The Use of Mixed Finite Element Methods for Viscoelastic Fluid Flow Analysis

Baaijens, Fpt Frank; Hulsen, MA Martien; Anderson, Patrick D.

doi:10.1002/0470091355.ecm066

Cited by 23 publications

(32 citation statements)

References 72 publications

(86 reference statements)

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“…For example, polymer melts and solutions behave as incompressible liquids in most processing flows. Diffusion and heat transfer seem usually negligible in fast processes like coating flows, and they are commonly neglected in viscoelastic flow modeling ( [122] and references therein). The most relevant reduced equation sets that apply in the simpler situations of incompressible flow, non-diffusing flows, and isothermal flows are reported in the next sections; other interesting cases are summarized in Appendices D and E.…”

Section: Simplified Modelsmentioning

confidence: 99%

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Pasquali

Scriven

2004

Journal of Non-Newtonian Fluid Mechanics

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A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two-and three-dimensional domains with complex shapes and free surfaces.

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Section: Simplified Modelsmentioning

confidence: 99%

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Pasquali

Scriven

2004

Journal of Non-Newtonian Fluid Mechanics

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“…If the exact solution is recovered, Equation (14) vanishes. However, in a ÿnite-element calculation this is generally not the case [20]. It must be emphasized that the stress splitting formulations are beneÿcial in maintaining the mathematical elliptic properties of the momentum and continuity equations, but the proper treatment of the hyperbolic constitutive equation remains an important part of the overall algorithm.…”

Section: Mathematical Modellingmentioning

confidence: 99%

“…After multiplying (19) by the weighting function u i ∈ V 0 ( ), (20) by q ∈ Q 0 ( ), (21) by ij ∈ V ( ) and (22) by d ij ∈ Q( ), and integrating by parts, we get the following variational formulation for (19)- (22) as…”

Section: Variational Formulationmentioning

confidence: 99%

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Spectral element method for viscoelastic flows in a planar contraction channel

Meng

Evans

2003

Numerical Methods in Fluids

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SUMMARYA new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic uid ows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four-to-one contraction channel for a uid governed by an Oldroyd-B constitutive equation. The behaviour of the Oldroyd-B uids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions.

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“…Computational methods used in viscoelastic flows remain an active area of research [3]. Several methods can be used to solve viscoelastic macroscopic constitutive equations in complex flows [4][5][6] and three-dimensional applications can be found in [7,8], but they did not take free surfaces into account. Two dimensional steady-state or transient viscoelastic flows with free surfaces have been studied for about 30 years, mainly applied to the die swell or filament stretching problems [9][10][11][12][13][14][15] and recently to extrudate swell or jet flow [16,17].…”

Section: Introductionmentioning

confidence: 99%

A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

et al. 2011

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The original publication is available at http://www.springer.comInternational audienceTA Mixed Finite Element (MFE) method for 3D non-steady flow of a viscoelastic compressible fluid is presented. It was used to compute polymer injection flows in a complex mold cavity, which involves moving free surfaces. The flow equations were derived from the Navier-Stokes incompressible equations, and we extended a mixed finite element method for incompressible viscous flow to account for compressibility (using the Tait model) and viscoelasticity (using a Pom-Pom like model). The flow solver uses tetrahedral elements and a mixed velocity/pressure/extra-stress/density formulation, where elastic terms are solved by decoupling our system and density variation is implicitly considered. A new DEVSS-like method is also introduced naturally from the MINI-element formulation. This method has the great advantage of a low memory requirement. At each time slab, once the velocity has been calculated, all evolution equations (free surface and material evolution) are solved by a space-time finite element method. This method is a generalization of the discontinuous Galerkin method, that shows a strong robustness with respect to both re-entrant corners and flow front singularities. Validation tests of the viscoelastic and free surface models implementation are shown, using literature benchmark examples. Results obtained in industrial 3D geometries underline the robustness and the efficiency of the proposed method

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The Use of Mixed Finite Element Methods for Viscoelastic Fluid Flow Analysis

Cited by 23 publications

References 72 publications

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Spectral element method for viscoelastic flows in a planar contraction channel

A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

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