We present a numerical scheme for viscoelastic flow based on a second-order central differencing method recently introduced in the context of incompressible newtonian flow; the incompressibility constraint is treated with the projection method. The result is a simple and efficient scheme that is readily adaptable to a wide class of differential constitutive equations and flow geometries. We implement the new method on Couette - Taylor flow for a fluid governed by the Oldroyd-B constitutive equations. We simulate transient flow in a domain that includes at least eight wavelengths during many hundreds of natural periods. For weak elasticity, a stationary instability leading to Taylor vortices is observed. For a regime of parameters where both inertia and elasticity are important, the instability is oscillatory. In both cases the early stage growth rates are compared to linear stability calculations, showing good agreement. The oscillatory instability is fourfold degenerate and gives rise to two bifurcating branches: an axially traveling wave and a standing wave; only one of these solutions is stable. In the early stages of the instability, there is generally a combination of traveling and standing waves, depending on the initial conditions. As nonlinearities become important, the flow spontaneously breaks into coexisting regions of upward- and downward-going waves. Such flow can persist for long times, until the globally stable traveling wave takes over and a limit cycle is reached.
Bibliographical noteFunding Information:
I have benefited from many fruitful discussions with Professor M. Denn. I am also thankful to Professors A. Chorin and S. Muller and Drs. A. Kast and D. Nolan for their advice. Dr. D. Adalsteinsson contributed to the graphical processing of the numerical data. This work was supported by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, Department of Energy, under Contract DE-AC03-76SF-00098.
- Couette - Taylor
- Finite differences
- Hopf bifurcation
- Viscoelastic flow