  The exact solution for a small falling drop is a classical result by Hadamard
and Rybczynski. But experiments show that small drops fall slower than
predicted, giving closer agreement with Stokes' result for a falling hard
sphere. Increasing the drop size, a transition between these two extremes is
found. This is due to surfactants present in the system, and previous work has
led to the stagnant-cap model. We present here an alternative approach which we
call the continuous-interface model. In contrast to the stagnant-cap model, we
do not consider a surfactant advection-diffusion equation at the interface.
Taking instead the normal and tangential interfacial stresses into account, we
solve the Stokes equation analytically for the falling drop with varying
interfacial tension. Some of the solutions thus obtained, e.g. the hovering
drop, violate conservation ofenergy unless energy is provided directly to the
interface. Considering the energy budget of the drop, we show that the terminal
velocity is bounded by the Stokes and the Hadamard-Rybczynski results. The
continuous-interface model is then obtained from the force balance for
surfactants at the interface. The resulting expressions gives the transition
between the two extremes, and also predicts that the critical radius, below
which drops fall like hard spheres, is proportional to the interfacial
surfactant concentration. By analysing experimental results from the
literature, we confirm this prediction, thus providing strong arguments for the
validity of the proposed model.
