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 she ar-thinning viscosity on the chaotic mixing of the flow between two eccentr ic, alternately rotating cylinders. We employ the well-developed h-p finite element method to achieve a high accuracy and efficiency in calculating st eady solutions, and a full unsteady algorithm for creeping viscoelastic how s to study the transient process in this periodic viscoelastic flow. Since the distribution of periodic points of the viscoelastic flow is not symmetr ic, we have developed a domain-search algorithm based on Newton iteration f or locating the periodic points. With the piecewise-steady approximation, o ur 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 distribu tion, remains exponential up to a Deborah number of 6.0, with only slight c hanges from Newtonian flow. On the other hand, the shear-thinning viscosity , modelled by the Carreau equation, has a large impact on both the advectio n of a passive tracer and the mean stretching of the fluid elements. The cr eeping, unsteady computations show that the transient period of the velocit y is much shorter than the transient period of the stress, and from a pragm atic 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 un derstanding the chaotic mixing patterns and the qualitative changes of the mean stretching of the fluid elements. The comparison between our computati ons and those of Niederkorn & Ottino (1993) reveals the importance of reduc ing the discretization error in the computation of chaotic mixing. The caus es of the discrepancy between our prediction of the tracer advection and Ni ederkorn & 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 flui d has a large effect on the chaotic mixing in this periodic flow.