Normal Steady shear

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

Bubbles and drops tend to deform when subject to external fluid fields until normal and shear stresses balance at the fluid-fluid interface. When compared with the infinite number of shapes possible for solid particles, fluid particles at steady state are severely limited in the number of possibilities since such features as sharp corners or protuberances are precluded by the interfacial force balance. 

The total drag on the sphere may be obtained, as in steady flow, by integrating the normal and shear stresses over the surface. In terms of the instantaneous velocity U the result is (L4) ... 

The Weissenbeig Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characterization of viscoelastic materials. Its capabilities include measurement of steady-state rotational shear within a viscosity range of 10-1 —13 mPa-s at shear rates of 10-4 — 104 s-1, of normal forces (elastic effect) exhibited by the material being sheared, and of an oscillatory shear range of 5 x 10-6 to 50 Hz, from which the elastic modulus and dynamic viscosity can be determined. A unique feature is its ability to superimpose oscillation on steady shear to provide dynamic measurements under flow conditions all measurements can be made over a wide range of temperatures (—50 to 400°C). 

In steady shear flow, the viscosity is independent of shear rate — for all y. This property alone represents a serious qualitative failure of the conventional bead-spring models. The normal stress functions are (108) ... 

From the additional group of results of Lodge s theory, only the expressions for the normal stresses in steady shear flow will be used in the following. These expressions read ... 

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. 

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...

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. 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

An important conclusion is that it is clear that Lodge s constitutive equation is not able to describe non-Newtonian behaviour in steady shear, because both the viscosity and the first normal stress coefficient appear to be no functions of the shear rate. 

It is well known in polymer rheology that a torsional parallel-plate flow cell develops certain secondary flow and meniscus distortion beyond some stress level [ 14]. For viscoelastic melts, this can happen at an embarrassingly low stress. The critical condition for these instabilities has not been clearly identified in terms of the shear stress, normal stress, and surface tension. It is very plausible that the boundary discontinuity and stress intensification discussed in Sect. 4 is the primary source for the meniscus instability. On the other hand, it is well documented that the first indication of an unstable flow in parallel plates is not a visually observable meniscus distortion or edge fracture, but a measurable decay of stress at a given shear rate [40]. The decay of the average stress can occur in both steady shear and frequency-dependent dynamic shear. 

Figure 1. Steady shear and extensional viscosities and normal stress difference versus strain rate for LLDPE at T=160°C.

One of the eharacteristics of viscoelastic foods is that when a shear rate is suddenly imposed on them, the shear stress displays an overshoot and eventually reaches a steady state value. Figure 3-43 illustrates stress overshoot data as a function of shear rate (Kokini and Dickie, 1981 Dickie and Kokini, 1982). The data can be modeled by means of equations which contain rheological parameters related to the stresses (normal and shear) and shear rate. One such equation is that of Leider and Bird (1974) ... 

In addition to relationships between apparent viscosity and dynamic or complex viscosity, those between first normal stress coefficient versus dynamic viscosity or apparent viscosity are also of interest to predict one from another for food processing or product development applications. Such relationships were derived for the quasilinear co-rotational Goddard-Miller model (Abdel-Khalik et al., 1974 Bird et al., 1974, 1977). It should be noted that a first normal stress coefficient in a flow field, V i(y), and another in an oscillatory field, fri(ct>), can be determined. Further, as discussed below, (y) can be estimated from steady shear and dynamic rheological data. 

Figure 1.10 Transient shear stress a and first normal stress difference A i after start-up of steady shearing for a low-density polyethylene melt, Melt I, at a shear rate y = 1 sec . (From Laun 1978, reprinted with permission from Steinkopff Publishers.)...

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

The time-dependent growth of Nx after start-up of steady shearing for a polyethylene melt is shown in Fig. 1-10. Note that at steady state the first normal stress difference is larger than the shear stress at this particular shear rate. The normal stress differences usually are more shear-rate-dependent than the shear stress. In fact, if the isotropic liquid belongs to a fairly general class known as viscoelastic simple fluids with fading memory (Coleman and Noll 1961), then at low shear rates the normal stress differences depend quadratically... 

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