Determine the bridge-width and curvature scaling relation

Determine whether the inverse bridge curvature scales as the bridge dissipation length, specifically whether $1/\kappa_b\sim R_b^2/R$ and whether the dissipation length satisfies $L\sim 1/\kappa_b$, in Darcy-governed droplet coalescence.

Background

The paper explains the late-time bridge-growth scaling by assuming that fluid motion and viscous dissipation are localized near the highly curved bridge. Particle Image Velocimetry supports a characteristic decay length proportional to the bridge width, while the theoretical analysis relates the matching length $x_m$ and dissipation length $L$ to $R_b^2/R$. However, the experimentally measured curvature is noisy, and the authors do not establish whether the inverse curvature is identical to the dissipation length or obeys the proposed scaling throughout the accessible Darcy regime.

Resolving this relation is important because competing assumptions lead to different bridge-growth exponents: the experimentally consistent $t^{1/5}$ law follows from one curvature-length estimate, whereas the self-similarity analysis yields a possible $t^{1/6}$ law. A definitive curvature and dissipation-length relation would therefore clarify the asymptotic scaling of Darcy-governed coalescence.