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TECHNICAL PAPERS

Galerkin Least-Squares Finite Element Approximations for Isochoric Flows of Viscoplastic Liquids

[+] 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, Porto Alegre, RS, CEP: 90050-170, 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, Porto Alegre, RS, CEP: 90050-170, Brazilfrey@mecanica.ufrgs.br

1

Corresponding author.

J. Fluids Eng 128(4), 856-863 (Dec 06, 2005) (8 pages) doi:10.1115/1.2201633 History: Received April 01, 2005; Revised December 06, 2005

Abstract

The flow of viscoplastic liquids is studied via a finite element stabilized method. Fluids, such as some food products, blood, mud, and polymer solutions, exhibit viscoplastic behavior. In order to approximate this class of liquids, a mechanical model, based on the principles of power expended and mass conservation, is exploited with the Papanastasiou approximation for Casson equation employed to model viscoplasticity. The approximation for the nonlinear set of partial differential equations is performed, using a stabilized finite element methodology. A Galerkin least-squares strategy is employed to avoid the well-known difficulties of the classical Galerkin method in isochoric flows. It circumvents the Babuška-Brezzi condition and handles the asymmetry of the advective operator in high advective flows. Some two-dimensional (2D) viscoplastic flows through a 4:1 planar expansion, for a range of Casson $(0⩽Ca⩽10)$ and Reynolds $(0⩽Re⩽50)$ numbers, have been investigated, paying special attention to the characterization of vortex length and unyielded regions. The numerical results show the arising of regions of unyielded material throughout the flow, strongly affecting the vortex structure, which is reduced with the increase of the Casson number even in flows with considerable inertia.

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Figures

Figure 1

Lid-driven cavity flow, for Re=400: (a) velocity and (b) pressure profiles

Figure 2

The lid-driven cavity flow of a Bingham liquid, for Bi=50: (a) eye of the vortex position, (b) unyielded zones and extra-stress contours, and (c) flow streamlines

Figure 3

Planar 4:1 expansion: problem statement

Figure 4

Details of unyielded zones (in black), Re=10: (a) Ca=0.1, (b) Ca=1, (c) Ca=3, and (d) Ca=10

Figure 5

Detail of viscoplastic flow streamlines, Re=50: (a) Ca=0, (b) Ca=0.1, (c) Ca=0.4, (d) Ca=0.8, (e) Ca=1 and (f) Ca=1.5

Figure 6

Structure of the unyielded zone (detail), Re=50: (a) Ca=0.1, (b) Ca=0.4, (c) Ca=0.8, (d) Ca=1, and (e) Ca=1.5

Figure 7

Extra stress and streamlines contours (detail), for Re=50 and Ca=1.5

Figure 8

Reattachment length versus Ca for Re=50

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