient, which imposes strict requirements on the temperature regime inside the droplet. This result offers a new vision of the critical Marangoni number, which describes the threshold for the transition of an evaporating droplet from capillary flow to developed Marangoni convection. The results of this work may attract the attention of experimenters to the study of the sensitivity of viscous flow in an evaporating droplet to the liquid-substrate boundary conditions, especially if the system under consideration passes into the Marangoni regime, when the no-slip condition changes to a partial or full slip condition due to the increase in viscous shear stress near the substrate.

研究动机与目标

• 研究一个小尺寸、缓慢蒸发的带固定接触线的自洽半球形滴中的黏性流动。
• 得到导致咖啡环效应的Deegan外向流的解析解。
• 表征由表面张力梯度引发的Marangoni流及其与基底边界条件的耦合。

提出的方法

• 将Stokes流理论应用于带蒸发驱动边界条件的半球顶盖几何。
• 在两种基底边界条件（无滑与全滑）下推导Deegan外向流和Marangoni流的解析解。
• 在无滑条件下分析蒸发速率、表面张力梯度与内部温度状态的耦合。
• 讨论向Marangoni对流转变的含义以及临界Marangoni数对流变转变的影响。

实验结果

研究问题

• RQ1 蒸发和表面张力梯度如何驱动带固定点的自洽半球形滴的内部流动？
• RQ2 在无滑与全滑基底条件下，Deegan外向流和Marangoni流的解析表达式是什么？
• RQ3 在不同边界条件下，蒸发速率如何与表面张力梯度相关？
• RQ4 本研究对从毛细对流转变为Marangoni对流的临界Marangoni数有何启示？

主要发现

• 在无滑条件下，蒸发速率与表面张力梯度之间存在一个刚性关系，限制了内部温度状态。
• 解析解揭示了Deegan外向流以及由表面张力梯度驱动的Marangoni流。
• 结果凸显蒸发滴流动对液体-基底边界条件的敏感性，尤其在Marangoni效应占主导时。
• 本工作为蒸发滴从毛细对流转变为Marangoni对流所涉临界Marangoni数提供了新视角。