Because Bernoulli’s equation relates pressure, fluid speed, and height, you can use this important physics equation to find the difference in fluid pressure between two points. All you need to know is the fluid’s speed and height at those two points.
Bernoulli’s equation relates a moving fluid’s pressure, density, speed, and height from Point 1 to Point 2 in this way:
Here are what the variables stand for in this equation (where the subscripts indicate whether you’re talking about Point 1 or Point 2):
The equation assumes that you’re working with the steady flow of an incompressible, irrotational, nonviscous fluid.
One thing you can take immediately from this equation is what’s called Bernoulli’s principle, which says that increasing a fluid’s speed can lead to a decrease in pressure.
Together, the equation of continuity
and Bernoulli’s equation allow you to relate the pressure in pipes to their changes in diameter. You often use the equation of continuity, which tells you that a particular volume of a liquid flows at a constant mass flow rate, to find the speeds you use in Bernoulli’s equation, which relates speed to pressure.
Here’s an example: The operating room is hushed as you’re ushered into it. On the operating table lies a very important person who has an aneurism in the aorta, the principal artery leading from the heart. An aneurysm is an enlargement in a blood vessel where the walls have weakened.
The doctors tell you, “The cross-sectional area of the aneurysm is 2.0A, where A is the cross-sectional area of the normal aorta. We want to operate, but first we need to know how much higher the pressure is in the aneurysm before we cut into it.”
Hmm, you think. You happen to know that the normal speed of blood through a person’s aorta is 0.40 meters/second, and that the density of blood is 1,060 kg/m3. But will that be enough information?
You’d like to use Bernoulli’s equation here because it relates pressure and velocity:
You can simplify Bernoulli’s equation because the patient is lying flat on the operating table, which means that y1 = y2, so Bernoulli’s equation becomes the following:
You want to know how much more pressure is in the aneurysm than in the normal aorta, so you’re looking for P2 – P1. Rearrange the equation:
That’s looking better; you already know
and v1 (the speed of blood in a normal person’s aorta). But what’s v2, the speed of blood inside the aneurysm, equal to? You think hard — and you have an inspiration: The equation of continuity can come to the rescue because it relates speeds to cross-sectional areas:
Because the density of blood is the same at Point 1 and Point 2, in the normal aorta and inside the aneurysm, you can divide out the density to get:
A1v1 = A2v2
Solving for v2 gives you the following:
Now plug in the numbers. Because the doctors told you A2 = 2.0A1 and you know that v1 = 0.4 m/s, you get
So now you’re ready to work with the equation you derived:
You can factor out both 1/2 and
the density, on the right side of the equation:
Plugging in the numbers gives you the following:
You tell the doctors that the pressure is 64 pascals higher in the aneurysm than in the normal aorta.
“How’s that?” ask the doctors. “Give that to us in units we can understand.”
“The pressure is about 0.01 pounds per square inch higher in the aneurysm.”
“How’s that? That’s nothing,” say the doctors. “We’ll operate immediately — you just saved a very important person’s life!”
All in a day’s work for a physicist.