We study a model inspired by the Oldroyd-B equations for viscoelastic fluids. The objective is to better understand the nonlinear coupling between the stress and velocity fields in viscoelastic flows, and thus gain insight into the reasons that cause the loss of accuracy of numerical computations at high Weissenberg number. We derive a model system by discarding the stress-advection and stress-relaxation terms in the Oldroyd-B model. The reduced (unphysical) model, which bears some resemblance to a viscoelastic solid, only retains the stretching of the stress due to velocity gradients and the induction of velocity by the stress field. Our conjecture is that such a system always evolves toward an equilibrium in which the stress builds up such to cancel the external forces. This conjecture is supported by numerous simulations. We then turn our attention to a finite dimensional model (i.e., a set of ordinary differential equations) that has the same algebraic structure as our model system. Numerical simulations indicate that the finite-dimensional analog has a globally attracting equilibrium manifold. In particular, it is found that subsets of the equilibrium manifold may be unstable, leading to a "peaking" behavior, where trajectories are repelled from the equilibrium manifold at one point, and are eventually attracted to a stable equilibrium point on the same manifold. Generalizations and implications to solutions of the Oldroyd-B model are discussed.
Bibliographical noteFunding Information:
We are grateful to Emmanuel Farjoun, Haim Sompolinsky, Edriss Titi and Benjamin Weiss for helpful advice. We also thank the anonymous reviewers for their very useful comments. RK was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. RF was supported by the Miller Foundation. GK was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
- Equilibrium manifold
- Global stability
- Nonlinear analysis
- Oldroyd-B model