The Stokes flow induced by the motion of an elastic massless filament immersed in a two-dimensional fluid is studied. Initially, the filament is deviated from its equilibrium state and the fluid is at rest. The filament will induce fluid motion while returning to its equilibrium state. Two different test cases are examined. In both cases, the motion of a fixed-end massless filament induces the fluid motion inside a square domain. However, in the second test case, a deformable circular string is placed in the square domain and its interaction with the Stokes flow induced by the filament motion is studied. The interaction between the fluid and deformable body/bodies can become very complicated from the computational point of view. An immersed boundary method is used in the present study. In order to substantiate the accuracy of the numerical method employed, the simulated results associated with the Stokes flow induced by the motion of an extending star string are compared well with those obtained by the immersed interface method. The results show the ability and accuracy of the IBM method in solving the complicated fluid-structure and fluid-mediated structure-structure interaction problems happening in a wide variety of engineering and biological systems.