Name: laminar flow profile between parallel plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 17 s
Channel: JoVE

2つの平行なプレート間の安定した層流は、狭くて閉じ込められたチャネル内の流体の動きを表し、単純な流体システムのモデル化によく使用されます。
2つの固定プレート間の流れの場合、速度はプレートに対して垂直にしか変化せず、流れ方向に沿った加速はないため、ナビエ・ストークス方程式は単純化されます。
これにより、放物線速度プロファイルが生じ、流体は中心で最も速く移動し、摩擦によりプレートでゼロまで減速します(滑り止め条件と呼ばれます)。
プレート間の流量は、圧力勾配、流体粘度、およびプレートの分離に依存します。ギャップを広げるか、圧力勾配を大きくすると、流量が増加します。
平均流速は圧力降下にも関連しており、チャネルの中心のピーク速度は平均速度の1.5倍です。
この解析は、層流状態を示す低いレイノルズ数に対してのみ有効であり、乱流の高レイノルズ数の流れに対しては失敗します。
この解析のアプリケーションには、層流状態が存在する水処理プラントの灌漑用水路、地下水流、および沈殿流域が含まれます。

平行板間の安定した層流

Steady, Laminar Flow Between Parallel Plates

平行板間の安定した層流を理解することは、さまざまな水輸送システムや排水システムでよく見られる狭い長方形のチャネル内の流れを解析および設計するために不可欠です。ナビエ-ストークス方程式は流体の動きを支配しますが、非線形性のため一般に解くのが困難です。ただし、平行板間の安定した層流など、特定のケースでは簡略化が可能です。このシナリオでは、安定した非圧縮性層流を想定しています。

 2 つの水平プレートが x 軸に沿って整列し、距離「2h」の間隔を置いており、y 軸がプレートに垂直である場合、流体の動きは平行であると想定されます。ここで、連続方程式は x 軸に沿った速度の変化がないことを示しており、ナビエ-ストークス方程式は次のように簡約されます。

<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;">

 y 方向と z 方向の方程式はこれらの軸に沿った圧力勾配を示していないため、圧力は y 方向に静水圧的に変化し、積分により次の式が得られます。

<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;">

 x 軸に沿って圧力勾配がある平行プレートの場合、方程式は次のように簡略化されます。

<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;">

 2 回積分し、境界条件 (y = +h および y = -h で u = 0) を適用すると、次の式が得られます。

<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;">

 この式から、プレート間の差全体の速度プロファイルは放物線状であり、プレート間の中間の y = 0 で最大速度になることがわかります。プレートの単位幅を通る体積流量は、差全体の速度プロファイルを積分することで得られます。

 平行プレート間の層流を理解することは、地下水の流れ、廃水処理、潤滑に実用的に応用できます。たとえば、このモデルは、プレート間の流れのように水が土粒子間を移動する土壌層内の流れを近似できます。流量と速度分布を知ることで、エンジニアは効率的な排水システムを設計し、地下水の動きを予測できます。

2つの平行なプレート間の安定した層流は、狭くて閉じ込められたチャネル内の流体の動きを表し、単純な流体システムのモデル化によく使用されます。
対して。。。

steady-laminar-flow-between-parallel-plates

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

laminar flow profile between parallel plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings.

The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem becomes simpler because the flow is steady, incompressible, and laminar. Two horizontal plates are placed along the x-axis, and the gap between them is 2h. The y-axis is perpendicular to the plates, and the fluid moves parallel to the plates.

With these assumptions, the continuity equation shows that the velocity does not change along the x-axis. The Navier-Stokes equations also simplify in the y and z directions. They show no pressure gradient along those axes, so pressure changes hydrostatically in the y-direction. After integration, the x-direction equation includes the pressure gradient along the plates.

Solving that equation requires integrating twice and applying the no-slip boundary conditions, where u = 0 at y = +h and y = -h. The result is a parabolic velocity profile across the gap. The fluid moves fastest at y = 0, the midpoint between the plates. The volumetric flow rate for a unit width of the plates is then found by integrating the velocity profile across the space.

This flow model has practical uses in groundwater flow, wastewater treatment, and lubrication. It can also represent water moving through soil layers, where the flow resembles movement between plates. Engineers use the velocity distribution and flow rate to help design drainage systems and predict groundwater movement.

Laminar Flow Profile Between Parallel Plates

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Learn how steady laminar flow between parallel plates leads to a parabolic velocity profile, pressure-gradient behavior, and flow-rate calculations.

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Laminar Flow Profile Between Parallel Plates

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