A computational fluid dynamics model is developed to study the dynamics of meniscus formation and capillary flow between vertical parallel plates. An arbitrary Lagrangian-Eulerian approach is employed to predict and reconstruct the shape of the meniscus with no need to employ implicit interface tracking schemes. The developed model is validated by comparing the equilibrium capillary height and meniscus shape with those predicted by available theoretical models. The model was used to predict the capillary flow of water in hydrophilic (silver) and hydrophobic (Teflon) vertical channels with wall spacings ranging from 0.5 mm to 3 mm. It is shown that the computational model accurately predicts the capillary flow regardless of the channel width, whereas the theoretical models fail at relatively large wall spacings. The model captures several important hydrodynamic phenomena that cannot be accounted for in the theoretical models including the presence of developing flow in the entrance region, time-dependent formation of the meniscus, and the inertial effects of the liquid in the reservoir. The sharp interface tracking technique enables direct access to the flow variables and transport fluxes at the meniscus with no need to use averaging techniques.