Mixing in viscoelastic fluids. Preliminary results (joint work with M. Shelley at the
Courant Institute).
The following simulations are of solutions to the Stokes-Oldroyd-B equations in a 2 pi periodic domain in the plane using a Fourier based spectral method. A small amount of numerical viscosity is added to control large stress gradient growth. The background force is a small time periodic perturbation from a 4 vortex flow. The Weissenberg number is an "elasticity" parameter. Weissenberg number equals zero corresponds to a purely Newtonian fluid, and infinite Weissenberg number corresponds to a purely elastic solid.
The first set of figures (and movies) below are of particle tracers in the fluid. The colors are for visualizing the mixing from the four separate vorices. With a steady background force these four vortices remain separated for all Weissenberg numbers.
1) Wi = 0.5, in this case the Weissenberg number is relatively low. The mixing observed is restricted to a small set near the stable and unstable manifold of the hyperbolic point. The behavior here is qualitatively similar to the behavior for a purely Newtonian fluid (in the zero-Reynolds number limit). The picture below shows t=0, on the left and t=400 on the right.The avi file below is a movie of the particle tracers over time.
2) Wi = 10.0, in this case the Weissenberg number is sufficiently high to influence the mixing of the fluid. The picture below shows t=0, on the left and t=400 on the right.The avi file below is a movie of the particle tracers over time.
3) Vorticity contour plots for Wi=0.5 (left) and Wi=10.0 (right) at t=400. Movies below (avi files) show vorticity over time.
4) The trace of the stress represents the amount of stretching of polymer coils. On the left is a contour plot of the trace of the stress for Wi=0.5 and on the left for Wi=10 at t=400. Initially the stress is the identity matrix with trace S =2. The maximum value of the trace for Wi=0.5 is about 3 and for Wi=10 about 200. The movie shows contour plots of the trace over time.
