12.2 Bernoulli’s Equation

Bernoulli's equation connects pressure, velocity, and height in a moving fluid. It's really just conservation of energy applied to fluids, showing how energy shifts between pressure, kinetic, and potential forms as fluid moves through pipes, airways, or blood vessels.

This section covers the equation itself, how to use it alongside the continuity equation, and why it matters for biological and medical systems like blood flow and respiration.

Bernoulli's Equation

Components of Bernoulli's equation

The equation states that along a streamline in an ideal fluid, the following sum stays constant:

$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Each variable:

• $P$ = static pressure (the force per unit area the fluid exerts on its surroundings)
• $\rho$ = fluid density (assumed constant for an incompressible fluid)
• $v$ = fluid velocity at that point
• $g$ = acceleration due to gravity (9.81 m/s²)
• $h$ = height above a chosen reference level

The three terms each represent a different form of energy per unit volume:

• Pressure term ($P$): energy stored as pressure within the fluid
• Dynamic pressure term ($\frac{1}{2}\rho v^2$): kinetic energy from the fluid's motion
• Hydrostatic pressure term ($\rho gh$): gravitational potential energy from the fluid's elevation

Because the sum is constant, if one term increases, at least one of the others must decrease. That tradeoff is the core of Bernoulli's principle.

Bernoulli's equation and energy conservation

Bernoulli's equation is conservation of energy applied to a fluid flowing along a streamline. The total energy per unit volume (pressure + kinetic + potential) stays the same between any two points, as long as the assumptions hold.

You can write this in a two-point form that's often more useful for problem solving:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$

This directly shows how changes at one point affect another:

• If velocity increases (say, fluid enters a narrower pipe), pressure must drop to compensate. This is the venturi effect.
• If height decreases (water flowing downhill), velocity or pressure (or both) must increase.

Applications of Bernoulli's principle

Most problems combine Bernoulli's equation with the continuity equation, which expresses conservation of mass for an incompressible fluid:

$A_1 v_1 = A_2 v_2$

where $A$ is the cross-sectional area and $v$ is the fluid velocity at each point. This tells you that fluid speeds up when it enters a narrower section and slows down when the section widens.

Here's how to approach a typical problem:

1. Identify two points along the flow where you know (or want to find) pressure, velocity, or height.
2. Use the continuity equation to relate the velocities at those two points if the pipe changes diameter.
3. Plug everything into the two-point form of Bernoulli's equation and solve for the unknown.

Common scenarios you'll see:

• Constriction (nozzle): Area decreases → velocity increases → pressure drops
• Expansion (diffuser): Area increases → velocity decreases → pressure rises
• Tank with a hole (Torricelli's theorem): Fluid exits a hole near the bottom of a tank. The exit speed is $v = \sqrt{2gh}$, where $h$ is the height of the fluid surface above the hole. This comes directly from Bernoulli's equation when you set the pressure at both the surface and the hole equal to atmospheric pressure.

Bernoulli's principle in biology and medicine

Circulatory system:

Blood flow through your vessels follows Bernoulli's principle (approximately). Two important clinical examples:

• Atherosclerosis (narrowed arteries): The constriction forces blood to speed up, which drops the local pressure. This pressure drop can promote turbulence and further damage to vessel walls, worsening the condition.
• Aneurysms (bulging vessel walls): The widened section slows blood flow, raising local pressure. That increased pressure pushes outward on the already-weakened wall, increasing the risk of rupture.

Respiratory system:

• Air speeds up and pressure drops in the narrower portions of the trachea and bronchi during breathing. This is the same venturi effect you see in pipes.
• Your vocal cords use Bernoulli's principle to vibrate. As air rushes through the narrow gap between the cords, the pressure drop pulls them together, then air pressure builds up and pushes them apart again, creating the oscillation that produces sound.

Medical devices:

• Venturi masks deliver precise oxygen concentrations by using a constriction to entrain room air at a controlled rate.
• Aspirators use the venturi effect to create suction for clearing fluids.
• Flow-regulating valves and pumps rely on pressure differences created by changes in fluid velocity.

Limitations of Bernoulli's equation

Bernoulli's equation rests on four simplifying assumptions:

1. Incompressible fluid — density stays constant (good for liquids, less accurate for gases at high speeds)
2. Steady flow — velocity and pressure at any given point don't change over time
3. Inviscid flow — no energy lost to friction or viscosity
4. Laminar flow — fluid moves in smooth, parallel layers with no turbulence

Real fluids violate these assumptions to varying degrees:

• All real fluids have viscosity, which causes energy loss due to friction, especially in long or narrow pipes.
• Turbulent flow (common at high Reynolds numbers) introduces chaotic mixing that Bernoulli's equation can't account for.
• Gases at high velocities become compressible, meaning density changes and the incompressible assumption breaks down.

For practical engineering, modified versions of Bernoulli's equation add terms for frictional losses (the Darcy-Weisbach equation, for example). In this intro course, though, you'll mostly work with the ideal form and just be aware of when it starts to break down.