Comparison between a stabilized mixed finite element formulation for Hershel-Bulkley fluid and regularized models

Resumo

In this work, a mixed stabilized finite element formulation with continuous interpolations for velocity and descontinuous formulation for pressure is used to approach pseudoplastic materials with yield stress, that is, non linear viscoplastics. Here, Herschel-Bulkley model, [1], and regularized ones for the apparent viscosity are considered. This formulation is based on two well succeeded stabilized methods, separately, one for pseudoplastic problems (non linear) without yield stress [2], and the other for linear problems with yield stress (ideal plastic), [3],which are modeled by linear constitutive equations with inequality restriction.
Firstly it is presented the difficulties encountered by classical formulations in solving the problems separately, as well as solutions proposed before for each of those problems.
Regularized generalized alternatives (based on simple, Papanastasiou [4] and Bercovier-Engelman [5] schemes) are presented to deal with the discontinuity of the constitutive relations for the materials with yield stress and results are presented showing their limitations. They are compared with the proposed stabilized formulation which allows obtaining stable results without the necessity of regularizations, applied directly to the Herschel-Bulkley constitutive relation.
References
[1] A.H.P. Skelland. Non-Newtonian Flow and Heat Transfer. John Wiley & Sons, Oxford, 1967.
[2] Bortoloti, M. A. A. and Karam F., J. A Stabilized Finite Element Analysis for a Power-Law Pseudoplastic Stokes Problem, Applicable Analysis, V. 95, n.2, pp. 467-482, 2016.
[3] Faria, C. O. ; Karam F., J. . A Regularized Stabilized Mixed Finite Element Formulation for Viscoplasticity of Bingham Type, Computers and Mathematics with Applications, 2013. V66, no.6, pp. 975-995, 2013.
[4] T. C. Papanastasiou e A. G. Boudouvis. Flows of viscoplastic materials: Models and Computations. Computers & Structures, 64(1-4):677694, 1997.
[5] M. Bercovier e M. Engleman. A Finite-element method for incompressible non-Newtonian flows. Journal of Computational Physics, 36:313326, 1980.