Measures of mixing for extrusion by averaging concepts

Abstract

This paper proposes modified Deformation Characteristics (DC) as a new deformation measure for three-dimensional flows in extrusion. In contrast to the previous DC, the modified DC is defined by an averaged length stretch rate with the aid of two subsequent averaging concepts: firstly, average of length stretch rate of all the local elements in a shell, and secondly, average of a fluctuation of the averaged length stretch rate with time using a probability function. The modified DC was applied to the case of a two-dimensional flow approximation based on the analysis of kinematics during one circulation including the ''turning effect.'' Consequently, we have proposed useful formulae to determine the modified DC for the two-dimensional flow approximation. This involves extensive numerical calculations of the modified DC, both for the three-dimensional flow and for the two-dimensional flow approximation. Typical results are discussed with comparisons between the modified DC, the previous DC, and a strain proposed by Pinto and Tadmor. It is interesting to note that the Weighted-Average Deformation Characteristics (WADC) obtained from the three-dimensional analysis was found to be close to that from the two-dimensional approximate analysis. Therefore, the formulae proposed for the modified DC of the two-dimensional flow approximation seems to be useful in analyzing the mixing behavior on extrusion with a large aspect ratio, avoiding the complicated evaluation of the averaging process of the length stretch rate for the three-dimensional flow.