We study the motion of a fluid layer confined by a horizontally oriented, axially rotating, circular cylinder. This physical system facilitates a simple framework for investigating the dynamics of a viscous flow with a fluid-fluid interface or that of a freesurface that makes a trijunction with a solid, all in the presence of gravity. The periodicity of the boundary conditions simplifies the analysis. Surface tension is included, and disjoining pressure is applied to avoid a contact line. Two simple limiting cases exist for zero and for infinite rotation speed, and represent that of a fluid pool on the cylinder bottom and that of a uniform film experiencing solid-body rotation, respectively. Two dimensionless quantities are introduced and used in perturbation analyses: Г, the ratio of average film thickness to cylinder radius that represents the fullness of the cylinder; and Г, the ratio of the Reynolds number to Froude number that therefore represents the ratio of gravitational to viscous forces. Three sets of approximations and their analytic/numeric solutions are presented: steady and unsteady lubrication approximations through three orders in ä; steady high-speed flow approximations through two orders of small Г; and steady creeping-flow approximations in the limit of large Г. Also a draining, thin film for zero rotation speed will be presented.
Lubrication theory is used with perturbation expansion of dependent variables in
terms of δ to study a thin film for steady
and unsteady flow. Г is assumed order one,
and the expansion is conducted and solved through second order (i.e. zeroth,
first, and second orders) for the first time. The lowest-order, thin-film
thickness profile is evaluated numerically for varying
Г, and the results for the steady solution
agree with previously published results. Evaluating the expressions at the next
order in Г, some differences are apparent.
In figure 1 we present the results for the time invariant case and first order.
Although the profile steepens (and exhibits an asymmetric dimple due to surface
tension for Г=0.47) in the vicinity of the
maximum thickness with increasing Г, the
position of the maximum remains at 0=90o, in contrast with
experiments. At the same order, with the initial condition of a uniform film, we
present the temporal evolution; however, as can be seen in figure 2 for
Г=0.47, the unsteady profile exhibiting a
dimple evolves until a singularity occurs, and the solution becomes
indeterminate—and never
achieves a resemblance to the steady solution. Prior to this, the maximum
thickness position can be seen to differ significantly from the steady case, and
is around 30o gradually shifting in time toward 45o. The
last surface is presented for dimensionless time 2.25 (scaled by rotation rate).
Solutions for increasing rotation rate (i.e. decreasing
Г) exhibit smaller maximum thicknesses with
position shifting from 30o toward 90o (not shown). Results
with and without surface tension display interesting differences and will be
presented through second order.
We next abandon the thin-film assumption, include inertial effects, and
investigate the steady system for Г« 1, a
cylinder rotating at high speed (or equivalently in reduced gravity or with
increased viscosity). This case represents the departure from solid-body
rotation. We solve through order Г by the
method of Frobenius. Twelve coefficients are determined to complete the series
solution, and the results are presented in figure 3. As
Г is increased, initially the position of
maximum thickness tends toward the cylinder bottom consistent with physical
experiments, goes through a uniform thickness solution, and as
Г is increased further proceeds to a
solution with maximum thickness around 270o.
Our final investigation concerns slow rotation speed (equivalently large
gravitational to viscous force), or flow where Г
is large. In this case we define a small parameter, y
Г-1. Physically, a small pool of fluid is
located symmetrically along the bottom of the circular cylinder in the limit of
vanishing Г. Solutions are obtained through
the first two orders of y.
Liu, Z., Schultz, W., Perlin, M., Fluid Flow in a Rotating Circular Cylinder, Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 372-392, August, 9, 2000.