A note on the calculation of strain histories in orthogonal streamline coordinate systems

Abstract

An extension of a previous work concerning the calculation of strain histories along streamlines is made to get more complete and useful expressions of Finger's strain tensor in a cylindrical (or Cartesian) coordinate system as well as in an orthogonal streamline coordinate system. One of the results shows that Winter's tracking model is correct.Relations among the recent three results of Winter, Adachi and Crochet et al. are presented clearly. Moreover useful applications of Frenet-Serret's formula to the study of the deformation and flow kinematics along streamlines are shown in comparison with the ordinary tensor approach.

Simulation of planar welding flows: Part 2. Strain history, stress calculation, and experimental comparison

“…(20) derived the same tracking equations in different notation and applied them to a finite element program for memory fluids. Adachi (21) rederived the tracking equations via a third method and found agreement with the earlier results of Winter and Dupont, et al In this paper, we use Winter's scheme and Nordberg's program (22) to compute the strain history of material elements along their path lines. The basic equations of the tracking procedure are given in the following in terms of planar coordinates.…”

Numerical study of flows of complex fluids between eccentric cylinders using transformation functions

“…The Finger and Cauchy tensors to be considered in the constitutive equation (42) To evaluate the deformation gradient tensorF t (t), we start from Adachi's work [27,28] on Protean co-ordinates [29]. The approach, deÿned in relation to variables deÿned in global or local transformation systems, is similar to that previously followed in stream-tube analysis [30,31] for ows involving open streamlines.…”

Numerical simulations of non-Newtonian flows between eccentric cylinders by domain decomposition and stream-tube method

“…To evaluate the deformation gradient tensorF t (t), we applied the analysis already defined in [16,17], starting from Adachi's work [21,22]. The approach allows simple relations to be obtained for the components ofF t (t).…”

“…(22) are related to the deformation gradient tensorF t (t) = ∂x m t ∂x γ t by the following relations [18]:…”