The Viscous Region

The resistance to deformation, i.e., the fluid s viscosity, is defined in terms of the shear stress, x  

FIGURE 13.2 Schematic illustration of a simple shear experiment. 

Various instruments are available to measure the viscosity of polymer melts and solutions, and more generally their rheological behavior, which include capillary and rotational viscometers. The former can be used to measure parameters such as shear viscosity, melt fracture, and extensional viscosity, which are important for many polymer processes. The latter type of device can be used in either steady or oscillatory mode, thus providing a measure of the viscosity as well as viscoelasticity (G , G, and tan 5) as a function of frequency and temperature. 

These different geometries present advantages and disadvantages. For example, the cone-and-plate arrangement offers a constant shear rate across the sample y, whereas in the parallel plate geometry, though it is easy to set and vary the gap, the shear rate is zero at the center and a maximum at the rim. 

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Often, a pilot plant will operate in the viscous region while the commercial unit will operate in the transition region, or alternatively, the pilot plant may be in the transition region and the commercial unit in the turbulent region. Some experience is required to estimate the difference in performance to be expected upon scale-up. 

High-Viscosity Systems A axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in... 

At high the power number, P , stays reasonably constant, thus, viscosity has little effect on the power requirements. Wdien moving to lower through the laminar region into the viscous region, the viscosity effect increases. In the laminar range [29]... 

For the viscous region, Carpenter s results are reasonably well correlated by the equation ... 

Fig. 6.4 Streamlines for two axisymmetric Hiemenz stagnation flow situations having different outer velocity gradients, one at a = 1 s 1 and the other at a = 5 s—. Both cases are for air flow at atmospheric pressure and T = 300 K. The streamlines are plotted to an axial height of 3 cm and a radius of 10 cm. However, the solution itself has infinite radial extent in both the axial and radial directions. In both cases the streamlines are separated by 2jt A l = 2.0 x 10-5 kg/s. The shape of the scaled radial velocities V = v/r is plotted on the right of the figures. The maximum value of the scaled radial velocity is Vmax = a/2. Even though streamlines show curvature everywhere, the viscous region is confined to the boundary layer defined by the region of V variation. Outside of this region the flow behaves as though it is inviscid.

In the viscous region immediately adjacent to a wall, the calculations are improved if I is reduced, with... 

Hanjalic et al. (HI) have used a dissipation-model equation to study a variety of boundary-layer flows in an extended MTEN model. Their formulation is purported to work in the viscous region, eliminating the need for wall-solution patching [see Eq. (62)]. 

Equations (40) and (41) do not hold in the viscous region near the wall. One must either modifj these equations to include viscous effects, or else use special solutions, as discussed in Section II, in this region. Experiments reveal a nearly uniform distribution of q in the wall region, except very close to the wall yu fv < 20). Aloreover, the value of q/u seems to be nearly universal, with... 

High-Viscosity Systems All axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in the transition and low-viscosity system is largely a measure of fluid motion throughout the tank. For close-clearance impellers, the anchor and helical impellers provide blending by having an effective action at the tank wall, which is particularly suitable for pseudoplastic fluids. 

The equations of fluid motion inside and outside a circulating drop under viscous flow regime were solved by Hadamard (H2) and Rybczynski (R9) in 1911, and are quoted in hydrodynamics textbooks (L2). The complete derivation is also repeated by Levich (L8). Although Hadamard s stream functions are strictly applicable to the viscous region only, visual observations (GIO, S18) indicated that the function approximates actual flows (E2, H3). Hadamard s stream function inside the drop, as given in polar coordinates with the origin at the center of the drop (K5), is... 

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