Cats2D Multiphysics > Research Topics > Rim coater flow regimes

*All unpublished results shown here are *Copyright © 2016–2019 Andrew Yeckel, all rights reserved

## Rim coater flow regimes

Here is a fun time waster suggested by Goodwin. These are steady state simulations of a fixed volume of liquid inside a spinning cylinder, sometimes called the rim coater. The volume fraction is 0.19, rotation is counter-clockwise, and gravity points downward. Properties are approximately those of water: viscosity is 10^{-3} Pa-s, density is 10^{3} kg/m^{3}, and surface tension is 0.1 N/m. I really like round numbers, so I've set gravitational acceleration to 10 m/s^{2}. Cylinders of 1 and 2 mm radius, spinning at 10 cm/s (approximately 105 rpm), are shown first.

On the right hand side the liquid film thickens where it is carried upward against gravity, and on the left hand side the film thins where it accelerates downward with gravity. The effect is slight in the smaller cylinder, for which the Bond number is 0.1 and Reynolds number is 100. Doubling the cylinder radius increases the Bond number to 0.4 and Reynolds number to 200. At this larger size the drag induced by motion of the cylinder is not sufficient to haul all of the liquid to the top of the cylinder, therefore much of the liquid accumulates in a circulating pool near the bottom of the cylinder.

Next shown is a much larger cylinder of 1 cm radius spinning at 10 cm/s (105 rpm) and 40 cm/s (419 rpm). The Bond number is 10, sufficient to trap nearly all the liquid in the pool. The Reynolds number based on cylinder radius is 1000 and 4000 in these cases, large enough to suggest inertial instability. Indeed, at the higher rotation rate, the simple vortex in the pool has begun to spawn a second recirculation, and surface waves have developed where the film enters the pool, seen in the image on the right.

Around 574 rpm the solution becomes unstable at a turning point (saddle-node bifurcation), shown below. The surface waves have become pronounced and the film has nearly pinched off where it enters the pool. An interesting flow structure, a vortex pair connected by a saddle point in the stream function, has emerged in the pool.

Not surprisingly, the turning point is sensitive to mesh refinement. Meshes of 10x1600, 20x1600, and 10x3200 all agree closely at 574 rpm, which I am calling the converged value. A coarse mesh of 20x400 elements predicts 456 rpm, well short of this value, and meshes of 10x800 and 20x800 predict 536 rpm, still appreciably low. But note that what I call a coarse mesh here yields 108,000 unknowns, and the finest meshes have 432,000 unknowns, quite large problems in two dimensions. Fortunately it's Bambi meets Godzilla when Goodwin's solver dispenses a factorization in less than 5 seconds for the largest of them.

These simulations make a lot of sense intuitively. Before I ran them, Goodwin showed me some simulations he had obtained at high values of Reynolds number. His results looked weird because the liquid collects at the top of the cylinder, somehow suspended there against gravity. I'm showing a similar solution below that I computed for a 10 cm radius cylinder spinning at 1.4 m/s (1466 rpm), much larger and faster than anything shown above. I've reduced the surface tension by a factor of five but left the other properties the same, which is similar to a 1 centipoise silicone oil. The Bond number is 5000 and Reynolds number is 1.4 X 10^{5}.

Inertia is sufficient to hoist the liquid to the top and hold it there. Unlike the situation depicted earlier where it collects at the bottom, there is no recirculation in this pool. The liquid decelerates sharply but continues around in its entirety with the spinning cylinder. This solution is a converged steady state, but it is not stable. Time integration shows an unstable evolution of the free surface, easily set off by numerical noise, that manifests as dripping.

Capillarity and inertia both tend to keep the film uniform in thickness and curvature, and predominate in systems that are small or spinning rapidly, respectively. Either of these limits is convenient to obtain a first converged solution, since the position of the free surface is known to be a circle. I've shown here that these limits behave quite differently in terms of flow stability, however. A small system in which capillarity dominates inertia and gravity is always stable, whereas a rapidly spinning system in which inertia dominates capillarity and gravity is likely to be unstable. Completely different solution regimes can be accessed starting from these limits.

Another interesting flow regime pointed out by Goodwin resembles more a classic hydraulic jump induced by inertia than it does a stream entering a pool. To observe a jump I've reduced the volume fraction of liquid in the cylinder by a factor of four. The result shown below is for a 2 centipoise silicone oil in a 1 cm radius cylinder spinning at 0.35 m/s (366 rpm).

The jump observed here remains fixed in the laboratory reference frame, but viewed relative to the spinning cylinder it moves upstream, making it akin to a tidal bore in a river. Whether this phenomenon is closely related to the hydraulic jump of a stream flowing over a level surface is debatable. In the rim coater, gravity causes an adverse pressure gradient that results in run back downstream of the jump, and rotational forces strongly interact with the pressure field as well. There are no analogies to these effects in the classic hydraulic jump of a stream.

Just for fun here are some more images of an unstable rim coater starting to drip. This one reminds me of a desert sunrise over distant hills:

This one seems more like a sunset:

I think the mesh is probably getting a bit distorted in that second image.