Planar Stagnation Flows

The flow at high Rep approaches the planar, finite-gap, stagnation flow between parallel plates. In this case, the injection velocity V dominates over the initial velocity U that enters the channel. The system of equations developed here are essentially the same as those for finite-gap planar stagnation flow. Indeed, it is only the relationship between K and the axial pressure gradient that distinguishes the two flows. 

The discussion in this chapter has been dominated by axisymmetric flow. However, there is analogous behavior for planar stagnation flow in two-dimensional cartesian coordinates. In fact Hiemenz s original work was for planar stagnation flow in a semi-infinite region. The planar flow illustrated in Fig. 6.18 is for a finite domain. 

Following the general approach discussed in Section 6.2 for axisymmetric flows, derive the general equations for planar stagnation flow. The planar equations are summarized, but not derived, in Section 6.9. Discuss the differences and similarities between the two stagnation flows. 

Steady extensional deformations can be created by impinging two liquid streams, creating a stagnation flow. Figure 7.7.1 illustrates both axisymmetric and planar stagnation flow. These flows are not homogeneous. A material element near the central part of the flow... 

The four-roller experiment of Taylor (1934) produces planar stagnation flow only near the center the roller radius is chosen to be a radius of curvature of one of the hyperbolic streamlines of steady planar extension. 

Although a nearly planar premixed turbulent flame is maintained in the stagnation-flow burning configuration (3), the divergence of flow-field streamlines results in mean strain rates which also modify the turbulent flame structure and burning rates. 

Fig. 6.18 Comparison of general flow patterns for planar and axisymmetric, finite-gap, stagnation flow.

Winter et al. [119, 120] studied phase changes in the system PS/PVME under planar extensional as well as shear flow. They developed a lubrieated stagnation flow by the impingement of two rectangular jets in a specially built die having hyperbolic walls. Change of the turbidity of the blend was monitored at constant temperature. It has been found that flow-induced miscibility occurred after a duration of the order of seconds or minutes [119]. Miscibility was observed not only in planar extensional flow, but also near the die walls where the blend was subjected to shear flow. Moreover, the period of time required to induce miscibility was found to decrease with increasing flow rate. The LCST of PS/PVME was elevated in extensional flow as much as 12 K [120]. The shift depends on the extension rate, the strain and the blend composition. Flow-induced miscibility has been also found under shear flow between parallel plates when the samples were sheared near the equilibrium coexistence temperature. However, the effect of shear on polymer miscibility turned out to be less dramatic than the effect of extensional flow. The cloud point increased by 6 K at a shear rate of 2.9 s. ... 

From Eq. (8.3.24) for the Stokes stream function near a cylinder, the undisturbed velocity field is easily shown to be resolvable into two flows. One is a planar stagnation-type flow shown in Fig. 8.4.IB that is associated with the velocity component at infinity along the line of centers of the cylinder and particle the other is a shear flow normal to the line of centers shown in Fig. 8.4.IG. The respective expressions valid for the cylinder radius a > x + where a is the particle radius, are... 

Fig. 5 Schematic diagram of planar elongational flow using cross-slot. Streamlines are marked red and the stagnation point is marked as X. The velocity in the z direction is zero...

Fiber spinning (Figure 7.5.2) approximates one end of the axisymmetric stagnation flow. The tubeless siphon is a little closer. But neither has a stagnation point. Ideally we want to confine a fluid to flow within the stream surfaces indicated in Figure 7.7.1. For planar stagnation these surfaces are defined (Winter et al., 1979) by the relation... 

Planar viscosities determined by measuring birefringence in unlubricated stagnation flow in both directions. The same polystyrene melt as Figure 7.4.4 was used in the same die. From Macosko et al. (1980). 

B.4 Lineal Stretch Efficiency of Planar Elongational Flow. Prove that the time average lineal stretch efficiency, z, of a two-dimensional stagnation flow (planar elongational flow Fig. 6.35) is... 

FIGURE 6.35 Streamlines and velocity profile of a two-dimensional stagnation (plane hyperbolic planar extensional) flow. 

This flow field can be maintained in a steady state, at least in the Eulerian sense, either by use of a four-roll mill [18] as in Figure 2.8.4(a) or by means of opposed jet flow as in Figure 2.8.4(b). However, it is important to note that the flow is still transient in the Lagrangian sense. That is, pure planar extension is confined to the central stagnation... 

Bentley and Leal have measured droplet shapes and critical conditions for droplet breakup over a wide range of capillary numbers, viscosity ratios, and flow types. The flow type is conveniently controlled in an apparatus called a four-roll mill, in which a velocity field is generated by the rotation of four rollers in a container of liquid (see Fig. 1-15). By varying the rotation rate of one pair of rollers relative to that of a second pair, velocity fields ranging from planar extension to nearly simple shear can be produced near the stagnation point. 

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