Radial flow between parallel disks

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following  

From the continuity equation, it is clear that the radial velocity must scale as 1 /r, 

Substituting the functional form of the velocity (Eq. 5.54) into the radial momentum equation produces a second-order nonlinear equation as 

Even though p(r) is function of r alone and /(z) is a function of z alone, this equation is not separable. 

Pressure may be eliminated from Eq. 5.55 by differentiating with respect to z, yielding 

---

Fig. 5.9 Illustrative geometry for the radial flow between parallel disks.

In Section 5.5 it was shown that for radial flow between parallel disks that the convective term could not be retained. It appears here that for 0 0 the convective term can be retained, since the cross-stream momentum equation provides a relationship between / and the pressure gradient. In the limiting case of 0 = n/2 the convective terms vanishes in any case. 

As a partial check on the derivations in the conical coordinates, it should be possible to recover two, easily identified, special cases—the radial flow between parallel disks and the axial Poiseuille flow in an cylindrical annular gap. The parallel-disk flow (Section 5.5) is the case where 0 = 0, with x taking the role of r and y taking the role of z. In this case, h = De/2 + x = r. The momentum equations become... 

Problem 4-8. Pressure-Driven Radial Flow Between Parallel Disks. The flow of a viscous fluid radially outward between two circular disks is a useful model problem for certain types of polymer mold-filling operations, as well as lubrication systems. We consider such a system, as sketched here ... 

Fig. 5.11 The T( 2) function for radially inward flow between parallel disks.

Radial Pressure Flow between Parallel Disks Solve the problem of radial pressure flow between two parallel disks. The flow is created by a pressure drop (P r 0 P r R). Disregard the entrance region, where the fluid enters from a small hole at the center of the top disk. 

Problem 3-37. Taylor Dispersion with Streamwise Variations of Mean Velocity. We consider steady, pressure-driven axisymmetric flow in the radial direction between two parallel disks that are separated by a distance 2h. We assume that the volumetric flow rate in the radial direction is fixed at a value Q and that the Reynolds number is small enough that the Navier-Stokes equations are dominated by the viscous and pressure-gradient terms. Finally, the flow is ID in the sense that u = [nr(r, z), 0, 0]. In this problem, we consider flow-induced dispersion of a dilute solute. We follow the precedent set by the classical analysis of Taylor for axial dispersion of a solute in flow through a tube by considering only the concentration profile averaged across the gap, ( ) = f h dz. 

Example 2.4. Radial Flow of a Newtonian Fluid Between Two Parallel Disks... 

A lubricant flows radially between two parallel circular disks from a radius ri to another f2 because of AP. 

Figure 7.3.14 illustrates the trajectory of a particle in between two consecutive disks (dashed lines). Define two coordinate axes the positive z-axis parallel to the disks, and in the direction of main liquid flow toward the main vertical device axis, and the positive r-axis radially... 

---