Exact solutions in the 2-dimensional viscoelastic channel flow
Seminar Room 1, Newton Institute
Recently, it has been discovered that flows of polymer solutions can become unstable and exhibit turbulent-like behaviour at very small Reynolds numbers. As a rule, viscoelastic flows with curved streamlines are linearly unstable, while parallel shear flows are believed to exhibit a subcritical transition to a turbulent state. In the absence of inertia, these instabilities are driven by anisotropic elastic stresses.
Here I try to identify exact solutions in the 2D viscoelastic channel flow. Starting from the exact solutions of the Navier-Stokes equation found by Th. Herbert, solutions for the Oldroyd-B viscoelastic model are obtained by analytic continuation from the Newtonian case. It is found that these solutions persist at relatively small Reynolds numbers if the normal-stress difference is large enough. Nevertheless, so far I was unsuccessful in tracking these solutions down to the Re=0 limit. Other types of analytic continuation will be discussed as well.