Continuity and Bernoulli 
Laminar and Turbulent Flow
So far we have dealt with fluids at rest. This next section deals with moving fluids. You will be solving for pressures and speeds with the equations you learn.
When a fluid flows, the quality of the flow can be described as laminar or turbulent. Examine the diagram of the bird in flight to see the difference between these two qualities.
Because the mathematics required to fully analyze turbulent flow is beyond the scope of this course, we will only ever deal with the smooth, laminar flow of fluids.
The Equation of Continuity
In the section that follows, we will develop an equation that allows for the prediction of the speed of a fluid in a pipe.
Consider the section of pipe that has laminar flow from left to right:
As the fluid moves through the pipe, the mass flow rate (measured in kg/s) must be constant everywhere in the pipe. This means that the fluid will slow in the larger portions of the pipe and speed up in the narrow portions. A fireman's hoze nozzle takes advantage of this idea - it is very narrow at the end of the nozzle giving the exiting water a high speed.
The mass flow rate of water in a pipe can be given by the expression:
where V is the volume, A is cross sectional area, l is the length, and l/t = v which is the speed.
Since the mass flow rate is constant, we can state:
And since the density of a fluid is constant, this can be simplified to what we call the equation of continuity:
Example 1 - the Heating Duct
Bernoulli's Principle
“Where the velocity of a fluid is low, pressure is high. Where the velocity is high, pressure is low.”
In simpler terms, if you increase the velocity of a fluid, the pressure in the fluid decreases. An airplane wing takes advantage of this principle. The wing's shape, known as an airfoil, is designed so that the air that passes over its top moves faster (it has further to go to reach the same point in the same time).
The faster moving air above the airfoil means that the pressure is less above the wing than below. The airplane is lifted by the higher pressure below the wing.
Bernoulli's Law
So far we have seen that the pressure in a fluid in a pipe depends upon several things:
- the pressure applied at one end of the pipe
- the difference in height, given by the equation ρgh
- and the speed of the fluid
Bernoulli takes all three of these ideas and gives us an expression that will allow us to find the pressure anywhere in a pipe that changes height and changes diameter.
Follow the derivation below.
Consider the pipe below, with flow from left to right.
We begin by finding the work (energy) necessary to move the fluid through the two distances labeled in the diagram as Δx1 and Δx2. Recall that Work = Force × distance
As the fluid moves through the pipe, both its kinetic energy (based on speed) and its potential energy (based on height) change. These changes are caused by the work done by the fluid.
By putting in the expressions derived above we arrive at Bernoulli's Law.
Some Hints and Tricks
You will often use Bernoulli's Law and the Equation of Continuity together to create a system of equations.
Where a portion of a pipe is open to the atmosphere, the pressure P at that point is equal to atmospheric pressure.
Example 2 - Drinking from the firehose
Example 3 - Heat rises
