Name: parabolic flow in circular tubes
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 23 s
Channel: JoVE

Hagen-Poiseuille 흐름으로 알려진 일정한 반경 R의 직선 원통형 튜브를 통한 안정적이고 압축되지 않는 흐름을 고려하십시오.
원통형 형상을 사용하면 좌표 r, θ 및 z를 방사형, 각도 및 축 방향 측정에 사용할 수 있습니다.
튜브의 축과 평행하게 흐르는 유체는 순전히 축 방향 속도 성분을 가지며 반경 방향 거리에 따라서만 변합니다.
이러한 조건은 Navier-Stokes 방정식을 단순화하여 속도가 중심에서 가장 높고 벽으로 갈수록 감소하여 경계에서 0에 도달함을 보여주는 포물선 속도 프로파일을 나타냅니다.
이것은 유체 입자가 조직화된 층에서 부드럽게 움직이는 층류를 보여줍니다.
이 프로파일을 튜브의 단면에 적분하면 유속 Q를 튜브 반경의 4제곱, 유체 점도 및 압력 구배에 의존하는 것으로 표현하는 Poiseuille의 법칙이 제공됩니다.
Q에서 계산된 평균 속도 V는 튜브 중심에서 관측된 최대 속도의 절반입니다.
이 속도 분포는 흐름이 중심에서 가장 높고 벽에서 포물선으로 0으로 감소한다는 것을 확인합니다.

원형 관 내의 정상, 층류

Steady, Laminar Flow in Circular Tubes

하겐-푸아수이유 흐름은 일정한 반경 R을 가진 원통형 관을 통과하는 점성 유체의 정상적이고 압축 불가능한 흐름을 설명합니다. 이 흐름 프로파일은 모세관과 같은 좁은 채널에서 유체 이동을 이해하는 데 자주 적용됩니다. 이는 층류의 기초적인 예입니다. 이 모델에서 원통형 좌표(r,θ,z)는 관 내의 반경(r), 각도(θ), 축(z) 치수를 설명하는 데 사용됩니다. 하겐-푸아수이유 흐름의 경우 속도 프로파일은 순전히 축 방향입니다. 즉, 속도 벡터는 z축을 따라서만 가리키고 관 중심에서 반경 거리 r에 따라서만 변합니다.

유체 운동의 지배 방정식인 나비에-스토크스 방정식은 이러한 조건에서 단순화됩니다. 각속도 또는 반경속도 성분이 없으면 축속도 v_z(r)은 r만의 함수로 나타납니다. 축소된 나비에-스토크 방정식을 풀면 포물선 속도 프로파일이 생성됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">

이 프로파일은 점성 저항이 중앙에서 최대에서 관 벽에서 0으로 유체 속도를 감소시킨다는 것을 보여줍니다. 미끄러짐 없는 조건이라고 알려진 경계에서의 이 0 속도는 유체와 관 벽 사이의 마찰로 인해 발생합니다. 결과적으로 흐름은 층류이며 유체 입자는 측면 혼합 없이 평행한 층으로 이동합니다.

관의 단면적에 대한 속도 프로파일을 적분하면 푸아죄유 법칙이라고 알려진 체적 유량 Q가 생성됩니다.

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">

여기서 ΔP는 관 길이 L에 걸친 압력 차이이고 μ는 유체의 동적 점도입니다. 이 관계는 Q가 관의 반경에 매우 민감하여 R의 4제곱에 따라 증가함을 보여줍니다. 따라서 반경이 약간만 증가하더라도 유량이 크게 증가하는데, 이는 유체 수송 응용 분야에서 중요한 원리입니다.

푸아죄유 법칙에 따르면 유체의 평균 속도 V는 다음과 같이 표현할 수 있습니다.

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

이 평균 속도는 관 중앙의 최대 속도의 정확히 절반으로, 포물선 속도 분포를 확인하고 원통형 관에서 층류의 예측 가능한 계층적 특성을 보여줍니다.

Hagen-Poiseuille 흐름으로 알려진 일정한 반경 R의 직선 원통형 튜브를 통한 안정적이고 압축되지 않는 흐름을 고려하십시오.
원통형 ...

Education

교육

JoVE Core

Civil Engineering

토목공학

Unit 1

제1단원

Differential Analysis of Fluid Flow

유체 흐름의 미분 분석

KEY TERMS AND CONCEPTS

핵심 용어 및 개념

parabolic flow in circular tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries.

Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves only along the z-axis and changes with distance r from the center of the tube.

Under these conditions, the Navier-Stokes equations become simpler. With no angular or radial velocity components, the axial velocity vz(r) depends only on r. Solving the reduced equations gives a parabolic velocity profile. The fluid moves fastest at the center and slows toward the wall because viscous drag reduces the speed.

At the tube wall, the velocity is zero. This is the no-slip condition, which comes from friction between the fluid and the wall. The result is laminar flow, where fluid particles travel in parallel layers with no lateral mixing.

Integrating the velocity profile over the tube cross section gives the volumetric flow rate Q. This is Poiseuille’s law. In this relation, ΔP is the pressure difference across tube length L, and μ is the fluid’s dynamic viscosity. The flow rate depends strongly on the tube radius and increases with the fourth power of R, so even a small increase in radius can greatly raise Q.

Poiseuille’s law also gives the mean velocity V of the fluid. The average velocity is exactly half of the maximum velocity at the tube center. This matches the parabolic profile and shows the predictable, layered nature of flow in a circular tube.

Parabolic Flow in Circular Tubes

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Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries. Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves...

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key terms and concepts

Explore how Hagen-Poiseuille flow creates a parabolic velocity profile in circular tubes, with no-slip at the wall and Poiseuille’s law for flow rate.

Parabolic Flow in Circular Tubes | 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

Parabolic Flow in Circular Tubes

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