Flow Classification and Its Application to Fluid Processing

Mathematically, the problem of flow field classification can be analyzed by the eigenanalysis of the deformation-rate tensor; however, such analysis technique have not been commonly applied in fluid processing. We derive a simplified objective flow classification scheme based on the invariants of the strain-rate tensor and the vorticity tensor. Multiaxiality of flow, which is related to the type of elongation, and converging/bifurcating flow, is characterized by the strain-rate tensor, while rotation contribution that protects fluid element from stretching is characterized by the relative intensity of an objective vorticity to the strain-rate. The spatial distributions of flow classification quantities offer an essential tool in understanding the flow pattern structure, and therefore can be useful to get insights into the connection between the geometry and the process performance.

A Study on a Grade-One Type of Hypo-Elastic Models

In Hypo-elastic constitutive models an objective rate of the Cauchy stress tensor is expressed in terms of the current state of the stress and the deformation rate tensor D in a way that the dependency on the latter is a homogeneously linear one. In this work, a type of grade-one hypo-elastic models (i.e. models with linear dependency of the hypo-elasticity tensor on the stress) is considered for isotropic materials based on the objective corotational rates of stress. A positive real parameter denoted by n is involved in the considered type. Different values can be selected for this parameter, each selection leads to a specific model within the class of grade-one hypo-elasticity. The spin of the associated corotational rate is also dependent on the parameter n. In the special case of n=0, the corresponding hypo-elastic model reduces to a grade-zero one with the logarithmic rate of stress; noting that this rate is a corotational rate associated with the logarithmic spin tensor. Moreover, by choosing n=2, the model reduces to a grade-one hypo-elastic model with the Jaumann rate, i.e. the corotational rate associated with the vorticity spin tensor. As case studies, the simple shear problem is investigated with utilizing the considered type of hypo-elastic models with various values for parameter n, and the curves for the stress-shear response are depicted.

Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate

It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.