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 ... 

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. 

The modulus sign is used because shear stresses within a fluid act in both the positive and negative senses. Gases and simple low molecular weight liquids are all Newtonian, and viscosity may be treated as constant in any flow problem unless there are significant variations of temperature or pressure. 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

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 above equation can be simplified assuming a Newtonian isothermal problem. For such a case Pawlowski reduced the above equations to a set of characteristic functions that describe the conveying properties of a single screw extruder under isothermal and creeping flow (Re < 100) assumptions. These are written as... 

In the problem we are addressing in this section, the two plates are assigned two different temperatures, To on the lower plate and T on the upper moving plate. In addition, we are assuming a Newtonian flow with an exponential temperature dependence, q = q0e a(T T° ... 

To simplify the problem, we can assume that the polymer bar moves at a constant speed Usy, and that a film of constant thickness, 5, exists between the bar and the heated plate. In addition, we assume that the polymer melt is Newtonian and that the viscosity is independent of temperature. The Newtonian assumption is justified by low rates of deformation that develop in this relatively slow flow problem. Furthermore, due to these low rates of deformation we can assume that the convective and viscous dissipation effects are negligible. 

The isoparametric element works quite well to formulate the finite element equations for flow problems, such as flows with non-Newtonian shear thinning viscosity. Due to the flexibility that exists to integrate variables throughout the elements, the method lends itself... 

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. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

The former vanishes when the velocity of the moving plate is zero, and the latter vanishes in the absence of a pressure gradient, (a) Explain on physical and mathematical grounds why the solution of the same flow problem with a non-Newtonian fluid, for example, a Power Law model fluid, no longer leads to the same type of expressions, (b) It is possible to define a superposition correction factor as follows... 

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