Name: fluid stress and flow equations
Uploaded: 2025-02-13T15:43:26.000Z
Duration: 1 min 28 s
Channel: JoVE

粘性条件下での流体運動では、せん断応力は流体の変形に正比例します。非圧縮性ニュートン流体では、この応力は変形速度に比例して変化します。
垂直応力は、特定の方向の圧力と変形速度に依存し、さまざまな圧力下での流体の流れの挙動を定義します。
せん断応力は接線方向に作用し、さまざまな流体層が互いに滑り落ちる方法を表し、応力と層間の速度変化を結び付けます。
これらの応力関係を微分運動方程式に代入すると、粘性流体の慣性、圧力、重力などの力のバランスをとるナビエ・ストークス方程式が形成されます。
慣性項は流体の加速度を捉え、移動する流体が速度や方向の急激な変化にどのように抵抗するかを示します。
圧力勾配は高圧領域から低圧領域への流体の動きを駆動し、粘性項は流体内の内部摩擦を表します。
各方向方程式は、粘度などの内力と重力などの外力を捕捉し、さまざまな条件での流体応答を予測します。
Navier-Stokes方程式は、定常流れまたは層流のシナリオで簡略化できるため、境界層やクエット流れなどの制御されたシナリオでの解析が容易になります。

ナビエ・ストークス方程式

Navier&#8211;Stokes Equations

密度が一定である非圧縮性ニュートン流体の場合、応力は、法線応力とせん断応力によって定義される変形率と線形関係を示します。法線応力は、流体に及ぼされる圧力と特定の方向の変形率に依存し、これにより、さまざまな圧力下で流体がどのように流れるかが決まります。一方、せん断応力は、流体層を横切って接線方向に作用します。これらは、隣接する流体層が互いにどのようにスライドするかを説明し、接線応力を層間の速度差に結び付けます。法線応力とせん断応力は、流体の動きに影響を与える流体内の内部力を定義します。

 ナビエ・ストークス方程式は、これらの応力関係を微分運動方程式に組み込むことによって導き出されます。

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

 これらの方程式は、粘性流体内の慣性力、圧力、重力のバランスを取り、さまざまな条件下での流体の挙動を予測するための基礎となります。ナビエ・ストークス方程式の慣性項は、流体の加速を考慮し、速度の変化に対する流体の抵抗を表します。この項は、流体が突然の速度または方向の変化に抵抗する方法を示すため、移動する流体の運動量保存を捉える上で重要です。

 ナビエ・ストークス方程式内の圧力勾配項は、高圧領域から低圧領域への正味の力を作成することで流体の動きを駆動します。一方、粘性項は、流体内の分子相互作用から生じる内部摩擦を表します。存在する速度勾配に応じて、これらの粘性力は流体の動きを遅くする働きをします。

 各方向のナビエ・ストークス方程式は、粘性応力などの内部力と重力などの外部力を捉えます。これらの方程式により、単純な流れから複雑な流れまで、さまざまな条件での流体の動きを予測できます。流れが滑らかで整然としている定常または層流シナリオでは、方程式がより扱いやすくなり、より簡単な分析が可能になります。この簡略化は、表面近くの流れに構造化された層がある境界層流れや、異なる速度で移動する 2 つの平行な表面間の流体を記述するクエット流などの制御されたシナリオで役立ちます。

粘性条件下での流体運動では、せん断応力は流体の変形に正比例します。非圧縮性ニュートン流体では、この応力は...

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

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 differences between layers matter.

These normal and shear stresses create the internal forces that control fluid motion. The Navier-Stokes equations are formed by putting these stress relations into differential equations of motion. They balance inertial, pressure, and gravitational forces in a viscous fluid and help predict how the fluid will behave.

The inertia term describes fluid acceleration and its resistance to changes in velocity. It is important for momentum conservation in moving fluids. The pressure gradient term pushes fluid from regions of high pressure to regions of low pressure, while viscous terms come from internal friction between molecules and slow the flow when velocity changes across the fluid.

Each directional Navier-Stokes equation includes both internal forces, such as viscous stress, and external forces, such as gravity. This makes the equations useful for many flow conditions, from simple to complex. For steady or laminar flow, where motion is smooth and orderly, the equations are easier to work with.

That simplification is helpful in boundary layer flow near a surface, where the fluid forms a structured layer. It is also useful for Couette flow, which describes fluid between two parallel surfaces moving at different velocities. These cases show how the same equations can describe controlled fluid motion in practical settings.

Fluid Stress and Flow Equations

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...

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navier&#8211;stokes 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

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couette flow

Linear Shear Between Moving Plates

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steady, laminar flow in circular tubes

Parabolic Flow in Circular Tubes

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design example: flow of oil through circular pipes

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

Pressure Drop in Oil Pipe Design

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key terms and concepts

Learn how normal stress, shear stress, and the Navier-Stokes equations explain fluid motion, pressure forces, viscosity, boundary layers, and Couette flow.

Fluid Stress and Flow Equations | JoVE K12

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civil engineering

Fluid Stress and Flow Equations

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