Dealing with conflicting no-slip Navier-Stokes boundary value constraints?

[Peeter]

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.

So, for example, in a shearing flow where there is a moving surface, the fluid velocity at the fluid/surface interface has to match the velocity of the moving surface.

Similarily, in a channel flow problem, where the boundaries of the channel are not moving, we set the fluid velocities equal to zero at the fluid/surface interfaces.

Observe that this leads to an inconsistency when we have an interface moving on top of a fixed interface. For example, when we have something that is stirring the fluid (like a stir stick in a coffee cup that's scraping along the bottom of the cup). On the surface of the stir stick the "no-slip" condition requires the velocity of the fluid to match the stir stick speed, but at the bottom of the cup we require the velocity to be zero. In an example like this, we can't have both zero and non-zero velocities where the stir-stick touches the cup-bottom.

How is the no-slip condition modified to deal with this inconsistency? Do you have to delete a neighbourhood of the point of contact from the locations where the boundary value conditions are evaluated, and if so, how would the size of that neighbourhood be determined?

 

[Andy Resnick]

Resolving the problem of wetting and slip is an ongoing process- it is an open problem to date. It was introduced to maintain a finite stress tensor across an interface, but the no-slip condition is routinely violated- liquid droplets can move across a solid surface, for example- ever drive in the rain?

Dussan, AFAIK, had one of the first proposals by simply allowing slip (she credits Navier as introducing it) in J. Fluid Mech 209 191-226 (1989). Huh and Mason (J. Fluid Mech 81 401 1977) present another slip model. More recently, Shikhmurzaev's book "Capillary Flows with Forming Interfaces" is interesting, but I'm not sure what to think of his ideas.