A two-dimensional (2D) Arbitrary-Lagrangian-Eulerian (ALE) method of fluid flow in domains containing moving objects is presented in this study. The ALE method is based on a fixed mesh that is locally fitted at the moving objects. At any time in the simulation, fluid is occupied the reference domain and is discretized using a mesh of bilinear isoparametric finite elements. The moving objects are described using sets of marker points or nodes which can slide over the basic mesh. Once the moving object has gone through the stationary element, the element is restored to its original form. Therefore, the mesh adaptation is local both in space and time and is performed only in those elements intersected by an object. As a result, the method does not require interpolation. There are an only three possible modifications to the intersected elements, when quadrilateral elements are used for 2D domain. The possibility of mesh entanglement is eliminated, because of the global mesh is independent of object movement. The mesh never becomes unsuitable due to its continuous deformation, thus eliminating the need for repeated re-meshing and interpolation. A validation is presented via a problem with an exact analytical solution to the case of 2D flow between two parallel plates separating with a prescribed velocity. The method's capabilities and accuracy are illustrated through application in realistic geometrical settings which show the robustness and flexibility of the technique.