The study concerns the unsteady flow of viscoelastic fluids in rigid cylindrical tubes, where one of the ends is either open or submerged in a viscoelastic sink. This shows the possibilities offered by analysis of axial speed or flow rate measurements and by pressure gradient measurements. In both cases, the model serves as a “rheometer” and allows the properties of the fluid to be determined automatically; in the second case, knowledge of the behavior of the fluid and use of axial speed analysis enables rheological modeling of the sink to be effected. This possibility allows “remote” investigations of complex mechanical systems to be done, and leads to numerous applications in various areas.