Viscoelastic flow

Isoparametric mapping removes tlie geometrical inflexibility of rectangular elements and therefore they can be used to solve many types of practical problems. For example, the isoparametric C continuous rectangular Hermite element provides useful discretizations in the solution of viscoelastic flow problems. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. To give a brief explanation of this problem we note that in a non-isothennal flow field the evolution of stresses will be affected by the... 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Marchal, J. M. and Crochet, M.J., 1987. A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian Pluid Mech. 20, 77-114. 

Petera, J. and Nassehi, V., 1996. Finite element modelling of free surface viscoelastic flows with particular application to rubber mixing. Int. J. Numer. Methods Fluids 23, 1117-1132. 

Another example of time-dependent adhesion has been reported by Rimai et al. [110], In this study, adhesion-induced stresses were found to cause an alternating block polyester/PDMS copolymer was found to flow over stacks of particles during a 2-week-long interval, suggestive of the occurrence of viscoelastic flow. 

Johnson [111] addressed the problem of viscoelastic flow by attempting to modify the JKR equation. In his approach, he postulated a creep compliance function... 

B. H. Zimm. Dynamics of polymer molecules in dilute solutions Viscoelasticity, flow birefringence and dielectric loss. J Chem Phys 24 269-219, 1956. 

Polyflow, a finite element program for calculating viscous and viscoelastic flows, Polyflow SA, Place de l Universite 16, B-1348 Louvain-la-Neuve, Belgium (1988)... 

U. Kei.kar. J. V.. Mashelkar, R. a., arid DLBRECHT, J. Trans. Inst. Chem. Eng. 50 (I972i 343. On ihe rotational viscoelastic flows around simple bodies and agitators. 

Zimm, BH, Dynamics of Polymer Molecules in Dilute Solution Viscoelasticity, Flow Birefringence and Dielectric Loss, Journal of Chemical Physics 24, 269, 1956. 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

Molecular Constitutive Equations for Polymer Melts in Viscoelastic Flow Solvers... 

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