Name: linear shear between moving plates
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 22 s
Channel: JoVE

クエット流れは、2つの平行なプレート間の流体運動を表し、一方は静止し、もう一方は一定の速度で移動し、安定した層流せん断駆動の流れを作り出します。
このセットアップにより、ナビエ・ストークス方程式が単純化され、無次元形式で表すことができます。
圧力勾配に関連する無次元パラメータは、Couetteフロープロファイルの変動を定義します。
流れ方向に沿った圧力勾配がない場合、このパラメータはゼロになり、2つのプレート間の線速度変動を示します。
プレート間の流体速度プロファイルは線形で、静止プレートのゼロから移動プレートの速度まで増加します。
クエット流は、潤滑剤の薄い層が固定ハウジング内の回転シャフトを分離するウォーターポンプなどの軽負荷ジャーナルベアリングの流体挙動をモデル化するために一般的に使用されます。
これらの軽負荷ベアリングでは、圧力の影響が最小限に抑えられるため、流動特性は純粋なクエット流と非常によく似ています。
潤滑剤層のせん断応力と流量は、シャフトの回転速度とシャフトとハウジングの間のギャップ幅に依存します。

クエット流

Couette Flow

クエット流は、2 つの平行プレート間の流体の流れを表します。一方のプレートは固定され、もう一方のプレートは一定速度で移動します。この構成により、粘性と非圧縮性の条件下での流体の動きを規定するナビエ・ストークス方程式を使用した簡略化された解析が可能になります。クエット流の場合、流れ方向に圧力勾配がゼロの、安定した層流の非圧縮性流が前提となります。この流れタイプは、せん断駆動流を理解するのに役立ち、堆積物の輸送や浸食などといった潤滑や土木工学のシナリオでの応用に関連しています。

 クエット流の設定では、x 軸を上部プレートの移動方向に揃え、y 軸を両方のプレートに垂直にします。流れは安定しており、層流で、完全に発達しており、変化はありません。

 クエット流の場合、速度分布方程式は次のように表されます。

<img alt="Equation 1" src="/files/ftp_upload/18064/18064_Equation_1.svg" style="text-align: center; height: 30px;">

 ここで:

<ul style="list-style-type: disc; display: flex; flex-direction: column; gap: 10px; padding: 0 0 0 40px;">
 <li>U は上部プレートの一定速度、</li>
 <li>b はプレート間の距離、</li>
 <li>μ は流体の動粘性です。</li>
</ul>

 無次元形式では、両辺を U で割ると次のようになります。

<img alt="Equation 2" src="/files/ftp_upload/18064/18064_Equation_2.svg" style="text-align: center; height: 30px;">

 ここで、u/U は無次元速度、y/b は固定プレートからの正規化された距離を表します。特定の条件を分析するために、無次元パラメータ P を次のように定義します:

<img alt="Equation 3" src="/files/ftp_upload/18064/18064_Equation_3.svg" style="text-align: center; height: 30px;">

 これらの条件では、x 方向に圧力勾配がない場合 (∂p/∂x=0)、P はゼロになり、速度方程式は次のように簡略化されます:

<img alt="Equation 4" src="/files/ftp_upload/18064/18064_Equation_4.svg" style="text-align: center; height: 26px;">

 この線形プロファイルは、流体速度が静止プレートのゼロから移動プレートの U まで線形に増加し、流体層全体で均一なせん断速度になることを示しています。

クエットフローは、2つの平行なプレート間の流体運動を表し、一方は静止しており、もう一方は一定の速度で移動し、安定した...

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

linear shear between moving plates

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations.

The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates. These assumptions make the flow fully developed and easier to analyze.

For this case, the velocity changes with distance from the stationary plate. The upper plate speed is U, the plate spacing is b, and the fluid’s dynamic viscosity is μ. In dimensionless form, the velocity is written as u/U, and the position is written as y/b.

A dimensionless parameter P is used to describe pressure effects. When there is no pressure gradient in the x-direction, P becomes zero because ∂p/∂x = 0. The velocity equation then reduces to a straight line, so the fluid speed rises from zero at the fixed plate to U at the moving plate.

This linear profile creates a uniform shear rate across the fluid layer. Shear-driven flow like this is useful for understanding lubrication and civil engineering problems. It also helps explain processes such as sediment transport and erosion.

Linear Shear Between Moving Plates

euler&#39;s equations of motion

Pressure and Velocity in Inviscid Flow

Pressure and velocity in inviscid flow are linked by Euler's equations of motion. In fluid mechanics, viscosity creates shear stress, which is a force that acts along a surface. For low-viscosity fluids like water, these shear stresses are very small. That lets the fluid be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent. Only normal stresses remain, and these act perpendicular to a fluid element. Pressure is the negative of the normal stress. It stays the...

stream function

Stream Function for Incompressible Flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

irrotational flow

Irrotational Flow Near Boundaries

Irrotational flow is fluid motion in which particles do not spin around their own axes. It has zero vorticity, which means the curl of the velocity field is zero. That condition places limits on the velocity gradients in the flow. A simple example is uniform flow. In this case, the velocity u stays constant, while v and w are both zero. This satisfies the irrotational conditions and gives a basic model of non-rotating flow. When a solid object is added, irrotational flow usually remains only...

velocity potential

Velocity Potential in Pipe Flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes. A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V...

plane potential flows

Plane Potential Flow Patterns and Models

Plane potential flows describe fluid motion using simple flow patterns and models. They assume the fluid is irrotational and incompressible. These assumptions let the flow be written with a velocity potential function, φ, which gives the flow speed in a direction, and a stream function, ψ, which shows the flow path. Both are governed by Laplace's equation and are useful for estimating flow patterns, velocity distributions, and pressure fields around hydraulic structures. Uniform flow is the...

navier&#8211;stokes equations

Fluid Stress and Flow Equations

Fluid stress and flow equations describe how incompressible Newtonian fluids move under pressure, shear, and gravity. In these fluids, density stays constant, and stress changes linearly with the rate of deformation. The main stress types are normal stress and shear stress. Normal stress depends on pressure acting on the fluid and on deformation rate in specific directions. Shear stress acts along the surface of a fluid layer. It shows how neighboring layers slide past one another, so velocity...

steady, laminar flow between parallel plates

steady-laminar-flow-between-parallel-plates

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...

couette flow

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations. The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates.

steady, laminar flow in circular tubes

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...

design example: flow of oil through circular pipes

design-example-flow-of-oil-through-circular-pipes

Pressure Drop in Oil Pipe Design

Pressure drop in oil pipe design depends on pipe size, oil properties, and the flow rate needed through a circular pipe. For steady, incompressible, laminar flow, Hagen-Poiseuille's law gives an exact solution from the Navier-Stokes equations. It is used to find the pressure difference needed to keep oil moving through a pipeline section. The pressure drop equation uses the oil's dynamic viscosity, the pipeline length, the desired volumetric flow rate, and the pipe's inner radius. This...

key terms and concepts

Learn how fluid moves between two parallel plates when one plate is fixed and the other moves. See the linear velocity profile, shear rate, and key assumptions.

Linear Shear Between Moving Plates | 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

Linear Shear Between Moving Plates

Description

Transcript