Stability of the steady flow of a viscous liquid through the thick-walled multilayered viscoelastic tubes with different rheological properties of the layers for no stress boundary conditions at the outer surface of the tube is investigated. The eigen values of the system are found. The influence of the material parameters of the layers and the Reynolds number on the spatial and temporal amplification rate of the most unstable mode is investigated. It is shown that the system can be stabilized by increasing the shear modules of the inner and middle layers. The temporal amplification rate significantly decreases and becomes negative with increasing the shear modules of the outer layer, that corresponds to temporal stability of the system. The comparative study of the obtained dependences with the solution of the same problem at no displacement boundary conditions is carried out. The results may be applied to the fluid flows in the ducts of different technical devices as well as to the blood flow in the arteries and veins which are composed of three layers with different functions, structure and material parameters.

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