A Flow show is a device on a drilling rig

Drilling rig (petroleum)

This article lists the main components of a petroleum onshore drilling rig.Offshore drilling rigs have similar elements, but are configured with a number of different drilling systems to suit drilling in the marine environment....

 that is attached to the flow line

Flow line

A flow line, used on a drilling rig, is a large diameter pipe that is connected to the bell nipple and extends to the possum belly and acts as a return line, , to the mud tanks.-Flow Line Components:The flow line will typically have equipment attached to it,...

 and has a paddle that swings out as the flow of drilling fluid

Drilling fluid

In geotechnical engineering, drilling fluid is a fluid used to aid the drilling of boreholes into the earth. Often used while drilling oil and natural gas wells and on exploration drilling rigs, drilling fluids are also used for much simpler boreholes, such as water wells. Liquid drilling fluid...

 passes by it. The angle of the paddle increases as the volume of drilling fluid increases and pushes it out further and vice versa.

The purpose of the flow show is to allow the driller (and other rig personnel) to monitor the flow of drilling fluid as it comes out of the hole. The reason for monitoring the flow:

• If a high pressure zone is drilled and overcomes the hydrostatic pressure of the column of drilling fluid, the fluid or gas in the high-pressure zone will begin to enter the borehole
  Borehole
  A borehole is the generalized term for any narrow shaft bored in the ground, either vertically or horizontally. A borehole may be constructed for many different purposes, including the extraction of water or other liquid or gases , as part of a geotechnical investigation, environmental site...
  
   and increase the amount of fluid coming out of the hole. By monitoring the flow, rig personnel will know in advance before the flow becomes critical.
• If a low pressure zone is drilled and is overcome by the hydrostatic pressure of the column of drilling fluid, the low pressure zone may begin to take fluid and decrease the amount of fluid coming out of the hole. By monitoring the flow, rig personnel will know in advance before losing too much drilling fluid (which is typically very expensive) down the hole and/or losing so much fluid that the hydrostatic pressure of the (decreasing) column of mud becomes so low that a blowout situation becomes imminent.

See Drilling rig (petroleum)

Drilling rig (petroleum)

This article lists the main components of a petroleum onshore drilling rig.Offshore drilling rigs have similar elements, but are configured with a number of different drilling systems to suit drilling in the marine environment....

for a diagram of a drilling rig.