Flow instability and flow control in compliant tubes are studied in application to biological systems and technical devices. Common features of the flows are determined by fluid—structure interaction that may lead to absolute and convective instabilities. The stability of a steady flow of an incompressible viscous fluid through a compliant tube composed of three anisotropic viscoelastic layers with different material properties is studied. The eigenvalues of the system are calculated for the no-stress and no-displacement boundary conditions at the outer surface of the tube. It is shown that the flow can be stabilized by a proper choice of the thickness, viscosity and elastic moduli of the layers. The multilayered viscoelastic coatings can be used to control flows in compliant ducts.