In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decomposition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence.