Asymptotic analysis of an oblate spheroidal particle floating at a fluid interface

The matched asymptotic expansion previously developed for a spherical floating particle is extended to oblate spheroidal particles of aspect ratio k=cpol/a≤1, where a is the equatorial radius and cpol is the polar semi-axis. The geometric content of the oblate generalization is captured by a single substitution, βeff(θe)=α − atan2(k sin θe,  cos θe), in the contact-line slope condition, where α is the contact angle and θe is the eccentric angle parametrizing the contact line. With a corresponding redefinition of the integration constant c=sin θe sin βeff, the meniscus differential equation and the matching machinery of the spherical case are inherited unchanged, while the contact-line geometry, the vertical force balance, and the linear stability operator are replaced by their oblate analogues. The aspect ratio k enters only as a multiplicative factor on the buoyancy and gravity terms in the force balance and through the chain rule δy=−k sin θe δθe in the linear stability analysis. The framework yields explicit closed-form predictions across k∈(0,1]. The maximum density ratio satisfies an approximate power law Dmax∝k−1 and hydrophilic and hydrophobic flotations are shown to operate by buoyancy- and capillary-dominated mechanisms, respectively, with Dmax saturating in the strongly hydrophobic regime. Direct numerical solution of the Young–Laplace equation confirms the asymptotic predictions for ϵ=a/lc≤0.4.