Shear rate streamline flow

For such flows, the range of flow rates is such that the variations in normal stresses are inconsequential relative to the shear stresses. Even when the fluid is highly elastic, the flow geometry and confinement may be such that the fluid elasticity has only a minor influence on the streamline orientations. On the other hand, shear-thinning effects upon the flow characteristics and the pressure distribution may appear to be quite important, even for low values of the shear rate. Such flow problems can be solved by considering constitutive equations that are of analogous (although, not necessarily the same) forms as those adopted for Newtonian flows as follows ... 

From Fig. 5.3 it may be seen that this exceeds the shear rate for non-laminar flow (approximately 30 s ) so that the entry to this region would need to be streamlined. Fig. 5.3 also shows that the extensional strain rate, k, in the tapered entry region should not exceed about 15 s if turbulence is to be avoided. 

In the previous sections of this chapter, the calculation of frictional losses associated with the flow of simple Newtonian fluids has been discussed. A Newtonian fluid at a given temperature and pressure has a constant viscosity /r which does not depend on the shear rate and, for streamline (laminar) flow, is equal to the ratio of the shear stress (R-,) to the shear rate (d t/dy) as shown in equation 3.4, or ... 

The streamlines of this flow are shown by Peters and Smith (12). In this case, the effective thickness of this layer appears to be about equal to the gap with the wall, indicating a pressure flow about equal to the drag flow. It can be calculated that this would increase the maximum shear rate on the fluid passing under the agitator blade by a factor of seven. 

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

The following example examines the SDF in drag flow between parallel plates. In this particular flow geometry, although the shear rate is constant throughout the mixer, a rather broad SDF results because of the existence of a broad residence time distribution. Consequently, a minor component, even if distributed at the inlet over all the entering streamlines and placed in an optimal orientation, will not be uniformly mixed in the outlet stream. 

The flow of a slurry in a CMP process has been investigated. This wafer-scale model provides the three dimensional flow field of the slurry, the spatial distribution of the local shear rate imposed on the wafer surface, and the streamline patterns which reflect the transport characteristics of the slurry. 

Where flow in the tube is streamline or turbulent, an infinitesimally thin stationary layer is found at the wall. The velocity increases from zero at this point to a maximum at the axis of the tube. The velocity profile of streamline flow is shown in Fig. 9A. The velocity gradient du/dr varies from a maximum at the wall to zero at the axis. In flow through a tube, the rate of shear is equal to the velocity gradient, and Eq. (1) dictates the same variation of shear stress. 

Using the methods of Bowen [1961] and of Dodge and Metzner [1959], construct the wall shear stress-apparent shear rate plots for turbulent flow of this material in 101.6mm and 203.2 mm diameter pipes. Also, calculate the velocity marking the end of the streamline flow. 

Attention is drawn to the fact that the values of m and n for use in turbulent region are deduced from the data in the laminar range at the values of (SVp/D) which is only the nominal shear rate at the tube wall for streamline flow, and thus this aspect of the procedure is completely empirical. Dziubinski [1995] stated that equation (4.26) reproduced the same experimental data as those referred to earlier with an average error of 15%, while equation (4.28) correlated the turbulent flow data with an error of 25%. Notwithstanding the marginal improvement over the method of Dziubinski and Chhabra [1989], it is reiterated here that both methods are of an entirely empirical nature and therefore the extrapolation beyond the range of experimental conditions must be treated with reserve. 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

---