The equation is for an idealized situation. It would probably easier to explain Bernoulli principle by stating that pressure differentials coexist with accelerations within a fluid or a gas. A fluid or gas will accelerate from a higher pressure zone to a lower pressure zone. In an idealized case where no work is done within the transition, then Bernoulli's equation defines a relationship between the pressure and speed of the fluid or gas at any point within that flow between the pressure zones.
Bernoulli doesn't cover interactions between a solid and fluid or gas, or other situations when work is done. From this Nasa link:
at the exit, the velocity is greater than free stream because the propeller does work on the airflow. We can apply Bernoulli's equation to the air in front of the propeller and to the air behind the propeller. But we cannot apply Bernoulli's equation across the propeller disk because the work performed by the engine (by the propeller)
violates an assumption used to derive the equation. :
propeller_analysis.htm
In the case of a propeller, there is a low pressure zone fore of the prop disk, and air accelerates towards this low pressure zone via a Bernoulli like reaction. Across the prop "disk", the speed remains about the same, but the pressure increases due to mechanical interaction between air and propeller, called a pressure jump. Just aft of the prop disk, the air has higher pressure, and continues to accelerate aft of the propeller. Part of the propeller's job is to reduce the amount of high pressure air just aft of the prop disk that would otherwise accelerate forwards to the low pressure zone just fore of the prop disk.
A wing does essentially the same thing, but the amount of induced wash is less (since the wing moves perpendicular to lift), and the pressure jump is less.
Any venturi based device is a good example of Bernoulli principle. Flow through a pipe is reduced in pressure as it flows due to friction between the pipe and fluid or gas, and viscosity within the fluid or gas. If the pipe has a narrowing section, then the flow has to speed up or otherwise mass would be accumulating. The faster moving flow in the narrower section has lower pressure and if this pressure is lower than some external pressure, it can be used to draw in an external source of fluid or gas.
Although carburetors are sometimes used as examples of Bernoulli principle, you have the additional effect of a nozzle perpendicular to a flow, which creates a vortice, lowering the pressure further still. Some spray pumps also operate on this same nozzle effect, and home experiments using a straw perpendicular to a flow also rely on this nozzle effect and are not good examples of Bernoulli principle. In order to "sense" the pressure of a horizontal flow, you need something like a static port, which is a flush mounted pipe that "hides" inside a boundary layer between a surface and a horiztonal flow.
A good example of Bernoulli principle is an venturi based pump, connected to a water tap and used to drain water from aquarium. Here is an example:
http://andysworld.org.uk/aquablog/?postid=247
If you follow the USA patent, there's a link to images, but you'll need to install a TIFF viewer browser add-on to view it. If you follow the Candian patent:
http://brevets-patents.ic.gc.ca/opic-cipo/cpd/eng/patent/1245129/summary.html
Then figure 4 on drawing page 2 shows the flow when operating as a drain via venturi effect:
python_syphon_drawing_page_2.htm
Note that this device is essentially the same device as the one patented in 1933, I'm not sure why it was granted a patent.
