Asymptotic Solution on Spherical Floating Particle

The method of matched asymptotic expansions is applied to analyze the meniscus around a floating particle. The asymptotic solutions, derived while considering Fraenkel’s warning, are expressed as functions of the filling angle and contact angle. The analytical solutions accurately predict both the filling angle and the position of the three-phase contact line, even for larger ϵ values. The deformed free surface profile around a single floating particle shows excellent agreement with numerical results obtained from transformed equations using the inclination angle of the deformed surface. The matched asymptotic solutions reveal two distinct steady-state interface configurations, with one stable configuration identified through vertical stability analysis. This asymptotic approach enables the construction of stability diagrams for floating conditions, helping determine the maximum floatable particle size or density ratio for sustained flotation. The methodology developed here provides a foundation for future research on buoyancy for particles of various shapes and stability under dynamic conditions.