3.3.4.3 Bernoulli's Equation

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Flowing energy in a fluid

Bernoulli's equation is one of the most useful ideas in fluid dynamics. It connects pressure, speed, and height in a moving fluid. The equation shows that along a streamline, a fluid can exchange pressure energy, kinetic energy, and gravitational potential energy.

For an ideal fluid, meaning a fluid that is incompressible and has negligible viscosity, Bernoulli's equation is

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

Here, $P$ is the pressure, $\rho$ is the fluid density, $v$ is the flow speed, $g$ is the gravitational acceleration, and $h$ is the height above a chosen reference level.

This means that if one of these terms increases, one or both of the others must decrease, as long as the fluid moves along the same streamline.

For steady flow of an incompressible, nonviscous fluid along a streamline,
$$
P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}
$$
This is Bernoulli's equation.

Meaning of the three terms

Each term in Bernoulli's equation has the units of pressure, or energy per unit volume.

The term $P$ is the pressure energy per unit volume.

The term

$$
\frac{1}{2}\rho v^2
$$

is called the dynamic pressure. It is related to the kinetic energy of the moving fluid.

The term

$$
\rho g h
$$

represents gravitational potential energy per unit volume.

A useful way to read the equation is that the total mechanical energy per unit volume stays constant along the flow, if there are no losses.

Term	Expression	Physical meaning
Pressure term	$P$	Pressure energy per unit volume
Kinetic term	$\frac{1}{2}\rho v^2$	Energy associated with fluid speed
Gravitational term	$\rho g h$	Energy associated with height

Comparing two points in a flow

Bernoulli's equation is often written for two different points, labeled 1 and 2:

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

This form makes it easy to compare what happens as the fluid moves from one place to another.

If the height stays the same, then the equation becomes

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

So where the fluid moves faster, the pressure is lower.

At the same height, faster fluid flow means lower pressure, for ideal steady flow:
$$
P + \frac{1}{2}\rho v^2 = \text{constant}
$$

Physical interpretation

Bernoulli's equation can be understood as an energy balance. If a fluid speeds up, its kinetic energy increases. That extra energy must come from somewhere. In ideal flow, it comes from a drop in pressure, or from moving to a lower height, or from both.

If fluid rises upward, the gravitational term $\rho g h$ increases. Then pressure or speed must decrease if the total remains constant.

This is why narrow regions in a pipe often have faster flow and lower pressure, and why fluid flowing downward can gain speed.

Special cases

A very common special case is horizontal flow. If $h_1 = h_2$, the height terms cancel:

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

Another important case is when the fluid is nearly at rest at one point, so $v \approx 0$. Then pressure can be converted into speed as the fluid starts moving.

A third case is when the speed is the same at both points. Then Bernoulli's equation reduces to

$$
P_1 + \rho g h_1 = P_2 + \rho g h_2
$$

which shows how pressure changes with height in that flow situation.

Relation to continuity

Bernoulli's equation is often used together with the continuity equation. The continuity equation relates speed and cross-sectional area in incompressible flow. If the area becomes smaller, the speed becomes larger. Bernoulli's equation then tells us that the pressure becomes smaller in that narrower region.

So continuity explains why speed changes, and Bernoulli explains how pressure changes when speed changes.

Common applications

Bernoulli's equation helps explain many important devices and effects in physics and engineering.

In a venturi tube, fluid enters a narrow section, speeds up, and the pressure drops.

In a spray bottle, fast-moving air above a tube lowers the pressure, allowing liquid to rise and spray out.

In airplane wings, pressure differences are part of the explanation of lift, though the full physics of lift is more complex than a simple Bernoulli-only argument.

In a pitot tube, the flow is brought to rest at one point, and the pressure difference is used to determine fluid speed.

Example idea

Suppose water flows through a horizontal pipe. At one point the speed is $2 \, \text{m/s}$, and at a narrower point it is $5 \, \text{m/s}$. Since the pipe is horizontal, the height terms are the same. Bernoulli gives

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

Rearranging,

$$
P_1 - P_2 = \frac{1}{2}\rho \left(v_2^2 - v_1^2\right)
$$

For water, $\rho \approx 1000 \, \text{kg/m}^3$, so

$$
P_1 - P_2
=
\frac{1}{2}(1000)(25 - 4)
=
10500 \, \text{Pa}
$$

So the pressure at the faster section is lower by $10500 \, \text{Pa}$.

Limits of Bernoulli's equation

Bernoulli's equation is powerful, but it does not apply in every situation. It works best when the flow is steady, the fluid density is constant, and frictional effects are negligible.

Real fluids have viscosity. Because of viscosity, mechanical energy can be lost to thermal energy. In such cases, Bernoulli's equation in its simple form is only an approximation.

It is also not generally valid between arbitrary points if the flow is turbulent or if energy is added by pumps or removed by turbines.

Bernoulli's equation in the simple form is valid only under ideal conditions, typically steady flow, incompressible fluid, negligible viscosity, and along a streamline.

Visual picture

The picture below shows fluid moving through a pipe that narrows. As the pipe narrows, the speed increases and the pressure decreases.

Rendering scientific sketch…

Bernoulli effect in a narrowing pipe

Pressure, speed, and height together

The next picture shows two points on a streamline at different heights. It illustrates how pressure, speed, and height all contribute.