This paper investigates the ion transport and electroosmotically induced flow around the cylindrical electrodes under both direct current (DC) and alternating current (AC) fields. The Poisson-Nernst-Plank (PNP) equations governing the ion transport around the ideally polarizable electrodes are solved numerically by neglecting the Stem layer effect. The fractional-step (FS) based decoupled solver is used in time integration of the ion-transport equations. A new immersed boundary (IB) methodology is described for imposing no-flux boundary conditions of ion concentration on the electrodes. A fully implicit coupled solver is also developed for calculating the ion transport around a pair of rectangular electrodes. The validity of the decoupled solver is verified by comparing its results with those obtained from the coupled solver. For further confirmation of the validity, the results are also compared with those obtained from the Poisson-Boltzmann model and both results are found to be in excellent agreement. The electroosmotically induced flow field is studied by numerically solving the Stokes equations. The system attains a steady state under DC, where the conduction term of ion transport is balanced by the diffusion term. Until the system attains a steady state for a few ms for the case of DC, fluid flow is induced. The electroosmotic flow under AC is more interesting, in that instantaneous flow oscillates with the frequency double of the applied field and a nonzero steady velocity field persists.