What is the boundary condition for a moving solid boundary in a viscous fluid?

[n9e9o9]

I am confused on the definition of the "no-slip" boundary condition because of two seemingly contradicting definitions.

Definition 1: The no-slip condition for viscous fluid states that at a solid boundary, the fluid will have zero velocity relative to the boundary.
Definition 2: The fluid velocity at all liquid–solid boundaries is equal to that of the solid boundary.

What is the velocity at a solid boundary if its moving? This would contradict the zero velocity definition.

Take the example of a air-liquid-solid system, with air on top, liquid in the middle, and the solid on the bottom. Suppose the bottom plate is pulled with a velocity V, at steady-state, to the right-hand side of the system. What would the boundary condition be and/or what would the velocity and shear stress profile look like? (Cartesian coordinates with y in the "north" direction and x in the "east direction")

My guess for the boundary conditions would be that the v=V at y=0 and v=0 at y=\delta.

Is this the correct logic?

 

[arildno]

The RELATIVE velocity of the fluid with respect to the solid boundary, i.e, their difference, is to be zero.

Thus, if with respect to some frame the boundary is moving with velocity V, so will the fluid at its boundary.

 

[Andy Resnick]

The no-slip boundary condition represents a 200-year old unsolved problem.

The official boundary condition is that the tangential component of velocity is continuous across a solid-liquid interface. So, if the solid is not moving, the tangential component of fluid velocity as the surface must also be zero. Because the solid-fluid interface does not deform, the normal component must also be zero. If the solid is moving, the tangential component of velocity of the fluid at the interface is the same as the velocity of the solid, and the normal component of velocity at the interface is still zero. "Stokes first problem" solves how a moving solid creates a velocity field in an initially stationary fluid.

Now, for a fluid-fluid interface, all bets are off becasue the interface can deform. There can be a velocity jump across a fluid-fluid interface.

The no-slip condition exists to ensure the stress tensor does not diverge. However, the no-slip condition is routinely violated all the time- web printing processes, droplet migration across my car windshield during a rainstorm, etc. etc. As I mentioned above, the no-slip boundary condition is a 200-year old unsolved problem.

 

[dameyawn]

I don't think droplet migration on your windshield is an example of no-slip violation. The droplets are actually rolling I think, so no-slip is always satisfied, like the tire of a moving car.

 

[Andy Resnick]

I don't think droplet migration on your windshield is an example of no-slip violation. The droplets are actually rolling I think, so no-slip is always satisfied, like the tire of a moving car.

Davis and Dussan did show, in a beautiful 1974 paper, that some droplets roll along the surface. But that is not a general result.