Derivation of Bernoulli's Equation

What is Bernoulli’s Equation?

Derivation of Bernoulli’s Equation: Bernoulli’s theorem states that the sum of pressure, potential energy per unit volume, and kinetic energy per unit volume of an ideal fluid that is flowing in a steady state is constant at each point along a streamlined path in a pipe.

Additionally, Bernoulli’s equation is a fundamental principle in fluid dynamics that explains how the pressure, velocity, and elevation of a flowing fluid are related. It helps us understand how fluids, like water or air, move in pipes or over surfaces.

Imagine you are holding a hose and water is flowing out. Bernoulli’s equation tells us that as the water moves faster (higher velocity), its pressure decreases. This is why you can feel the hose pushing back when you cover part of the opening (the pressure increases, slowing down the water). Additionally, if you lift the hose, the water flows slower at the top, and according to Bernoulli’s equation, its pressure increases compared to the bottom where it’s flowing faster.

This principle is used in our daily activities. For example, airplanes stay in the air because the shape of their wings makes air move faster above them, reducing pressure and creating lift. Formula 1 cars use it to stay glued to the track by designing their shapes to control airflow. Hence, Bernoulli’s equation helps us understand how fluid speed and pressure changes are interconnected.

Bernoulli’s Equation Formula

There is no way we will talk about fluid dynamics without having a look at Bernoulli’s equation. This equation, derived from the principle of conservation of energy, describes the behaviour of a fluid as it flows through different sections of a pipeline or conduit. In its simplest form, the equation can be expressed as follows:

P + ½ρv² + ρgh = constant

Where:

P represents the pressure exerted by the fluid,
ρ denotes the fluid density,
v symbolizes the fluid velocity,
g stands for the acceleration due to gravity, and
h represents the height above a reference point

The Components of Bernoulli’s Equation

Breaking down Bernoulli’s equation reveals its components. These components contribute to the overall understanding of fluid dynamics:

Pressure Term (P): The pressure exerted by the fluid on its surroundings helps in determining the fluid’s behavior. As the fluid’s speed changes, its pressure also varies, and this will influence the dynamics of the system.
Kinetic Energy Term (½ρv²): This term represents the kinetic energy possessed by the fluid due to its velocity. It highlights the connection between the fluid’s speed and its energy state.
Potential Energy Term (ρgh): The potential energy term takes into account the fluid’s height above a reference point. It demonstrates the influence of gravitational potential energy on the fluid’s overall energy.

Derivation of Bernoulli’s Equation

At the heart of fluid dynamics lies Bernoulli’s equation, which describes the relationship between pressure, velocity and elevation in a fluid flowing along a streamline. This equation is a manifestation of the conservation of energy principle applied to fluid flow. By considering various factors, we can derive Bernoulli’s equation step by step.

Conservation of Energy:

The derivation of Bernoulli’s equation starts with the concept of conservation of energy. This fundamental principle asserts that the total mechanical energy of a fluid particle remains constant along its path, neglecting external forces like friction.

Kinetic Energy Term

The kinetic energy term in Bernoulli’s equation accounts for the energy associated with the fluid’s velocity. As a fluid particle moves, its kinetic energy changes, which later influences the overall energy balance.

Potential Energy Term

The potential energy term introduces the elevation of the fluid particle in a gravitational field. Changes in potential energy contribute to variations in the total energy of the fluid.

Pressure Energy Term

The pressure energy term represents the force exerted by the fluid’s pressure. It plays a significant role in the derivation as pressure differences contribute significantly to fluid flow behavior.

Simplifying Assumptions

To facilitate the derivation process, certain assumptions are made. These include steady flow conditions, incompressibility of the fluid, and negligible viscous effects.

Applying Euler’s Equation

Euler’s equation, derived from Newton’s second law, is a fundamental equation in fluid dynamics. It forms the basis for deriving Bernoulli’s equation by considering pressure, velocity, and acceleration terms.

Integrating Euler’s Equation

Integrating Euler’s equation along a streamline leads to the development of Bernoulli’s equation. This integration combines the kinetic energy, potential energy, and pressure energy terms, which offers a comprehensive insight into fluid flow behavior.

Applications of Bernoulli’s Equation

The derivation of Bernoulli’s equation opens the door to many practical applications in various fields. From aviation to hydraulics, this equation plays a pivotal role in understanding and predicting fluid behavior.

Aerodynamics and Flight:

In aviation, Bernoulli’s equation elucidates the principles of lift generation. The faster-moving air above an aircraft’s wing corresponds to lower pressure, creating an upward force that allows flight.

Blood Flow and Physiology:

The circulatory system’s functioning can be explained using Bernoulli’s equation. Blood flow through arteries and veins involves pressure differences, which influence the circulation of blood in the body.

Hydraulic Engineering:

Hydraulic systems heavily rely on Bernoulli’s equation to design pipelines, water supply networks, and dams. This equation aids in optimizing fluid transport and distribution.

Meteorology and Weather Patterns:

Bernoulli’s equation contributes to understanding weather phenomena such as wind patterns and air pressure variations. It explains how differences in atmospheric pressure drive air movements on a global scale.

Sports and Recreational Activities:

Sports like baseball and golf exploit the principles of Bernoulli’s equation. The phenomenon of a curveball in baseball and the lift of a golf ball are both influenced by fluid dynamics.

Practical Examples of Bernoulli’s Equation

To solidify our understanding, we will now look into practical scenarios where Bernoulli’s equation takes center stage:

1. Airplane Lift and Flight

Picture yourself seated in an airplane, soaring through the sky. Have you ever wondered how this massive machine defies gravity? The answer lies in Bernoulli’s equation. As air flows over the curved shape of the wing, it accelerates, leading to a decrease in pressure above the wing and a higher pressure below. This pressure difference creates lift, and it allows the aircraft to achieve and maintain flight.

2. Straw Sippers

Are you amazed at the physics behind a straw sipper? When you sip a beverage through a straw, you create a low-pressure area within the straw. Bernoulli’s equation explains this phenomenon: the increased fluid velocity within the narrow straw results in lower pressure, causing the liquid to rise into the straw and into your mouth.

3. Squirt Guns and Fluid Dynamics

Engage in some summertime fun with a squirt gun, and you are also experimenting with Bernoulli’s equation. As you squeeze the trigger, water is forced through a narrow nozzle. The increased velocity of the water results in a drop in pressure, this in turn, allows atmospheric pressure to push water into the gun, which will now be ready for your next squirt.

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Reference:

Common Applications of Bernoulli’s Principle