Abstract: Effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing between two
alternately rotating cylinders has been studied. The h-p finite element method is used to
obtain high accurate solutions of the steady flow. The unsteady, periodic flow is simulated
using the piecewise-steady approximation. Characteristics of the chaotic mixing are
analyzed by examining the asymptotic coverage of a passive tracer and the lineal stretching
of the fluid elements in the annulus. For the viscoelastic fluids modeled by the upper-
convected Maxwell constitutive equation (UCM), our computation predicts little effect of
the fluid elasticity on the mixing patterns. On the other hand, the shear-thinning
viscosity, modeled by the Carreau equation, has a large impact on the advection of a
passive tracer and the distribution of lineal stretching. We find that the zones of the
lowest stretching match remarkably well with the regular zones in the tracer-coverage
plotting. Our study reveals the vital importance of reducing the discretization errors of
the velocity field in the numerical simulation of chaotic flows.

Key words: chaotic mixing, viscoelasticity, shear-thinning viscosity

摘要： Effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing between two
alternately rotating cylinders has been studied. The h-p finite element method is used to
obtain high accurate solutions of the steady flow. The unsteady, periodic flow is simulated
using the piecewise-steady approximation. Characteristics of the chaotic mixing are
analyzed by examining the asymptotic coverage of a passive tracer and the lineal stretching
of the fluid elements in the annulus. For the viscoelastic fluids modeled by the upper-
convected Maxwell constitutive equation (UCM), our computation predicts little effect of
the fluid elasticity on the mixing patterns. On the other hand, the shear-thinning
viscosity, modeled by the Carreau equation, has a large impact on the advection of a
passive tracer and the distribution of lineal stretching. We find that the zones of the
lowest stretching match remarkably well with the regular zones in the tracer-coverage
plotting. Our study reveals the vital importance of reducing the discretization errors of
the velocity field in the numerical simulation of chaotic flows.

关键词: chaotic mixing;viscoelasticity;shear-thinning viscosity