Name: pressure and speed in bernoulli's equation
Uploaded: 2025-02-13T15:43:14.000Z
Duration: 1 min 30 s
Channel: JoVE

Bernoulli&#39;s equation describes the behavior of fluids in motion by relating pressure, velocity, and elevation along a streamline.
It relies on the principle of energy conservation, assuming steady flow, incompressible fluids, and no viscosity, meaning no friction forces are present.
The pressure term represents the force, that the fluid exerts at a specific point.
The velocity term represents the fluid&#39;s kinetic energy, indicating that pressure decreases as fluid velocity increases.
The elevation term reflects the fluid&#39;s potential energy based on its vertical position relative to a reference point.
This principle applies to real-world situations like water distribution systems, dam spillways, and Venturi meters.
In water distribution systems, Bernoulli&#39;s equation predicts how pressure drops as water flows through pipes of varying diameters, ensuring efficient supply across urban networks.
Bernoulli&#39;s equation explains how water velocity increases and pressure decreases as it flows downward in dam spillways, helping engineers design structures that can safely manage varying flow rates.
Venturi meters, commonly used in sewage systems, measure flow rates by detecting pressure differences between pipe sections, ensuring effective wastewater management.

bernoulli's equation for flow along a streamline

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:
<img alt="Bernoulli's equation formula P+(1/2)&#961;v&#178;+&#961;gh=constant, fluid dynamics principle." src="/files/ftp_upload/18033/18033_Equation_1.svg" style="text-align: center; height: 26px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
Here, P is the pressure, &#961; is the density of the fluid, v is velocity, g denotes the acceleration due to gravity, and elevation is denoted by h. Each term in this equation represents energy per unit volume of the fluid.
Consider water flowing through a horizontal pipe with two sections of different diameters. At point 1, the pipe has a large diameter, while at point 2, the pipe narrows. Bernoulli's equation tells us that if the velocity of the water increases at the narrower section (point 2), the pressure must decrease to conserve energy along the streamline. Applying Bernoulli's equation to the pipe sections gives:
<img alt="Bernoulli's principle equation; fluid dynamics formula; pressure and velocity relationship." src="/files/ftp_upload/18033/18033_Equation_2.svg" style="text-align: center; height: 26px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Bernoulli's principle is crucial in understanding and designing fluid flow systems. For instance, in water distribution networks, variations in pipe diameters cause pressure changes that affect flow efficiency. As the equation dictates, a narrower pipe increases velocity while reducing pressure, ensuring consistent supply across urban systems. In dam spillways, the fluid's potential energy due to elevation is converted into kinetic energy as water descends. This results in an increase in velocity and a corresponding drop in pressure, a phenomenon used to design spillways capable of safely managing varying discharge rates.

Bernoulli&#39;s equation describes the behavior of fluids in motion by relating pressure, velocity, and elevation along a streamline.
It relies on the ...

bernoulli-s-equation-for-flow-along-a-streamline

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JoVE Core

Civil Engineering

Unit 1

Fluid Dynamics

KEY TERMS AND CONCEPTS

pressure and speed in bernoulli's equation

Bernoulli's equation links pressure, speed, and height in a moving fluid along a streamline. It is used for incompressible, inviscid fluids under steady flow. The idea comes from applying Newton's second law to a small fluid element. That element feels forces from pressure differences, gravity, and changes in velocity.

The resulting equation shows how fluid energy is shared. In the equation, P is pressure, ρ is fluid density, v is velocity, g is the acceleration due to gravity, and h is elevation. Each term represents energy per unit volume of the fluid. This makes Bernoulli's equation a useful way to track how energy changes as fluid moves.

A simple pipe example shows the main effect. Water flows through a horizontal pipe with two sections. Point 1 has a large diameter, and point 2 has a narrower diameter. When the water speeds up at the narrow section, the pressure must drop to keep energy conserved along the streamline.

Bernoulli's principle also helps explain real flow systems. In water distribution networks, changes in pipe diameter create pressure changes that affect flow efficiency. A narrower pipe raises velocity and lowers pressure, which helps support a steady supply across urban systems. In dam spillways, water loses potential energy as it falls. That energy turns into kinetic energy, so velocity increases and pressure decreases. Engineers use this effect to design spillways that can handle changing discharge rates safely.

