Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids

V.P. Vallala (Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA)
J.N. Reddy (Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA)
K.S. Surana (Department of Mechanical Engineering, University of Kansas, Lawrence, Kansas, USA)

Engineering Computations

ISSN: 0264-4401

Publication date: 11 October 2011

Abstract

Purpose

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

Design/methodology/approach

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

Findings

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Originality/value

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

Keywords

Citation

Vallala, V., Reddy, J. and Surana, K. (2011), "Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids", Engineering Computations, Vol. 28 No. 7, pp. 828-852. https://doi.org/10.1108/02644401111178785

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Publisher

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Emerald Group Publishing Limited

Copyright © 2011, Emerald Group Publishing Limited

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