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Exact solutions in the 2-dimensional viscoelastic channel flow

Date: 
Friday 12th September 2008 - 10:20 to 10:40
Venue: 
INI Seminar Room 1
Session Chair: 
T Mullin
Abstract: 

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.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons