Meniscus on a shaped fibre: singularities and hodograph formulation

Area of Science:

• Physics
• Fluid Dynamics
• Materials Science

Background:

• Capillary rise is crucial in various scientific and industrial applications.
• Understanding meniscus behavior on complex geometries is challenging.
• Existing models often simplify fiber shapes, limiting applicability.

Purpose of the Study:

• To develop a novel mathematical framework for analyzing capillary rise on complex-shaped fibers.
• To investigate the formation of singularities in the contact line.
• To provide a new interpretation of the meniscus problem as fluid flow through a porous medium.

Main Methods:

• Matched asymptotic expansions were employed to simplify the complex problem.
• The problem was reformulated as finding a minimal surface with specific boundary conditions.
• Chaplygin's hodograph transformation was used to analyze an oval cross-section fiber.
• The meniscus problem was analogized to non-Newtonian fluid flow in porous media.

Main Results:

• The capillary rise problem was successfully reduced to a nonlinear minimal surface determination.
• A novel interpretation of the meniscus problem as fictitious fluid flow was established.
• Singularities in the contact line were identified, even for smooth fiber profiles.
• Analysis of an oval fiber revealed singularities at endpoints with infinite curvature.

Conclusions:

• The minimal surface formulation provides a powerful tool for studying capillary rise on complex geometries.
• The study highlights the potential for unexpected contact line behavior and singularities.
• The fluid flow analogy offers a new perspective for understanding meniscus dynamics.