Dynamics of finger arrays in a diffusion-limited growth model with a drift

We propose a phenomenological model of fingering dynamics in the presence of an external drift, motivated by recent experiments on quasi-two-dimensional electrodeposition with hydrodynamic convection. We study the dynamical transition between a Laplacian growth regime and a weak finger competition regime. The model proposed defines a wavelength-selection problem of the finger array which is coupled in a nontrivial way to the single-finger selection problem in a channel. The wavelength-selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles, as the coarsening spatial scale of the pattern matches the appropriate diffusion length. This yields specific predictions for the selected wavelength. The dynamics of the model are studied numerically with boundary-integral methods in the quasistatic approximation and, complementarily, using a biased random walk Monte Carlo scheme on a lattice. Results are in qualitative agreement with the theoretical predictions.