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

ハーゲン・ポアズイユ流れとして知られる、一定の半径Rの直線円筒形チューブを通る安定した非圧縮性の流れを考えてみましょう。
円筒形の形状では、半径、角度、および軸方向の測定に座標r、θ、およびzを使用できます。
チューブの軸に平行に流れる流体は、純粋に軸方向の速度成分を持ち、半径方向の距離によってのみ変化します。
これらの条件により、ナビエ・ストークス方程式が単純化され、速度が中心で最も高く、壁に向かって減少し、境界でゼロに達することを示す放物線速度プロファイルが明らかになります。
これは、流体粒子が整然とした層内を滑らかに移動する層流を示しています。
このプロファイルをチューブの断面に積分すると、ポアズイユの法則が得られ、流量Qはチューブの半径、流体粘度、および圧力勾配の4乗に依存すると表されます。
Qから計算される平均速度Vは、チューブの中心で観測された最大速度の半分です。
この速度分布は、流れが中心で最も高く、壁で放物線状にゼロに減少することを確認します。

円管内の定常層流

Steady, Laminar Flow in Circular Tubes

ハーゲン・ポアズイユ流れは、一定の半径 R を持つ円筒管を通る粘性流体の定常非圧縮流を表します。この流れプロファイルは、毛細管などの狭いチャネル内の流体輸送を理解するためによく適用されます。これは層流の基本的な例として役立ちます。このモデルでは、円筒座標 (r、θ、z) を使用して、管内の半径 (r)、角度 (θ)、および軸方向 (z) の寸法を表します。ハーゲン・ポアズイユ流れの場合、速度プロファイルは完全に軸方向です。つまり、速度ベクトルは z 軸に沿ってのみ指し、管の中心からの半径距離 r のみで変化します。

流体運動の支配方程式であるナビエ・ストークス方程式は、これらの条件下では単純化されます。角速度成分または半径速度成分がない場合、軸方向速度 v_z(r) は r のみの関数として現れます。簡約ナビエ・ストークス方程式を解くと、放物線状の速度プロファイルが得られます。

<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;">

このプロファイルは、粘性抵抗によって流体の速度が中央の最大値から管壁のゼロまで低下することを示しています。境界でのこのゼロ速度は、滑りなし状態として知られ、流体と管壁の間の摩擦によって生じます。その結果、流れは層流となり、流体粒子は横方向の混合なしに平行層内を移動します。

管の断面積にわたって速度プロファイルを積分すると、ポアズイユの法則として知られる体積流量 Q が得られます。

<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;">

ここで、ΔP は管の長さ L にわたる圧力差、μ は流体の動粘性です。この関係は、Q が管の半径に非常に敏感で、R の 4 乗で増加することを示しています。したがって、半径がわずかに増加するだけでも流量が大幅に増加します。これは、流体輸送アプリケーションで重要な原則です。

ポアズイユの法則から、流体の平均速度 V は次のように表すことができます。

<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;">

この平均速度は管の中心での最大速度のちょうど半分であり、放物線状の速度分布を裏付け、円筒形管内の層流の予測可能な層状の性質を示しています。

ハーゲン・ポアズイユ流れとして知られる、一定の半径Rの直線円筒形チューブを通る安定した非圧縮性の流れを考えてみましょう。
円筒形の...

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Differential Analysis of Fluid Flow

流体流れの微分解析

KEY TERMS AND CONCEPTS

主要用語と概念

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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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.

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civil engineering

Parabolic Flow in Circular Tubes

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