In this talk, we consider an inverse Stokes problem in a bounded domain with a discontinuous viscosity coefficient. By analyzing the singularity of the Dirichlet Green's function near the interface in combination with constructing a well-posed Stokes-Brinkman system in a small domain, we prove a global uniqueness theorem that the discontinuous viscosity coefficient can be determined by the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary. Moreover, we also consider an inverse Stokes problem of reconstructing a bounded solid in an unbounded Stokes fluid, where the well-known LSM, FM and GLSM are extended from the wave equation into the Stokes equation by only taking the velocity fields in a certain domain. Finally, some numerical results are presented to illustrate the effectiveness of the inversion algorithms.