A numerical study of viscoelastic effects in chaotic mixing between eccentric cylinders

In this paper, we are concerned with the effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing of the flow between two eccentric, alternately rotating cylinders. We employ the well-developed h-p finite element method to achieve a high accuracy and efficiency in calculating steady solutions, and a full unsteady algorithm for creeping viscoelastic hows to study the transient process in this periodic viscoelastic flow. Since the distribution of periodic points of the viscoelastic flow is not symmetric, we have developed a domain-search algorithm based on Newton iteration for locating the periodic points. With the piecewise-steady approximation, our computation for the upper-convected Maxwell fluid predicts no noticeable changes of the advected coverage of a passive tracer from Newtonian flow, with elasticity levels up to a Deborah number of 1.0. The stretching of the fluid elements, quantified by the geometrical mean of the spatial distribution, remains exponential up to a Deborah number of 6.0, with only slight changes from Newtonian flow. On the other hand, the shear-thinning viscosity, modelled by the Carreau equation, has a large impact on both the advection of a passive tracer and the mean stretching of the fluid elements. The creeping, unsteady computations show that the transient period of the velocity is much shorter than the transient period of the stress, and from a pragmatic point of view, this transient process caused by stress relaxation due to sudden switches of the cylinder rotation can be neglected for predicting the advective mixing in this time-periodic flow. The periodic points found up to second order and their eigenvalues are indeed very informative in understanding the chaotic mixing patterns and the qualitative changes of the mean stretching of the fluid elements. The comparison between our computations and those of Niederkorn & Ottino (1993) reveals the importance of reducing the discretization error in the computation of chaotic mixing. The causes of the discrepancy between our prediction of the tracer advection and Niederkorn & Ottino's (1993) experiment are discussed, in which the influence of the shear-thinning first normal stress difference is carefully examined. The discussion leads to questions on whether small elasticity of the fluid has a large effect on the chaotic mixing in this periodic flow.