Finite elements method for viscoelastic flows

Let us consider the set of equations governing the flow of an Oldroyd-B fluid  

Substituting equation (15) in equation (19), and using equation (17), we obtain  

The non-homogeneous Stokes problem (18)-(20) in velocity and pressure is mathematically coupled to transport equations (16) through Ty. In this case the elimination of the tensor Ty is not possible, it has to be considered as a primitive variable. Two basic ideas (introduced by Marchal and Crochet) guide these developments. 

The viscoelastic constitutive equations are of hyperbolic type and it is well-known that numerical solutions require special care. The classical Galerkin method was very shown to be inadequate for such problems. Special techniques were developed based on the very general 

The three-field formulation should reduce to a convenient approximation of the Stokes problem when applied to a Newtonian flow. Hence a second inf-sup condition is necessary to obtain stability. If the approximation (Tv)h of the extra-stress tensor is continuous, this supplementary condition can be satisfied by using a sufficient number of interior nodes in each element. On the contrary if this approximation is discontinuous, this can be done by imposing that the derivatives DUh of the approximated velocity field are in the space of (Tv)h- Various possible choices concerning the satisfaction of the inf-sup condition and the introduction of upwinding have been explored since 1987. In the following we will recall the basic steps (see [10], [24] and [38] for details). 

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Baaijens FPT (1998) Mixed finite element methods for viscoelastic flow analysis a review. J Non-Newtonian Fluid Mech 79 361-85. 

Baaij ens, P. T. F. Mixed Finite Element Methods for Viscoelastic Flow Analysis A Review. In Faculty of Mechanical Engineering. Eindhoven University of Technology Center for Polymers and Composites Eindhoven, 2001, p. 37. 

Swarbrick, S. J. and Nassehi, V., 1992a. A new decoupled finite element algorithm for viscoelastic flow. Part 1 numerical algorithm and sample results. Int. J. Numer. Methods Fluids 14, 1367-1376,... 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

Chang P W, Patten T W and Finlayson B A (1979), Collocation and galerkin finite element methods for viscoelastic fluid flow—I. Description of method and problems with fixed geometry . Computers and Fluids, 7,267-283. 

Robert A. Brown is Warren K. Lewis Professor of Chemical Engineering and Provost at the Massachusetts Institute of Technology. He received his B.S. (1973) and M.S. (1975) from the University of Texas, Austin, and his Ph.D. from the University of Minnesota in 1979. His research area is chemical engineering with specialization in fluid mechanics and transport phenomena, crystal growth from the melt, microdefect formation in semiconductors and viscoelastic fluids, bifurcation theory applied to transitions in flow problems, and finite element methods for nonlinear transport problems. He is a member of the National Academy of Engineering, the National Academy of Sciences, and the American Academy of Arts and Sciences. 

M. Kawahara and N. Takeuchi. Mixed finite element method for analysis of viscoelastic fluid flow. Comput. Fluids., 5 33, 1977. 

C. Braudo, A. Fortin, T. Coupez, Y. Demay, B. Vergnes, and J. F. Agassant, A Finite Element Method for Computing the Flow of Multi-mode Viscoelastic Fluids Comparison with Experiments, J. Non-Newt. Fluid Meek, 75, 1 (1998)... 

IIOJ. Baranger and D. Sandri, Finite element method for the approximation of viscoelastic fluid flow with a differential constitutive law. First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025. 

J. Baranger and S. Wardi, Numerical analysis of a finite element method for a transient viscoelastic flow, Comput. Meth. Appl. Mech. Engrg., 125 (1995) 171-185. 

With the recent advance in computational technology, efforts have also been devoted to numerical simulations of drag reduction, such as for viscoelastic polymers via constitutive equations and finite element methods [Dimitropoulos et al., 1998 Fullerton and McComb, 1999 Mitsoulis, 1999 Beris et al., 2000 Yu and Kawaguchi, 2004] and for DR flow with surfactant additives via second-order finite-difference direct numerical simulation (DNS) studies [Yu and Kawaguchi, 2003, 2006]. 

Recently, significant progress has been made in the development of numerieal algorithms for the stable and accurate solution of viseoelastio flow problems, whieh exits in processes like electrospinning process. A limitation is made to mixed finite element methods to solve viscoelastic flows nsing constitutive equations of the differential type [31]. 

To help understand and quantitatively evaluate the secondary movement shown above, Debbaut et al. [75, 77] augmented this experimental work with a three-dimensional flow simulation that incorporated viscoelastic effects. The finite element method, using a 4-mode Giesekus model as the viscoelastic constitutive equation, was used for the simulation. The polymer used for the experiment and simulation was a low-density polyethylene. Figures 12.20 and 12.21 show the experimental observations and the numerical predictions of the deformations of the interface for the rectangular straight channel [78], and for the teardrop channel [75], respectively. 

Housiadas KD, Tanner RI (2009) On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid. J Non-Newtonian Huid Mech 2009 162 88-92 Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29 329-349 Huilgol RR (2006) On the derivation of the symmetric and asymmetric Hele-Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function. J Non-Newtonian Fluid Mech 138 209-213... 

The present study attempted to numerically predict residual stress and birefringence in injection molded PC specimens with different thickness, 2.0mm and 6.5mm. Numerical simulations have been done based on a viscoelastic fluid model and commercial software MOLDFLOW by three dimensional finite element methods. The former is used to compute flow-induced residual stress, while the latter for combined residual stresses, including thermal-induced and flow-induced stresses. Effects of processing conditions on the residual are considered by the numerical simulations. As for 2.0mm PC injection molded parts, the predicted residual stresses of viscoelastic model show quite precise in accordance with experimental results. But for 6.5mm PC specimen, Moldflow simulated results have less error. 

Understanding the free surface flow of viscoelastic fluids in micro-channels is important for the design and optimization of micro-injection molding processes. In this paper, flow visualization of a non-Newtonian polyacrylamide (PA) aqueous solution in a transparent polymethylmethacrylate (PMMA) channel with microfeatures was carried out to study the flow dynamics in micro-injection molding. The transient flow near the flow front and vortex formation in microfeatures were observed. Simulations based on the control volume finite element method (CVFEM) and the volume of fluid (VOF) technique were carried out to investigate the velocity field, pressure, and shear stress distributions. The mesoscopic CONNFFESSIT (Calculation of Non-Newtonian How Finite Elements and Stochastic Simulation Technique) method was also used to calculate the normal stress difference, the orientation of the polymer molecules and the vortex formation at steady state. 

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