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RESEARCH PAPERS: Non-Newtonian Behavior and Rheology

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

[+] Author and Article Information
Flávia Zinani

Laboratory of Applied and Computational Fluid Mechanics (LAMAC), Mechanical Engineering Department, Federal University of Rio Grande do Sul, Rua Sarmento Leite 425, 90050-170 Porto Alegre (RS), Brazil

Sérgio Frey1

Laboratory of Applied and Computational Fluid Mechanics (LAMAC), Mechanical Engineering Department, Federal University of Rio Grande do Sul, Rua Sarmento Leite 425, 90050-170 Porto Alegre (RS), Brazilfrey@mecanica.ufrgs.br

1

Corresponding author.

J. Fluids Eng 130(8), 081507 (Jul 29, 2008) (14 pages) doi:10.1115/1.2956514 History: Received July 12, 2007; Revised January 15, 2008; Published July 29, 2008

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 2

Horizontal velocity profiles in x1=0.5L. (a) Re=1. (b) Re=400.

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Figure 3

(a) Horizontal velocity versus x2, Re=50, n=0.5. (b) Vertical velocity versus x1, Re=50, n=0.5. (c) Horizontal velocity versus x2, Re=100, n=1.5. (d) Vertical velocity versus x1, Re=100, n=1.5.

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Figure 4

Streamlines for Re=100. (a) n=0.25. (b) n=0.5. (c) n=0.75. (d) n=1. (e) n=1.5.

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Figure 5

Vortex eye position. (a) Detail for various Re. (b) A general view in the cavity geometry.

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Figure 6

Problem statements for the abrupt contraction flows

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Figure 7

Carreau viscosity function

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Figure 8

Pressure elevation plots for Re=100, Cu=100, and n=0.5. (a) Planar and (b) axisymmetric.

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Figure 9

τ12 fields: Cu=1 and n=0.1 for (a) planar and (b) axisymmetric. Cu=100 and n=0.1 for (c) planar and (d) axisymmetric. τ11 fields: Cu=1 and n=0.1 for (e) planar and (f) axisymmetric. Cu=100 and n=0.1 for (g) planar and (h) axisymmetric.

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Figure 10

Pressure drop along the symmetry line. Re=1 for (a) planar and (b) axisymmetric. Re=100 for (c) planar and (d) axisymmetric.

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Figure 11

Profile of u* along the symmetry line: for Newtonian, (a) planar and (b) axisymmetric; for Cu=100 and n=0.5, (c) planar and (d) axisymmetric; for Re=1 and n=0.5, (e) planar and (f) axisymmetric; for Re=1 and Cu=10, (g) planar and (h) axisymmetric

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Figure 12

Streamlines. Re=100, Cu=100, and n=0.5: (a) planar and (b) axisymmetric.

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Figure 13

Streamlines. Re=1, Cu=1, and n=0.1; (a) planar and (b) axisymmetric. Re=1, Cu=100, and n=0.1; (c) planar and (d) axisymmetric.

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Figure 14

Profile of u* versus y and r in the contraction plane: (a) Re=1, Cu=100, planar; (b) Re=1, Cu=100, axisymmetric; (c) Re=1, n=0.1, planar; (d) Re=1, n=0.1, axisymmetric; (e) Cu=10, n=0.1, planar; (f) Cu=10, n=0.1, axisymmetric

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Figure 15

Comparison between the multifield GLS results and results of Kim (32))

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Figure 1

Lid-driven cavity flow, Re=400. (a) Pressure elevation, (b) τ11 contours, and (c) τ12 contours.

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