[論文レビュー] 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被引用数 0

ひとこと要約

Analytical Stokes-flow solutions are derived for a small evaporating sessile hemispherical droplet with a pinned contact line, showing Deegan outward (coffee-ring) flow and Marangoni flow under surface-tension gradients for no-slip and full-slip substrate conditions.

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.

研究の動機と目的

• Investigate viscous hydrodynamic flow in a small, slowly evaporating, sessile hemispherical droplet with a pinned contact line.
• Obtain analytical solutions for Deegan outward flow responsible for the coffee-ring effect.
• Characterize Marangoni flow arising from a surface tension gradient and its interaction with substrate boundary conditions.

提案手法

• Apply Stokes flow theory to a hemispherical cap geometry with evaporation-driven boundary conditions.
• Derive analytical solutions for Deegan outward flow and Marangoni flow under two substrate boundary conditions: no-slip and full-slip.
• Analyze the coupling between evaporation rate, surface tension gradient, and internal temperature regime under no-slip conditions.
• Discuss the transition implications to Marangoni convection and the critical Marangoni number for regime change.

実験結果

リサーチクエスチョン

• RQ1 How do evaporation and surface tension gradients drive internal flows in a pinned, sessile hemispherical droplet?
• RQ2 What are the analytical forms of Deegan outward flow and Marangoni flow under no-slip vs full-slip substrate conditions?
• RQ3 How does the evaporation rate relate to the surface tension gradient under different boundary conditions?
• RQ4 What does the study imply about the critical Marangoni number for transition from capillary to Marangoni convection?

主な発見

• Under no-slip, a rigid relationship exists between evaporation rate and surface tension gradient, constraining the internal temperature regime.
• Analytical solutions reveal the Deegan outward flow and a Marangoni flow driven by surface-tension gradients.
• The results highlight sensitivity of evaporating droplet flows to liquid-substrate boundary conditions, especially when Marangoni effects dominate.
• The work offers a new perspective on the critical Marangoni number governing the shift from capillary to Marangoni convection in evaporating drops.