摘要：Electrodeposition in thin cells of different orientations relative to gravity leads to
complex stable and unstable physicochemical hydrodynamic flows. Here we present a theoretical
macroscopic 3D model and numerical simulations describing such flows. The model
consists in the Nernst-Planck equations for ion transport, the Poisson equation for the electric
field and the Navier-Stokes equations for the fluid flow. These equations are solved in a
uniform grid using finite differences and a strongly implicit iterative scheme. Our model predicts
that when the cathode is above the anode the flow becomes stable as long as there is no
growth. When growth is present, the model predicts zones of lowered concentration, adjacent
to a downwards growing finger, inducing a gravity driven convective vortex roll wrapped to the
finger. In a vertical cell with the anode above the cathode, the model predicts the existence of
an unstable flow in the form of plumes emanating from each cathode, expanding toward one
another and mixing. For both cases, in the presence of growth, the model predicts the existence
of an electrically driven vortex ring at the dendrite tip; it allows fluid to penetrate the dendrite
tip and to be ejected from its side. Such behavior has been observed in experiments.
