The bulk motion of an aqueous solution induced by the application of DC electric field is studied numerically. The physical model consists of a micro-cavity with two completely polarizable cylindrical electrodes. The electric double layer (EDL) model coupled with Navier-Stokes equations governing the electroosmotic flow has been described. The Nernst-Planck model uses two extra equations for the prediction of ion concentration. We employed IB (immersed boundary) technique for the implementation of boundary conditions and semi-implicit fractional-step method for solving the governing equations. A new method is described for implementing concentration boundary conditions on the electrodes. The bench mark problems, driven cavity flow and flow over a cylnder were used for the validation of our present code. The numerical results are compared with the analytical results obtained using Gouy-Chapman-Stern model for the one dimensional case. For the two dimensional case the flow field and the ionic concentration distributions obtained shows that the electoosmotic effect is predominant in the thin region around the electrode.