Fluid dynamics has been used to simulate blood flow through major arteries of the human body (e.g. aorta and carotid) for advancement of medical technology. This dissertation is directed towards blood flow through a saccular aneurysm attached to a cerebral artery and the effects that the velocity and force of the blood flow have on the aneurysm wall. The two-dimensional nonlinear incompressible Navier-Stokes equations are solved on a staggered Eulerian grid to determine the flow of blood through the artery and aneurysm. An immersed boundary method is utilized to enforce solid boundaries. Subsequently, these nonlinear equations are coupled with the dynamic equation for the motion of an elastic body using an implicit second-order finite-difference scheme on a Lagrangian grid. An efficient and effective numerical program is created that simulates blood flow through a moving artery and its adjoining aneurysm.