On a scenario of transition to turbulence for polymer fluid flow in a circular pipe

Equations describing non-stationary and stationary flows of an incompressible polymer fluid through a pipe are derived on the basis of rheological mesoscopic Pokrovskii–Vinogradov model. Exact stationary solutions of them are obtained, and conditions providing their existence are outlined. Numerical simulation of stabilization of non-stationary flow is done, as well, and the restrictions on the values of parameters that ensure stabilization are computed. In a number of cases these restrictions coincide with the conditions of existence of stationary solutions. The obtained results enable one to describe constructively the process of destruction of laminar Poiseuille-type flows, which usually initiates onset of turbulence. The key role in mechanics of this process is played by the size and orientation of macromolecules of the polymer fluid. Mathematical description of the process uses essentially the solutions' singular points.