Theory of the coalescence of two spherical liquid drops at early times

We consider the dynamics of two equal, spherical liquid drops of Newtonian fluid, brought into contact at a negligible speed, and merging driven by surface tension. The effects of an outer fluid or that of gravity are disregarded, so that the entire motion is determined uniquely by the Ohnesorge number Oh equals eta divided by StartRoot rho gamma upper R EndRoot Oh = η / ρ γ R $\textit{Oh }= \eta /\sqrt {\rho \gamma R}$ , where eta η $\eta$ is the viscosity, rho ρ $\rho$ the density, gamma γ $\gamma$ the surface tension and upper R R $R$ the drop radius. Using methods of matched asymptotics, in the limit that the minimum radius r 0 r 0 $r_0$ of the bridge connecting the drops is much smaller than the initial drop radius, we compute the dynamics for r 0 left parenthesis t right parenthesis r 0 ( t ) $r_0(t)$ , as well as the shape of the gap between the drops. For very early times, the translational motion of the drops along the axis of symmetry is arrested by inertia, which affects the shape of the scaling function describing the gap on the scale of r 0 r 0 $r_0$