An innovative computation procedure is developed to solve the external flow problems for viscous fluids. The method is able to handle the infinite domain so that it is convenient for the external flows. The code is based on the projection method of the Navier-Stokes equations. We use the three-step explicit finite element method to solve the momentum equation by extracting the boundary effects from the finite computation domain. The pressure Poisson equation for the external field is treated by the boundary element method. The arbitrary Lagrangian-Eulerian (ALE) scheme is employed to incorporate the present algorithm to deal with the moving boundary, such as the motion of an impulsively moving circular cylinder in a viscous fluid. The model demonstrates that drag force is well predicted for a circular cylinder moving in a still viscous fluid starting from rest, to a constant acceleration, and then maintaining at a uniform velocity. In the constant acceleration phase, the drag force is closed to the added mass effect from the ideal flow theory. On the other hand, the drag force is equal to viscous flow theory in the constant velocity phase.