Fluid Flow between parallel infinite plates

[hawaiifiver]

Hello

I am trying to understand some fluid mechanics

I have two parallel infinite plates with fluid between the plates and some fluid above the plates.
The fluid above the plates has a free surface exposed to the atmosphere. And we can neglect body forces.

The fluid flow (steady and laminar) is two dimensional in both regions, and velocity doesn't depend on x and y: u = u_1 (z) i

What can i say about the value of u_1 (z) at the free surface. is u_1(surface) = 0 if the top plate is moving with a velocity U i. The bottomn plate is stationary.

Also how would you use the Navier Stokes quation to find u_1 (z) in both regions.

 

[Andy Resnick]

Between the plates is straightforward: it's plane Couette flow.

The fluid above the top plate is significantly more complex: if it were unbounded, it would be similar to Stokes' first problem- except you have suppressed the time dependence that makes the solution finite.

At the fluid-fluid interface, it gets very complicated. There are so-called jump balance equations for mass, momentum, and energy which are quite horrendous and I'm not going to try and write them here. They can be found in Slattery's "Interfacial Transport Phenomena", and relate the motion of the fluid on either side of the dividing surface to the dynamics of the interface.