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

ベルヌーイの方程式は、流線に沿って圧力、速度、および仰角を関連付けることにより、運動中の流体の振る舞いを記述します。
これは、エネルギー保存の原理に依存しており、定常流、非圧縮性流体、および摩擦力が存在しない、つまり粘度がないことを前提としています。
圧力項は、流体が特定の点で及ぼす力を表します。
速度項は流体の運動エネルギーを表し、流体の速度が増加すると圧力が減少することを示します。
標高項は、参照点に対する垂直位置に基づく流体の位置エネルギーを反映します。
この原則は、配水システム、ダムの余水吐、ベンチュリーメーターなどの実際の状況に適用されます。
配水システムでは、ベルヌーイ方程式により、水がさまざまな直径のパイプを流れるときに圧力がどのように低下するかを予測し、都市ネットワーク全体での効率的な供給を確保します。
ベルヌーイの方程式は、ダムの余水吐で水が下向きに流れるときに水の速度がどのように増加し、圧力が減少するかを説明し、エンジニアがさまざまな流量を安全に管理できる構造を設計するのに役立ちます。
下水道で一般的に使用されるベンチュリメーターは、パイプセクション間の圧力差を検出して流量を測定し、効果的な廃水管理を保証します。

流線に沿った流れに関するベルヌーイの方程式

Bernoulli's Equation for Flow Along a Streamline

ベルヌーイの方程式は、流線に沿って移動する流体のエネルギー保存を関連付けます。この方程式は、定常流下の非圧縮性および非粘性流体に適用されます。このような流れでは、ニュートンの第 2 法則が小さな流体要素に適用され、圧力差、重力、および速度変化による力を受けます。力のバランスにより、次の形式のベルヌーイの方程式が導き出されます。

<img alt="Equation 1" src="/files/ftp_upload/18033/18033_Equation_1.svg" style="text-align: center; height: 26px;">

 ここで、P は圧力、ρ は流体の密度、v は速度、g は重力による加速度、h は高度を表します。この方程式の各項は、流体の単位体積あたりのエネルギーを表します。

 直径が異なる 2 つの部分がある水平管を流れる水について考えます。点 1 では管の直径が大きく、点 2 では管の直径が狭くなります。ベルヌーイの方程式によれば、水の流速が狭い部分 (点 2) で増加すると、流線に沿ってエネルギーを保存するために圧力を低下させる必要があります。ベルヌーイの方程式を管部分に適用すると、次のようになります。

<img alt="Equation 2" src="/files/ftp_upload/18033/18033_Equation_2.svg" style="text-align: center; height: 26px;">

 ベルヌーイの方程式は、流体の流れシステムを理解して設計する上で非常に重要です。たとえば、配水管網では、管の直径が変化すると圧力が変化し、流量効率に影響します。法則が示すように、管が狭くなると流速が上昇し、圧力が低下するため、都市システム全体で一貫した供給が保証されます。ダムの放水路では、水が下降するにつれて、高さによる流体の位置エネルギーが運動エネルギーに変換されます。その結果、流速が増加し、それに応じて圧力が低下します。この現象は、さまざまな放水量を安全に管理できる放水路を設計するために使用されます。

ベルヌーイの方程式は、流線に沿って圧力、速度、および仰角を関連付けることにより、運動中の流体の振る舞いを記述します。
それは...

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

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Fluid Dynamics

流体力学

KEY TERMS AND CONCEPTS

主要用語と概念

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 for flow along a streamline

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 provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Pressure and Speed in Bernoulli's Equation

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