Simple shear flow steady solutions

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

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. 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

To explore viscoelasticity in liquids, consider the simple shear flow experiment shown in Figure 33.9. Begin the experiment by applying a shear force to the solution until it reaches steady state. At that time, the total force /, that is being applied to the top plate (divided by the number of molecules) equals the sum of the viscous and elastic forces exerted by the molecule. 

If the particular product is not very non-Newtonian, a single-point measurement might be sufficient. Even then however we must be sure that the flow type is the same. For instance if our experience off-line is built up from steady-state, simple-shear flow curves, then a too-fast measurement might mean the measurement is not made under steady-state conditions. Equally if the on-hne flow has an appreciable extensional component, then problems can arise for some hquids, especially pol3nner solutions. Also if a vibrational mode is used, then it is probable that some non-hnear oscillatory function is being measured. Ah these facts could mean that we end up with no simple one-to-one correlation between on-hne and off-hne. Hence the safest way is to duphcate on-hne what is done off-hne. This is possible nowadays for most situations. 

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. 

It can be shown that for most dilute solutions there exists a simple correlation between dynamic and steady state flow characteristics (16). For most detergent solutions the magnitude of the complex viscosity 1 n I at a certain angular frequency CO coincides with the steady state viscosity n, at the corresponding shear rate "f (12, 17). 

In the steady, unidirectional flow problems considered in this section, the acceleration of a fluid element is identically equal to zero. Both the time derivative du/dt and the nonlinear inertial terms are zero so that Du/Dt = 0. This means that the equation of motion reduces locally to a simple balance between forces associated with the pressure gradient and viscous forces due to the velocity gradient. Because this simple force balance holds at every point in the fluid, it must also hold for the fluid system as a whole. To illustrate this, we use the Poiseuille flow solution. Let us consider the forces acting on a body of fluid in an arbitrary section of the tube, between z = 0, say, and a downstream point z = L, as illustrated in Fig. 3-4. At the walls of the tube, the only nonzero shear-stress component is xrz. The normal-stress components at the walls are all just equal to the pressure and produce no net contribution to the overall forces that act on the body of fluid that we consider here. The viscous shear stress at the walls is evaluated by use of (3 44),... 

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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