Slow flows of yield stress fluids: yielding liquids or flowing solids?

Yield stress fluids (YSF) exhibit strongly non-linear rheological characteristics. As a consequence, they develop original flow features (as compared to simple fluids) under various boundary conditions. This paper reviews and analyzes the characteristics of a series of slow flows (just beyond yielding) under more or less complex conditions (simple shear flow, flow through a cavity, dip-coating, blade-coating, Rayleigh-Taylor instability, Saffman-Taylor instability) and highlights some of their common original characteristics: (i) a transition from a solid regime to a flowing regime which does not correspond to a true “liquid state,” the flow in this regime may rather be seen as a succession of solid states during very large deformation; (ii) a strong tendency to localization of the yielded regions in some small region of the material while the rest of the material undergoes some deformation in its solid state; (iii) the deformation of YSF interface with another fluid, in the form of fingers tending to penetrate the material via a local liquefaction process. Finally, these observations suggest that slow flows of YSF are a kind of extension of plastic flows for very large deformations and without irreversible changes of the structure. This suggests that the field of plasticity and the field of slow flows of YSF could benefit from each other.     

Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation

This paper investigates the emergence of mixed-mode oscillations (MMOs) in the forced Van der Pol’s equation. It is found that the MMOs studied here can be classified as a slow passage through canard explosion, which is different from the usual fast-slow bursters. We first consider the external forcing as a control parameter and study its influence on the Van der Pol’s equation with constant forcing (VPCF). Then we briefly discuss the famous canard phenomenon in VPCF. The results of these analysis, together with the “transformed phase diagram,” are applied to the forced Van der Pol’s equation, which shows that the canard explosion and the external forcing plays an important role in the generation of MMOs, that is, the MMOs are created since the external forcing slowly and periodically visits the rest and spiking areas of VPCF.     

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

The cusp singularity—a point at which two curves of fold points meet—is a prototypical example in Takens’ classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens’ classification occurring for the nerve impulse example—folds and folded saddles—is included to make this article self-contained.