How to deal with conflicting "no-slip" Navier-Stokes boundary value constraints?

The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces.

Now that my class is done, I've been playing with solving an idealized fluid flow problem: figuring out the rate of "spin-down" of stirred fluid in a non-bottomless cylindrical container (ie: my coffee after being given a good stir).

For the non-time dependent portion of the problem, I'm not able to apply the no-slip condition as taught in class to my cup of coffee. Matching velocities to my stir stick (I've idealized that as a cylinder as in Acheson's bottomless tea example (pg 45) ), I've got a problem at the base of the cup: If I match velocities of the fluid to the "stir-stick", I can't also match velocities to the bottom of the cup (zero velocity) where the stir-stick touches the base.

I've been pointed to some journal articles to read, but won't be able to do so until the fall when I enroll in my next course (and get library access again). In the interim, I was wondering if I could get some pointers on an approach to deal with conflicting 'no-slip' conditions like this one (ie: any problem with a moving interface in contact with a static one has this trouble.)

EDIT: I believe that I can work around the inconsistency problem with no-slip constraints for this problem by "stirring without touching the bottom of the cup" ... ie: set the stir depth to 1cm (say). This makes the non-time dependent solution uglier, since I have to treat three different regions:

1. Below the stir stick (zero velocity no-slip condition on the inside of the cup in this region ; velocity matching the stir velocity along the circle at the stir radius at the stir depth).
2. Between the center of the cup and the stir stick. "no-slip" conditions: match velocities below the stir stick. match the stir velocity at the stir radius.
3. Between the stir stick and the edge of the cup. "no-slip" conditions: zero velocity on the cup edge. stir velocity at the stir radius.

This makes the problem much messier ... I already have Bessel functions to deal with, and now have to apply them separately in three regions.

I'm still interested to see if there's a better way to deal with the boundary value conditions ... ie: without forcing an artificial distance between a static and a dynamic interface so that they don't touch.

1 Answer

If you were to stir an ideal no-slip liquid, your spoon would get stuck if it is to touch the bottom. You can try taking 2 glass plates, dipping them in water, and moving them relatively to each other, then get them touching together and try moving again, the friction increases and for ideal no-slip liquid the force required to move surfaces at given speed would become infinite as the distance approaches zero.

• Do you know of a model that works better than "no-slip" for stirring that involves non-static interfaces in contact? Commented Jul 11, 2012 at 20:31