2002/04/30 by Michael E. Cates, M. E. Cates, D. A. Head +3 · 1 citation
Chemical Engineering · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Classical mechanics #Fluid Dynamics and Turbulent Flows #Geometry #Instability #Lambda #Material Dynamics and Properties #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Monotonic function #Nonlinear system #Physics #Quantum mechanics #Rheology #Rheology and Fluid Dynamics Studies #Scalar (mathematics) #Shear flow #Shear rate #Shear stress #Sigma #Sigma model #Simple shear #Thermodynamics #cond-mat.soft
paper · pdf · doi:10.1103/physreve.66.025202
published as Phys. Rev. E 66, 025202(R) (2002). · Reference added; typos corrected. To appear in PRE Rap. Comm
arxiv created 2002/07/08 · openalex publication_date 2002/08/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study a simple scalar constitutive equation for a shear-thickening material at zero Reynolds number, in which the shear stress \ensuremathσ is driven at a constant shear rate \stackrel\ifmmode \else \.\fi\ensuremathγ and relaxes by two parallel decay processes: a nonlinear decay at a nonmonotonic rate R(\ensuremathσ1) and a linear decay at rate \ensuremathλ\ensuremathσ2. Here \ensuremathσ1,2(t)=\ensuremathτ1,2^\ensuremath-1\ensuremath∫0t\ensuremathσ(t^\ensuremath')exp[\ensuremath-(t\ensuremath-t^\ensuremath')/\ensuremathτ1,2]dt^\ensuremath' are two retarded stresses. For suitable parameters, the steady state flow curve is monotonic but unstable; this arises when \ensuremathτ2>\ensuremathτ1 and 0>R^\ensuremath'(\ensuremathσ)>\ensuremath-\ensuremathλ so that monotonicity is restored only through the strongly retarded term (which might model a slow evolution of the material structure under stress). Within the unstable region we find a period-doubling sequence leading to chaos. Instability, but not chaos, persists even for the case \ensuremathτ1\ensuremath→0. A similar generic mechanism might also arise in shear thinning systems and in some banded flows.