Steady shear-free flow

The two functions and 2 depend on the parameter b as well as on the elongation rate e in steady shear-free flows. When b = 0, the function 2 is zero, and is replaced by the symbol tj, which is called the elongational viscosity . The elongational viscosity describes the resistance to elongational flow if 8 is positive and the resistance to biaxial stretching if e is negative (the terms extensional viscosity and Trouton viscosity have also been used for ). 

Example 3.2. Predictioiis of the PTT Model in Steady Simple Shear and Steady Shear-Free Flow... 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

There are two broad classes of rheometric experiments that have been developed shear flows and shear-free flows. Within each category one can speak of steady flows and various unsteady flows the latter can include step-function experiments, sinusoidal experiments and others. We now discuss these idealized flows and the material functions that are commonly defined. For a much more... 

Because of the difficulty of attaining steady-state shear-free flows, it is thought that it may be preferable to study some of the unsteady-state flows experimentally. One can, of course, study the shear-free analogs of any of the unsteady-state experiments depicted in Figure 2 for shear flows. For example, one may define growth functions analogously to equations (5)-(7)... 

FIGURE 3.2 The deformation of (a) a unit cube of material from time ti to f2 (t2 > fi) in (b) steady simple shear flow and (c) three kinds of shear-free flow. The volume of material is preserved in all of these flows. (Reprinted by permission of the publisher from Bird et al., 1987a.)... 

Similar flow histories for shear-free flows as described for shear flows in Figure 3.3 can also be used. Here we discuss only steady and stress growth shear-free flows. For steady simple (i.e., homogeneous deformation) shear-free flows two viscosity functions, 7Ji and 1)2, are defined based on the two normal stress differences given in Eq. 3.13 ... 

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 ... 

B.7 Predictions of Fiber Orientation in the Startup of Shear-Free Flow. From Eq. 3.101 obtain the system of ordinary differential equations for determining the components of the orientation tensor. A, at the start of shear-free flow and under steady state conditions. In particular, find the equations for determining Al 1, A22, and A33. The initial orientation of the fibers is taken as random which means An = A22 = A33 = 1/3. 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

At this point it is of interest to note the physical meaning of the integral, occurring in eq. (2.2). This integral gives the free energy stored per unit of volume in steady shear flow, when the unity of shear rate is applied. 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

According to eq. (3.38) the left-hand side of eq. (3.41) can also be considered as twice the reduced stored free energy in steady shear flow. Using for this quantity the character FR and for the reduced shear stress the character ft, one obtains instead of eq. (3.41) ... 

Fig. 1 At the level of the approximation we use in this chapter, all experimental shear geometries are equivalent to a simple steady shear. We choose our system of coordinates such that the normal to the plates points along the z-axis and the plates are located at z = j. Between two parallel plates we assume a defect-free well aligned lamellar phase. The upper plate moves with the velocity in positive x direction, the lower plate moves with the same velocity in negative A direction. The y-direction points into the xz-plane. We call the plane of the plates Cry-plane) the shear plane, the x-direction the flow direction, and the y-direction the vortidty direction...

Since pressure driven viscometers employ non-homogeneous flows, they can only measure steady shear functions such as viscosity, 77(7). However, they are widely used because they are relatively inexpensive to build and simple to operate. Despite their simplicity, long capillary viscometers give the most accurate viscosity data available. Another major advantage is that the capillary rheometer has no free surfaces in the test region, unlike other types of rheometers such as the cone and plate rheometers, which we will discuss in the next section. When the strain rate dependent viscosity of polymer melts is measured, capillary rheometers may provide the only satisfactory method of obtaining such data at shear rates... 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

The concept of an apparent viscosity introduced for a Couette flow has been applied empirically to a variety of incompressible inertia free steady shear flows through the generalized relation... 

Disregarding this complication, the comparison of steady shear and DLS experiments shows that on addition of a free polymer the re-entry effect can also be observed in the flow behavior. This is highlighted in Fig. 4 by comparing... 

Of course, using the correlations (8)—(13) under circumstances different from those cited above is not allowed. This means, first of all, that the particle motion through the fluid or the fluid flow around the particle should be steady and strictly ID over a sufficiently long distance to allow for the development of boundary layer and wake toward a steady state and to permit the use of the standard drag curve. Lateral forces due an asymmetrical flow field should be absent. The development of both the boundary layer around and the wake behind a particle is affected by local accelerations or decelerations of the immersed particle and/or of the embedding fluid, by a steady shear field in the surrounding fluid, by the dynamics of free-stream turbulence, by particle rotations (either externally and deliberately imposed, or as the result of shear flow), by adjacent walls of a container, and by the presence... 

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