Modelling of Viscoelastic Flow

The basic procedure for the derivation of a least squares finite element scheme is described in Chapter 2, Section 2.4. Using this procedure the working equations of the least-squares finite element scheme for an incompressible flow are derived as follows  

Field unknowns in the governing flow equations are substituted using finite element approximations in the usual manner to form a set of residual statements. These statements are used to formulate a functional as 

FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

In the following section representative examples of the development of finite element schemes for most commonly used differential and integral viscoelastic models are described. 

---

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. 

L.E. Fraenkel, On a linear partly hyperbolic model of viscoelastic flow past a plate, Proc. Roy. Soc. Edinburgh, 114 A (1990) 299-354. 

As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hypeiholic) of the constitutive equation, to the propagation of "stress singularities" and to the so-called "High Weissenberg Number" problem. 

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

The complexity of viscoelastic flows requires a multidisciplinary approach including modelling, computational and mathematical aspects. In this chapter we will restrict ourselves to the latter and briefly review the state of the art on the most basic mathematical questions that can be raised on differential models of viscoelastic fluids. We want to emphasize the intimate connections that exist between the theoretical issues discussed here and the modelling of complex polymer flows (see Part III) and their numerical simulations (see Chapter II.3). 

The important problem of stability of viscoelastic flows is far more involved than for Newtonian flows, and many problems still remain open. However, it is clearly established that the difficulty lies in the relationship between various mathematical notions of stability. Some results have been obtained in this direction for restricted classes of flows and/or of models. Moreover several important studies of spectral stability have been performed. 

David and Augsburger (63) studied the decay of compressional forces for a variety of excipients, compressed with flat-faced punches on a Stokes rotary press. They found that initial compressive force could be subject to a fairly rapid decay and that this rate was dependent on the deformation behavior of the excipient for the materials studied, they found that maximum loss in compression force was for compressible starch and MCC, which was followed by compressible sugar and DCP. This was attributed to differences in the extent of plastic flow. The decay curves were analyzed using the Maxwell model of viscoelastic behavior. Maxwell model implies first order decay of compression force. 

Lee, H.S., "Turbulence Measurements and Modeling of Viscoelastic Fluid Flow in a Rectangular Open Channel", Ph.D. Thesis, Mechanical Engineering Dept., State University of New York, Stony Brook, N.Y. (1982)... 

Gotsis, A. D. 1987. Study of the Numerical Simulation of Viscoelastic Flow Effect of the Rheological Model and the Mesh (Ph.D. Thesis, Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA.)... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

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. 

Keunings, R., 1989. Simulation of viscoelastic fluid flow. Tn Tucker, C. L. HI (ed.), Computer Modeling for Polymer Proces.sing, Chapter 9, Hanser Publishers, Munich, pp. 403-469. 

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. 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

---