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A numerical method for a second-gradient theory of incompressible fluid flow

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Title: A numerical method for a second-gradient theory of incompressible fluid flow
Author(s): Kim, Tae-Yeon; Dolbow, John E.; Fried, Eliot
Abstract: This work concerns the development of a finite-element method for discretizing a recent second-gradient theory for the flow of incompressible fluids. The new theory gives rise to a flow equation involving higher-order gradients of the velocity field and introduces an accompanying length scale and boundary conditions. Finite-element methods based on similar equations involving fourth-order differential operators typically rely on C1-continuous basis functions or a mixed approach, both of which entail certain implementational difficulties. Here, we examine the adaptation of a relatively inexpensive, nonconforming method based on C0-continuous basis functions. We first develop the variational form of the method and establish consistency. The method weakly enforces continuity of the vorticity, traction, and hypertraction across interelement boundaries. Stabilization is achieved via Nitsche’s method. Further, pressure stabilization scales with the higher-order moduli, so that a classical formulation is recovered as a particular limit. The numerical method is verified for the problem of steady, plane Poiseuille flow. We then provide several numerical examples illustrating the robustness of the method and contrasting the predictions to those provided by classical Navier–Stokes theory.
Issue Date: 2006-04
Publisher: Department of Theoretical and Applied Mechanics (UIUC)
Series/Report: TAM Reports 1090, (2006)
Genre: Technical Report
Type: Text
Language: English
URI: http://hdl.handle.net/2142/343
ISSN: 0073-5264
Publication Status: published or submitted for publication
Peer Reviewed: is peer reviewed
Date Available in IDEALS: 2007-03-09
Is Version Of: Published as: Tae-Yeon Kim, John Dolbow, and Eliot Fried. A numerical method for a second-gradient theory of incompressible fluid flow. Journal of Computational Physics, Vol. 223, No. 2, 2007, pp. 551-570. DOI: 10.1016/j.jcp.2006.09.022. Copyright 2007 Elsevier Science.
 

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  • Theoretical and Applied Mechanics (TAM) Technical Reports
    TAM technical reports include manuscripts intended for publication, theses judged to have general interest, notes prepared for short courses, symposia compiled from outstanding undergraduate projects, and reports prepared for research-sponsoring agencies.

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