Bernoulli's Principle: Definition, Equation, Examples

How do airplanes fly? Why does a curveball follow such a strange path? And why do you have to board up the ​outside​ of your windows during a storm? The answers to all of these questions are the same: They're a result of Bernoulli's principle.

Bernoulli's principle, sometimes also called the Bernoulli effect, is one of the most important results in study of fluid dynamics, relating the speed of the fluid flow to the fluid pressure. This might not seem particularly important, but as the huge range of phenomena it helps to explain shows, the simple rule can reveal a lot about the behavior of a system. Fluid dynamics is the study of moving fluid, and so it makes sense that the principle and its accompanying equation (Bernoulli's equation) come up quite regularly in the field.

Learning about the principle, the equation that describes it and some examples of Bernoulli's principle in action prepares you for many problems you'll encounter in fluid dynamics.

Bernoulli's principle is named after Daniel Bernoulli, the Swiss physicist and mathematician who developed it. The principle relates the fluid pressure to its speed and elevation, and it can be explained through the conservation of energy. In short, it states that if the speed of a fluid increases, then either its static pressure must decrease to compensate, or its potential energy must decrease.

The relationship with the conservation of energy is clear from this: either the additional speed comes from the potential energy (i.e., the energy it possesses due to its position) or from the internal energy that creates the pressure of the fluid.

The Bernoulli principle therefore explains the main reasons for fluid flow that physicists need to consider in fluid dynamics. Either the fluid flows as a result of elevation (so its potential energy changes) or it flows because of pressure differences in different parts of the fluid (so fluids in the high-energy, higher-pressure zone move to the low-pressure zone). The principle is a very powerful tool because it combines the reasons why fluid moves.

However, the most important thing to take from the principle is that faster-flowing fluid has a lower pressure. If you remember this, you will be able to take the key lesson from the principle, and this alone is enough to explain many phenomena, including the three in the introductory paragraph.

The Bernoulli equation puts the Bernoulli principle into clearer, more quantifiable terms. The equation states that:

\(P + \frac{1}{2} \rho v^2 + \rho gh = \text{ constant throughout}\)

Here ​P​ is the pressure, ​ρ​ is the density of the fluid, ​v​ is the fluid velocity, ​g​ is the acceleration due to gravity and ​h​ is the height or depth. The first term in the equation is simply the pressure, the second term is the kinetic energy of the fluid per unit volume and the third term is the gravitational potential energy per unit volume for the fluid. This is all equated to a constant, so you can see that if you have the value at one time and the value at a later time, you can set the two to be equal to each other, which proves to be a powerful tool for solving fluid dynamics problems:

\(P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2\)

However, it's important to note the limitations to Bernoulli's equation. In particular, it assumes that there is a streamline between points 1 and 2 (the parts labeled by the subscripts), there is a steady flow, there is no friction in the flow (due to viscosity within the fluid or between the fluid and the sides of the pipe) and that the fluid has a constant density. This is generally not the case, but for slow fluid flow that can be described as laminar flow, the equation's approximations are appropriate.

The most common example of Bernoulli's principle is that of a fluid flowing through a horizontal pipe, which narrows in the middle and then opens up again. This is easy to work out with Bernoulli's principle, but you also need to make use of the continuity equation to work it out, which states:

\(ρA_1v_1= ρA_2v_2\)

This uses the same terms, aside from ​A​, which stands for the cross-sectional area of the tube, and given that the density is equal at both points, these terms can be ignored for the purposes of this calculation. First, re-arrange the continuity equation to give an expression for the velocity in the constricted portion:

\(v_2=\frac{A_1v_1}{A_2}\)

This can then be inserted into Bernoulli's equation to solve for the pressure in the smaller section of the pipe:

\(P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2\)
\(P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho \bigg(\frac{A_1v_1}{A_2} \bigg)^2 + \rho gh_2\)

This can be re-arranged for ​P​₂, noting that in this case, ​h​₁ = ​h​₂, and so the third term on each side cancels out.

\(P_2 = P_1 + \frac{1}{2} \rho \bigg( v_1^2 – \bigg (\frac{A_1v_1}{A_2} \bigg)^2 \bigg)\)

Using the density of water at 4 degrees Celsius, ​ρ​ = 1000 kg/m³, the value of ​P​₁ = 100 kPa, the initial velocity of ​v​₁ = 1.5 m/s, and areas of ​A​₁ = 5.3 × 10⁻⁴ m² and ​A​₂ = 2.65 × 10⁻⁴ m². This gives:

\(\begin{aligned}\)
\(P_2 &= 10^5 \text{ Pa} + \frac{1}{2} × 1000 \text{ kg/m}^3 \bigg( (1.5 \text{ m/s})^2 – \bigg (\frac{5.3 × 10^{−4} \text{ m}^2 × 1.5 \text{ m/s}}{2.65 × 10^{−4} \text{ m}^2 } \bigg)^2 \bigg)\)
\(&= 9.66 × 10^4 \text{ Pa}\)
\(\end{aligned}\)

As predicted by Bernoulli's principle, the pressure decreases when there is an increase in velocity from the constricting pipe. Calculating the other part of this process basically involves the same thing, except in reverse. Technically, there will be some loss during the constriction, but for a simplified system where you don't need to account for viscosity, this is an acceptable result.

Some other examples of Bernoulli's principle in action can help to clarify the concepts. The most well-known is the example comes from aerodynamics and the study of airplane wing design, or airfoils (although there are some minor disagreements about the details).

The top part of an airplane wing is curved while the bottom is flat, and because the air stream passes from one edge of the wing to the other in equal periods of time, this leads to a lower pressure on the top of the wing than on the bottom of the wing. The accompanying pressure difference (according to Bernoulli's principle) creates the lift force that gives the plane lift and helps it get off the ground.

Hydroelectric power plants also depend on the Bernoulli principle to work, in one of two ways. First, in a hydroelectric dam, water from a reservoir travels down some large tubes called penstocks, before striking a turbine at the end. In terms of Bernoulli's equation, the gravitational potential energy decreases as the water travels down the pipe, but in many designs, the water exits at the ​same​ speed. By the equation, it's clear that there must have been a change in pressure to balance the equation, and indeed, this type of turbine takes its energy from the pressure energy in the fluid.

Arguably a simpler type of turbine to understand is called an impulse turbine. This works by reducing the size of tube before the turbine (using a nozzle), which increases the velocity of the water (according to the continuity equation) and reduces the pressure (by Bernoulli's principle). The transfer of energy in this case comes from the kinetic energy of the water.