Steady simple shear flow, constitutive

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

The factor of 2 arises naturally because B = 2D, as we saw in Example 2.2.4, and also because viscosity is normally defined by the shear relation given in eq. 2.1.1. We can see this by applying the Newtonian constitutive relation to the steady simple shear flow of Example 2.2.2. Substituting eq. 2.2.21 for 2D into 2.3.2 gives... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Equations (4.8)-(4.10) have been solved in simple steady state shear flow using Mathematica software (Leonov and Chen 2010). The stress components are expressed as function of shear rate y with the values of constitutive parameters 00, a,p, r, r2,Xe, and t o. Here Oq and t]o represent relaxation time and zero shear viscosity respectively. The other parameters XgandXv represent the tumbling for elasticity and viscosity. Rest of the characteristic parameters a,p,ri,r2 represent anisotropic properties of liquid crystal polymers. Among the eight parameters, only relaxation time and zero shear viscosity are determined from experimental data. The other six parameters can be obtained from cinve fitting data using the Mathematica software. 

Note that since m(s) and a( i, 2) are functions only of time y, then t]q, y3, and v are constants. A material that can be represented by the constitutive equation given in Eq. (3.76) is called a Coleman-Noll second-order fluid (Coleman and Markovitz 1964 Truesdell and Noll 1965). For steady-state simple shear flow, Eq. (3.76) yields... 

In the preceding sections, we have presented the material functions derived from various constitutive equations for steady-state simple shear flow. During the past three decades, numerous research groups have reported on measurements of the steady-state shear flow properties of flexible polymer solutions and melts. There are too many papers to cite them all here. The monographs by Bird et al. (1987) and Larson (1988) have presented many experimental results for steady-state shear flow of polymer solutions and melt. In this section we present some experimental results merely to show the shape of the material functions for steady-state shear flow of linear, flexible viscoelastic molten polymers and, also, the materials functions for steady-state shear flow predicted from some of the constitutive equations presented in the preceding sections. 

The primary purposes of this chapter were first to introduce some representative phenomenological constitutive equations used to describe the viscoelasticity of flexible homogeneous polymeric liquids, and then to show how such constitutive equations may be used to describe relatively simple flow problems (e.g., steady-state shear flow and steady-state elongational flow). For such purposes, in this chapter we have presented only relatively simple constitutive equations. Owing to the space limitations here we have not presented other more complicated constitutive equations, which have been dealt with in the monographs by Bird et al. (1987) and Larson (1988). 

The comparison of the predictions given by these constitutive equations in steady-state shear and extensional simple flows are summarized in Figs. 1 and 2. 

This equation has the correct limiting behavior it reduces to an equation for a simple Newtonian fluid when dx/dt approaches to 0 for steady shear flow. When the stress changes rapidly with time, and X is negligible compared with dx/dt, it reduces to the constitutive equation of a Hookian solid. 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

This constitutive equation is referred to as the upper converted Maxwell (UCM) model. In Example 3.1 this time derivative will be written out for simple shear and shear-free flows. The predictions of this model for steady shear... 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

A model consisting of the codeformational MaxweU constitutive equation coupled to a kinetic equation for breaking and re-formation of micelles is presented to reproduce most of the nonlinear viscoelastic properties of wormlike micelles. This simple model is also able to predict shear banding in steady shear and pipe flows as well as the long transients and oscillations that accompany this phenomenon. Even though the model requires six parameters, all of them can be evaluated from single and independent rheological experiments, and then they can be used to predict other flow situations. The predictions of our model are compared with experimental data for aqueous micellar solutions of cetyltrimethylammonium tosilate (CTAT). 

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