Bernoulli's experiment demonstrates the principle of conservation of mechanical energy in a flowing fluid. As water flows through a pipe of varying cross-sectional area, its pressure and velocity continuously change. Bernoulli's theorem states that, under suitable conditions, the total mechanical energy of the fluid remains constant along a streamline.

The total energy of a flowing fluid consists of three components:

• Pressure energy,
• Kinetic energy,
• Potential energy.

If the velocity of the fluid increases, its pressure decreases, and vice versa, provided no external energy is added or removed from the system.

Bernoulli's principle is one of the fundamental laws of fluid mechanics and forms the basis for the design and analysis of many hydraulic structures and flow-measuring devices.

Everyday Intuition

The relationship between pressure and velocity can be observed in many everyday situations.

• Water emerging from a partially closed garden hose travels farther because reducing the outlet area increases the velocity of the water.
• Air moving rapidly over an aircraft wing creates a pressure difference that generates lift.
• Rivers flowing through narrow sections move faster than in wider sections.

In each case, part of the pressure energy is converted into kinetic energy while the total mechanical energy remains nearly constant.

The Bernoulli apparatus demonstrates this same phenomenon under controlled laboratory conditions.

Experimental Relevance

The objective of the Bernoulli experiment is to verify Bernoulli's theorem experimentally by measuring the pressure head at different sections of a converging-diverging flow channel.

As water passes through the converging section,

• Flow area decreases,
• Velocity increases,
• Pressure decreases.

As the channel diverges,

• Flow area increases,
• Velocity decreases,
• Pressure recovers.

By measuring the pressure head at different locations and calculating the velocity head, the total energy head can be evaluated and compared along the flow path.

In an ideal fluid,

$$\text{Pressure head}+\text{Velocity head}+\text{Elevation head} === \text{Constant}.$$

Small deviations from the ideal value occur because of frictional losses and experimental uncertainties.

Mathematical Formulation

Bernoulli's theorem for steady, incompressible, frictionless flow along a streamline is

$$\frac{P}{\rho g} + \frac{V^2}{2g} + Z === \text{Constant}.$$

where,

Symbol	Quantity	Unit
$P$	Pressure	$\mathrm{N/m^2}$
$\rho$	Density of fluid	$\mathrm{kg/m^3}$
$V$	Velocity of flow	$\mathrm{m/s}$
$g$	Acceleration due to gravity	$\mathrm{m/s^2}$
$Z$	Elevation above datum	$\mathrm{m}$

The three terms represent:

Pressure Head

$$\frac{P}{\rho g}$$

Energy possessed due to pressure.

Velocity Head

$$\frac{V^2}{2g}$$

Energy possessed due to fluid motion.

Elevation Head

$$Z$$

Energy possessed due to elevation.

Between two points,

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + Z_1 === \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + Z_2.$$

For the Bernoulli apparatus used in this experiment, all pressure tapping points are approximately at the same elevation. Therefore,

$$Z_1=Z_2,$$

and the elevation head remains constant.

Hence,

$$\frac{P}{\rho g} + \frac{V^2}{2g} === \text{Constant}.$$

Application to the Bernoulli Apparatus

The Bernoulli apparatus consists of

• Water reservoir,
• Centrifugal pump,
• Measuring tank,
• Converging-diverging test section,
• Pressure tapping points connected to piezometer tubes.

Water flows through the test section whose area gradually decreases and then increases.

The piezometer tubes indicate the pressure head at different locations.

At the throat,

• Area is minimum,
• Velocity is maximum,
• Pressure head is minimum.

At wider sections,

• Velocity decreases,
• Pressure head increases.

By measuring discharge and pressure head, the velocity head and total head can be calculated and compared to verify Bernoulli's theorem.

Engineering Significance

Bernoulli's theorem is widely used in hydraulic and civil engineering applications.

Some important applications include:

• Venturi meters,
• Orifice meters,
• Flow through nozzles,
• Pipe flow analysis,
• Jets issuing from openings,
• Flow over weirs,
• Hydraulic structures,
• Pumps and turbines.

Understanding the conversion between pressure energy and kinetic energy is essential for the design and operation of water supply systems, irrigation networks, hydroelectric plants, and hydraulic machinery.

The Bernoulli experiment provides a practical demonstration of this fundamental energy principle and forms the basis for many advanced topics in fluid mechanics.