What are examples of Couette flow in nature? What are examples of Couette flow in nature?
All I could find are theoretical discussions.

It seems to be the flow of viscous fluid between two moving plates. I am imagining slipping on an oily floor. Except perhaps this has an engineering application?
 A: Consider lubrication of bearings involving rotating shafts.  The gap between the shaft and the outer static surface of the bearing is small, so that the curvature can be neglected and locally, the flow can be considered as occurring between parallel plates.  In bearings, the shaft and static member are not concentric, so that it is not exactly Couette flow, and there are pressure variations in the circumferential direction in addition to the "drag flow."
In the screw extruder application discussed by @M. Enns, there is also pressure flow present because the pressure builds up from inlet to outlet of the extruder.  Only if the extruder is being run with no back pressure does one encounter pure Couette flow.  But, there would be no point in doing that (other than using the extruder as purely a conveying device).
A: Extrusion seems to be one application...

Couette flow is a classical problem of primary importance in the history of fluid mechanics
  (1-4), which is a typical example of exact solutions for Navier-Stokes equation. Couette flow is
  perhaps the simplest of all viscous flows, while at the same time retaining much of the same
  physical characteristics of a more complicated boundary-layer flow. One of the most important
  process in industry is extrusion. Since the gap between the barrel and the screw of extruder is
  small, assuming a fluid flowing between parallel plates leads to representative results. There exist
  a large number of parameters in extrusion process which influence significantly the production
  rate and the quality of the final product.

From Computational Study of Couette Flow Between Parallel Plates for
Steady and Unsteady Cases by 
Y. Rihan
Atomic Energy Authority, Hot Lab. Centre, Egypt
A: Wind-driven flow in a body of water is a situation where the Couette flow can be a useful approximation. Of course in this case there is no rigid plate at the top and the water surface becomes soon wavy. Nonetheless the wind exerts a shearing action on the surface that sets the underneath water in motion. This is the most fitting example that I could imagine to occur in nature.
