On the flow between a rotating and a stationary disk

Abstract

The analysis and experiments in this paper are restricted to the flow between two coaxial, infinite disks, one rotating and one stationary. The results of numerical calculations show that many solutions can exist for a given Reynolds number Ωl2/v (Ω is the angular velocity of the rotating disk and I is the spacing between the two disks). Out of a greater number of possible solutions, three solution branches have been identified; the branches correspond to one-, two- and three-flow cells in the meridional plane.The one-cell branch has been accorded detailed treatment. Within this branch there are two subbranches. The first, now well documented in the literature, includes solutions from zero to infinite Reynolds number. The latter limiting case is characterized by an inward-flowing boundary layer on the stationary disk and an outward-flowing boundary layer on the rotating disk. In between is a core flow rotating with a constant angular velocity. The second sub-branch of the single-cell flows, apparently unknown heretofore, begins with an infinite Reynolds number, decreases to a minimum and then increases to an infinite Reynolds number again. The first infinite Reynolds number limit again corresponds to two boundary-layer flows separated by a core flow with constantangular velocity opposite in direction to the angular velocity of the rotating disk. The second limiting case of infinite Reynolds number is the free-disk solution of von Kármán (1921). Asymptotic solutions have been obtained which more fully describe the nature of this flow as the Reynolds number increases.The second part of the paper presents experimental measurements corresponding to the Reynolds number range 0–100. Profiles were measured with a hot-wire anemometer. The measurements are in agreement with the first, one-cell branch of solutions. A semi-quantitative evaluation of edge effects is obtained.

Turbulence characteristics of the Bödewadt layer in a large enclosed rotor-stator system

“…Differences in turbulence characteristics were observed between the rotor and stator sides and attributed to the effects of the radial convective transport of turbulence. In stability experiments over a free rotating disk, Wilkinson and Malik [10] found the transition to turbulent flow to occur at the range 2.9×10 5 ≤ Re r ≤ 3.1×10 5 . In his review of the laminar-to-turbulent transition, Kobayashi [11] reported that the flow over a rotating disk remains laminar for values of the local Reynolds number Re r ≤ 4.5 × 10 4 and is fully turbulent for Re r greater than about 3.9 × 10 5 .…”

Note on porous rotating disk flow

“…The second class is due to the finite domain length, where the flow near ζ c is of the form W ∼ −(ζ − ζ c ) 2 /ζ c 2 and V = 0. For the impermeable case, solutions of this type were discussed by Mellor et al [7] in their study of two-disk flows where the second disk is stationary and located at ζ c . We do not report details here, other than to note that our computations show that this type of two-disk flow does in fact exist for a range of suction and injection rates through the rotating disk, and can be obtained on a range of domain lengths by using initial conditions that differ only slightly from those needed for the single disk solutions.…”

Surface sliding friction of negatively charged polyelectrolyte gels

“…(10) indicates that the shear flow can penetrate into the gel with a thickness of Next, the case of two gels rotating with relative angular velocity 0 ω is considered. For a liquid existing between two infinitely large coaxial plates rotating, its spatial velocity profile is very complicated [32,33]. But for the present case, the Reynolds number of motion,…”

“…, so only the angular velocity component is predominant [32]. The shear stress on the gel surfaces at a distance r from the rotation axis is…”