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

연속성 방정식은 질량 보존의 원리에서 파생되며, 시스템에 들어오고 나가는 유체의 질량이 일정하게 유지된다는 것을 나타냅니다.
비압축성 유체의 경우, 유동 면적과 속도의 곱은 유동 경로 전체에서 일관되게 유지되어 단면의 변화에 맞게 조정됩니다.
단면적이 감소함에 따라 유체 속도가 증가하여 일정한 유속을 유지합니다.
이 관계는 수도관, 우수 관리 및 개방 수로와 같은 시스템을 분석하는 데 매우 중요하며 다양한 유체 흐름 응용 분야에서 안정성과 효율성을 보장합니다.
급수 시스템에서 연속성 방정식은 파이프가 좁아질 때 속도 변화를 예측하여 안정적인 분포를 유지하는 데 도움이 되며, 이는 압력 강하를 방지하는 데 필수적입니다.
빗물 관리를 위한 저류조를 설계하는 데 도움이 됩니다. 폭풍우가 치는 동안 유출수가 유역으로 유입될 때, 유출 속도는 하류의 범람을 방지하기 위해 조정된 유입 속도와 일치해야 합니다.
강이나 운하와 같은 열린 수로의 경우 연속성 방정식은 너비 또는 깊이의 변화가 수위와 유속에 어떤 영향을 미치는지 이해하는 데 도움이 됩니다. 이러한 이해는 효과적인 홍수 통제 및 내비게이션 시스템을 설계하는 데 필수적입니다.

연속성 방정식

연속성 방정식은 밀폐된 시스템 내에서 비압축성 유체의 정상 흐름에 대해 질량 유량이 일정해야 한다고 주장합니다. 이 원리는 노즐, 주사기, 파이프와 같이 유체가 다양한 단면적을 통과하는 시스템에 적용됩니다.

질량 유량은 다음과 같이 표현됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;">

밀도가 일정하게 유지되는 비압축성 흐름의 경우 연속성 방정식은 다음과 같이 간소화됩니다.

<img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;">

이 방정식은 단면적이 감소하면 속도가 증가하고 그 반대의 경우도 마찬가지이므로 질량 보존이 보장됩니다. 예를 들어 주사기를 통과하는 흐름이 있습니다. 플런저가 노즐을 통해 유체를 압축할 때 연속성 방정식은 출구 면적이 입구 면적의 절반인 경우 출구 속도가 입구 속도의 두 배여야 함을 보여줍니다.

또 다른 예는 중력 구동 탱크의 흐름입니다. 베르누이 방정식과 연속 방정식을 흐름을 따라 있는 지점에 적용하면 출구 속도와 체적 유량을 결정하는 데 도움이 됩니다.

<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;">

여기서 h는 유체의 높이이고, D는 탱크의 직경이며, d는 출구 직경입니다. 이 관계는 출구 직경이 탱크 직경에 비해 작아짐에 따라 유량이 안정적인 값으로 수렴함을 보여줍니다. 이러한 파생은 다양한 제약 하에서 유체 거동을 예측하는 데 연속 방정식을 사용하는 것이 유용하다는 것을 보여 주며, 동적 시스템에서 질량 보존을 보장합니다.

연속성 방정식은 질량 보존의 원리에서 파생되며, 시스템에 들어오고 나가는 유체의 질량이 일정하게 유지된다는 것을 나타냅니다.
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Education

교육

JoVE Core

Civil Engineering

토목공학

Unit 1

제1단원

Fluid Dynamics

유체역학

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 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.

civil engineering

Fluid Flow and Area Changes

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