Non-Newtonian flow problems

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

The described application of Green s theorem which results in the derivation of the weak statements is an essential step in the formulation of robu.st U-V-P and penalty schemes for non-Newtonian flow problems. 

Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution. Calculations that assume a parabolic profile will be conservative in the sense that they will predict a lower conversion than would be predicted for the actual profile. The changes in velocity profile due to variations in temperature and composition are normally much more important than the fairly subtle effects due to non-Newtonian behavior. 

In Yeow YL, Uhlherr PHT (ed) Proc Fifth Nad Conf Soc Rheol, Melbourne, pp 141-144 Zheng R, Phan-Thien N, Tanner RI, Coleman CJ (1992) A boundary element/particular solution approach ftn non-Newtonian flow problems. In Moldenaers P, Keunings R (eds) Theoretical and applied rheology. Proceedings of the XI inteniational congress on rheology, Brussels, Belgium, p 311... 

For non-circular shapes, the equations of motion may result in nonlinear partial differential equations, which are difficult to solve analytically. Therefore, approximate methods such as the variational method (Kantorovich and Krylov, 1958) are generally used for solving non-Newtonian flow problems. Schechter (1961) used the application of the variational method to solve the non-linear partial differential equations of pressure drop and flow rate of the polymer for non-circular shapes such as a rectangle or square. Moreover, Mitsuishi and Aoyagi (1969 1973) used similar methods for other non-circular shapes such as an isosceles triangle. The results were based on the Sutterby model (1966), which incorporates a viscosity function based on the rheological constants. Flow curves with pressure drop and flow rate for both circular and non-circular shapes were generated and the results were compared with the power law model. 

B.7 Equivalent Newtonian Viscosity. It has been suggested by Broyer and co-workers (1975) that the solutions to non-Newtonian flow problems can be obtained by using the Newtonian solution with fx replaced by an equivalent Newtonian viscosity, fc. For isothermal flow between parallel plates carry out the following ... 

Problems of forced convection diffusion in non-Newtonian flow have to this author s knowledge not yet been attacked. The equations needed for solving such problems are given in this article. The equation of motion in terms of the stress tensor [Eq. (25)] can be used to describe non-Newtonian flow provided that a suitable form for the stress tensor is used examples of two non-Newtonian stress tensors are given in Eqs. (28a) and (28b). 

The final chapter on applications of optical rheometric methods brings together examples of their use to solve a wide variety of physical problems. A partial list includes the use of birefringence to measure spatially resolved stress fields in non-Newtonian flows, the isolation of component dynamics in polymer/polymer blends using spectroscopic methods, the measurement of the structure factor in systems subject to field-induced phase separation, the measurement of structure in dense colloidal dispersions, and the dynamics of liquid crystals under flow. 

The simulation of non-Newtonian fluid flow is significantly more complex in comparison with the simulation of Newtonian fluid flow due to the possible occurrence of sharp stress gradients which necessitates the use of (local) mesh refinement techniques. Also the coupling between momentum and constitutive equations makes the problem extremely stiff and often time-dependent calculations have to be performed due to memory effects and also due to the possible occurrence of bifurcations. These requirements explain the existence of specialized (often FEM-based) CFD packages for non-Newtonian flow such as POLYFLOW. 

Examples of CFD applications involving non-Newtonian flow can be found, for example, in papers by Keunings and Crochet (1984), Van Kemenade and Deville (1994), and Mompean and Deville (1996). Van Kemenade and Deville used a spectral FEM and experienced severe numerical problems at high values of the Weissenberg number. In a later study Mompean and Deville (1996) could surmount these numerical difficulties by using a semi-implicit finite volume method. 

The time-temperature superposition principle can be incorporated into isothermal constitutive equations (including the generalized Newtonian fluid model) to solve non-isothermal flow problems. If we define... 

Ottinger (1996) combined the Brownian dynamics simulation technique with finite elements to solve polymer flow problems. The approach has been introduced as the CONNFFESSIT (Calculation of Non-Newtonian Flow Finite Element and Stochastic Simulation Techniques) approach. Hulsen et al. (1997) extended the approach to the so-called Brownian configuration field (BCF) method, which treats the stochastic equation as the stochastic field equation, and hence avoids the difficulties associated with individual molecule tracking. If the BCF is applied to fiber suspension flows, the vectors p and q will be functions of space and time. The discrete equation for the time evolution is... 

The applicability of the CONNFFESSIT (Calculation Of Non-Newtonian Flows Finite Elements and Stochastic Simulation Technique) in its present form is limited to the solution of fluid mechanical problems of incompressible fluids under isothermal conditions. The method is based on a combination of traditional continuum-mechanical schemes for the integration of the mass- and momentum-conservation equations and a simulational approach to the constitutive equation. 

The viscoelastic fluid flow normal to cylinders is still very much at the forefront of outstanding problems in the numerical simulation of non-Newtonian flows [e.g., see Baaijens et al. (1994)]. 

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