Bernoulli's equation is a statement of energy conservation for flowing fluids. When a liquid or gas moves without significant friction, the sum of its pressure energy, kinetic energy, and gravitational potential energy remains constant along a streamline. Engineers and physicists apply this relationship to explain why airplane wings generate lift, how water speeds through pipes, and why chimney drafts rise. In its classic form, the equation can be written as
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The pressure term in the equation represents stored energy per volume due to molecular collisions. The kinetic term reflects how fast the fluid is moving, while the height term accounts for gravitational potential energy. As a parcel of fluid travels along a pipe, gains in one form of energy must be offset by losses in another so that their sum stays the same. If the pipe narrows, the fluid speeds up, increasing kinetic energy at the expense of pressure.
Daniel Bernoulli formulated this relationship in the 18th century while investigating fluid flow through tubes. His insight that faster flow corresponds to lower pressure shocked many contemporaries. Over time, the principle became a cornerstone of hydrodynamics. It paved the way for understanding lift on airplane wings, the Venturi effect in carburetors, and many other phenomena. Bernoulli's work stands as a prime example of how theoretical reasoning can illuminate real-world processes.
A streamlined derivation begins with Newton's second law applied to an infinitesimal fluid element. Along a streamline, the forces from pressure gradients and gravity must equal the change in momentum. Integrating this expression along the path and recognizing that energy is conserved yields Bernoulli's equation. Mathematically, one arrives at
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The equation suggests that where a fluid speeds up, pressure must drop if height stays constant. This concept underlies the operation of a Venturi meter: by measuring the pressure difference between a wide section and a narrow section of pipe, you can deduce the flow speed. Similarly, as air flows faster over the curved top surface of a wing, its pressure decreases compared to the slower air below, generating lift.
Beyond aerodynamics, Bernoulli's principle appears in medical devices such as the Venturi oxygen mask, which mixes air with oxygen using pressure differences. It governs the flow of water from reservoirs, the draw of fluids up a siphon, and the circulation of blood in arteries. The equation allows engineers to estimate pump requirements, design efficient hydraulic systems, and analyze natural flows like those in rivers and aquifers.
When a fluid flows uphill, part of its kinetic or pressure energy is converted into gravitational potential energy. Conversely, as it descends, gravity accelerates the fluid, increasing its velocity if pressure remains constant. In tall buildings and skyscrapers, these height variations lead to noticeable changes in water pressure between lower and upper floors. Engineers must account for such effects to ensure consistent supply.
While powerful, the equation assumes no energy is lost to friction, turbulence, or heat. Real fluids exhibit viscosity, which causes pressure drops along pipes. Rapid changes in cross-section or obstacles can create eddies and turbulence that deviate from ideal behavior. Nevertheless, Bernoulli's equation offers an excellent first approximation and is often combined with empirical corrections to model complex systems.
To employ this tool, enter the fluid density, initial pressure, initial velocity, and initial height for one point in the flow. Next provide the velocity and height at a second point. When you click the Compute button, the calculator rearranges Bernoulli's equation to solve for the unknown final pressure:
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The result illustrates how faster flow or higher elevation at the second point lowers its pressure relative to the first. Positive differences indicate a pressure increase, while negative differences reveal a drop.
Imagine water flowing through a pipe that rises from ground level to a second-floor tap 4 meters higher. Suppose the pipe narrows so the water velocity increases from 1 m/s at the entrance to 3 m/s near the tap. If the density is 1000 kg/m³ and the initial pressure is 200,000 pascals, plugging these numbers into the calculator shows that the pressure near the tap drops substantially. This explains why faucets on higher floors sometimes deliver lower pressure than those on the ground.
Bernoulli's principle not only explains everyday observations but also leads to sophisticated engineering designs. Airplane wings, carburetors, and even the atomizers in perfume bottles rely on pressure changes induced by varying flow speed. Understanding the equation reveals why fluid systems behave the way they do and how to manipulate them effectively.
Often, Bernoulli's equation is combined with the continuity equation, which states that the product of cross-sectional area and velocity remains constant for an incompressible fluid. This combination allows you to determine unknown areas or speeds at different points in a network of pipes, channels, or ducts.
Whether you're designing a hydraulic system, studying aerodynamic lift, or simply curious about why water pressure drops when you open multiple faucets, Bernoulli's principle provides clarity. By exploring different input values with this calculator, you will gain intuition about how energy moves within a fluid and why pressure, velocity, and elevation are so closely intertwined.
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