Name: fluid flow and area changes
Uploaded: 2026-01-30T03:08:26.000Z
Duration: 1 min 28 s
Channel: JoVE

連続方程式は、質量保存の原理に由来し、システムに出入りする流体の質量は一定であることを示しています。
非圧縮性流体の場合、流れの面積と速度の積は流路全体で一定に保たれ、断面積の変化に合わせて調整されます。
断面積が小さくなると、流体速度が増加し、流量が一定に保たれます。
この関係は、水道パイプライン、雨水管理、開水路などのシステムの解析に不可欠であり、さまざまな流体フローアプリケーションでの安定性と効率性を確保します。
水道システムでは、連続方程式は、圧力降下を回避するために不可欠なパイプが狭くなったときの速度変化を予測することにより、安定した分布を維持するのに役立ちます。
雨水管理のための貯水池の設計を支援します。嵐の際に流出水が流域に入る場合、流出速度は流入速度と一致し、下流の洪水を防ぐために調整する必要があります。
川や運河のような開水路の場合、連続方程式は、幅や深さの変化が水位や流速にどのように影響するかを理解するのに役立ちます。この理解は、効果的な洪水制御およびナビゲーションシステムを設計するために不可欠です。

連続方程式

連続方程式は、閉じ込められたシステム内で非圧縮性流体が安定して流れるためには、質量流量が一定でなければならないと主張します。この原理は、ノズル、注射器、管など、流体が断面積の異なる部分を通過するシステムに適用されます。

 質量流量は次のように表されます:

<img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;">

 密度が一定である非圧縮性の流れの場合、連続方程式は次のように簡略化されます:

<img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;">

 この方程式は、断面積が減少すると速度が増加し、断面積が減少すると速度が増加することを示しています。これにより質量が確実に保存されます。例として、注射器を通る流れがあります。プランジャーがノズルを通る流体を圧縮する場合、連続方程式は、出口面積が入口面積の半分である場合、出口速度は入口速度の 2 倍でなければならないことを示しています。

 もう 1 つの例として、重力駆動タンク内の流れがあります。ベルヌーイの方程式と連続方程式の両方を流れに沿った点に適用すると、出口速度と体積流量を決定するのに役立ちます:

<img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3a.svg" style="text-align: center; height: 75px;">
<img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3b.svg" style="text-align: center; height: 20px;">

 ここで、h は流体の高さ、D はタンクの直径、d は出口直径です。この関係は、出口直径がタンク直径に対して小さくなるにつれて、流量が安定した値に収束することを示しています。これらの導出は、さまざまな制約の下で流体の挙動を予測し、動的システムにおける質量保存を保証する連続方程式の有用性を例示しています。

連続方程式は、質量保存の原理に由来し、システムに出入りする流体の質量は一定であることを示しています。
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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

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 the area gets larger, the speed must decrease. This keeps mass conserved as the fluid moves.

A syringe shows this relationship clearly. When the plunger pushes fluid through a nozzle, the outlet speed rises if the outlet area is smaller than the inlet area. If the outlet area is half the inlet area, the outlet velocity is twice the inlet velocity.

A gravity-driven tank gives another example. Here, Bernoulli's equation and the continuity equation can be used together at points along the flow. They help find the outlet velocity and the volumetric flow rate. In this setup, h is the fluid height, D is the tank diameter, and d is the outlet diameter. The flow rate approaches a stable value as the outlet diameter becomes small compared with the tank diameter.

Fluid Flow and Area Changes

bernoulli's equation for flow along a streamline

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

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...

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 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 the continuity equation links area, velocity, and mass conservation in steady fluid flow, with examples from syringes and gravity-driven tanks.

Fluid Flow and Area Changes | 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

Fluid Flow and Area Changes

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