Pressure and Speed in Bernoulli's Equation

pressure-and-speed-in-bernoullis-equation

Bernoulli's equation links pressure, speed, and height in a moving fluid along a streamline. It is used for incompressible, inviscid fluids under steady flow. The idea comes from applying Newton's second law to a small fluid element. That element feels forces from pressure differences, gravity, and changes in velocity. The resulting equation shows how fluid energy is shared. In the equation, P is pressure, ρ is fluid density, v is velocity, g is the acceleration due to gravity, and h is...

bernoulli's equation for flow normal to a streamline

bernoulli-s-equation-for-flow-normal-to-a-streamline

Pressure Changes Across Curved Streamlines

Bernoulli's equation for flow normal to a streamline describes how pressure changes across curved streamlines. When a fluid follows a curved path, the outer side of the curve experiences centrifugal force, which pushes the fluid outward. The pressure is therefore higher on the inner side, near the center of curvature, and lower farther out. The size of this pressure difference depends on the fluid's velocity and the radius of curvature. In flows with nearly straight streamlines, the pressure...

bernoulli's equation: problem solving

Venturi Meter Flow Rate from Pressure

A Venturi meter measures fluid flow rate in a pipeline by using pressure changes and Bernoulli's equation. In a sewage system, it can be used to determine wastewater flow rate by reading the pressure difference across the meter. The narrowed throat makes the fluid speed up, and that change in speed is linked to pressure drop. The first step is to find the cross-sectional areas of the pipe and the Venturi throat. These areas are needed to interpret the pressure difference shown on the pressure...

static, stagnation, dynamic and total pressure

static-stagnation-dynamic-and-total-pressure

Pitot-Static Measurement in Bernoulli Flow

Static pressure, dynamic pressure, stagnation pressure, and total pressure are key ideas in fluid dynamics. They are often explained with Bernoulli's equation. In that equation, p is static pressure, 1/2ρV^2 is dynamic pressure, γz is hydrostatic pressure or potential energy pressure, and pT is total pressure. For incompressible flow, the total pressure stays constant along a streamline. Static pressure is the pressure a fluid exerts when it is at rest, or when flowing fluid is brought to rest...

free jet

Liquid Speed from Reservoir Outlets

Free jets show how liquid moves after leaving a reservoir through an opening and flowing into the air without resistance. The jet speed is found with Bernoulli’s principle, and it depends on the height of liquid above the opening. This relationship assumes atmospheric pressure at both the reservoir surface and the jet exit. The liquid column’s height controls the speed at the nozzle because the fluid’s potential energy changes into kinetic energy during flow. As the liquid leaves the...

continuity equation

Fluid Flow and Area Changes

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes. The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When...

energy line and hydraulic gradient line

Fluid Head Profiles in Pipes and Channels

Fluid head profiles in pipes and channels show how pressure, speed, and height change in a flowing fluid. These profiles come from Bernoulli's equation for steady, incompressible, inviscid flow. In that equation, total head includes pressure head, velocity head, and elevation head. The energy line (EL) shows the total head available in the flow. It sits above the datum and includes all three energy parts. Under ideal conditions, the EL stays horizontal because the total energy does not change...

design example: designing water slide

Water Slide Speed and Bernoulli's Principle

Water slide speed is controlled by Bernoulli's principle and the height of the slide. Engineers use these ideas to keep the ride exciting while making sure the water stays safe for riders. As water moves downhill, gravity makes it speed up. The water at the bottom depends on the height it starts from. A taller slide gives the water more potential energy at the top. That energy changes into kinetic energy as the water descends, so the flow becomes faster. Bernoulli's principle helps describe...

key terms and concepts

Learn how Bernoulli's equation links pressure, speed, and height in flowing fluids, with pipe and dam spillway examples for high school science.

Pressure and Speed in Bernoulli's Equation | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

Pressure and Speed in Bernoulli's Equation

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