Numerical Investigation of a Stokes Model for Flow down a Fiber

We numerically solve a fourth order nonlinear partial differential equation derived by Craster and Matar [Craster and Matar, J. Fluid Mech. 553, 85 (2006)] that models a viscous fluid flowing down the outside of a vertical fiber in order to investigate the initial formation of perturbations along the fluid free surface. We compare numerical results of their model to existing experimental data [Smolka et al., Phys. Rev. E 77, 036301 (2008)]. In the simulations, perturbations consistently coalesced with neighboring perturbations during their initial formation, whereas in the experiments no coalescence was observed during this time period. We find that the amplitude growth follows two distinct exponential functions (referred to as phases I and II) in the initial formation of perturbations; in the experiments the data follows only one exponential function. The switch in growth rates from phase I to II is influenced by the coalescence of neighboring perturbations. The wavelength varies throughout the time that a perturbation forms so that the flow is unstable to a range of wavenumber. We compare the growth rates of several perturbations during phase II from the simulations to experimental data. Of the six data sets, the range of growth rate and wavenumber were in excellent agreement in two of the data sets and in fair to good agreement with two other data sets. For the last two data sets, there was no overlap in growth rate and little overlap in wavenumber. We find that linear stability results developed from Craster and Matar’s Stokes flow model are in excellent qualitative agreement with the growth of the perturbations measured during phase II in simulations for all six data sets. Finally, we find that data from simulations consistently overpredict the final perturbation amplitude measured in experiments.