Analysis of the drop weight method

The drop weight method is an accurate yet simple technique for determining surface tension $\sigma$ . It relies on dripping a liquid of density $\rho$ at a low flow rate $\tilde{Q}$ from a capillary of radius $R$ into air and measuring the combined volumes of the primary and satellite drops that are formed. The method’s origin can be traced to Tate, who postulated that the volume ${\tilde{V}}_{\text{ideal}}$ of the drop that falls from the capillary should be given by $\rho g{\tilde{V}}_{\text{ideal}}=2\pi R\sigma$ , where $g$ is the gravitational acceleration. Since Tate’s law is only an approximation and the actual drop volume ${\tilde{V}}_f<{\tilde{V}}_{\text{ideal}}$ , in practice the surface tension of the liquid-air interface is determined from the experimental master curve due to Harkins and Brown (HB). The master curve is a plot of the fraction of the ideal drop volume, $\Psi \equiv {\tilde{V}}_f∕{\tilde{V}}_{\text{ideal}}$ , as a function of the dimensionless tube radius, $\Phi \equiv R∕{\tilde{V}}_f^{1∕3}$ . Thus, once the actual drop volume ${\tilde{V}}_f$ , and hence $\Phi$ , is known, $\sigma$ is readily calculated upon determining the value of $\Psi$ from the master curve and that $\Psi =\rho g{\tilde{V}}_f∕2\pi R\sigma$ . Although HB proposed their master curve more than 80 years ago, a sound theoretical foundation for the drop weight method has heretofore been lacking. This weakness is remedied here by determining the dynamics of formation of many drops and their satellites in sequence by solving numerically the recently popularized one-dimensional (1–d) slender-jet equations. Computed solutions of the 1-d equations are shown to be in excellent agreement with HB’s master curve when $\tilde{Q}$ is low. Moreover, a new theory of the drop weight method is developed using the computations and dimensional analysis. The latter reveals that there must exist a functional relationship between the parameter $\Phi$ , where ${\Phi }^{-3}$ is the dimensionless drop volume, and the gravitational Bond number $G\equiv \rho g{R}^2∕\sigma$ , the Ohnesorge number $\mathrm{Oh}\equiv \mu ∕{\left(\rho R\sigma \right)}^{1∕2}$ , where $\mu$ is the viscosity, and the Weber number $\mathrm{We}\equiv \rho {\tilde{Q}}^2∕{\pi }^2{R}^3\sigma$ . When $\mathrm{We}\to 0$ , the computed results show that