Name: pressure and velocity in inviscid flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

移動する流体では、粘度によってせん断応力が発生します。粘度の低い水の場合、これらの影響は無視できるため、流体運動解析が簡素化されます。
せん断応力が無視できる場合、流体は非粘性、または摩擦がないと見なされます。非粘性流体では、垂直応力はすべての方向で同じであり、圧力は応力方向に依存しません。
圧力は垂直応力の負であり、圧縮応力が正の圧力値になることを保証します。
オイラー方程式として知られる非粘性流れの一般的な運動方程式は、運動量保存則を説明しています。これらの方程式は、流体要素に適用されるニュートンの第 2 法則を表しており、速度の変化を流体に作用する力に関連付けます。
オイラー方程式は、圧力勾配、重力などの物体力、および流体速度の変化を駆動する局所的および対流的な加速度を考慮しています。
粘性力を無視して解析を簡略化しますが、偏微分方程式の非線形速度項のために解くのは依然として困難です。
オイラー方程式を流線に沿って統合すると、ベルヌーイ方程式が導き出され、非粘性流れの圧力、速度、仰角の変化に関する洞察が得られ、流体力学の重要なツールになります。

オイラーの運動方程式

Euler&#39;s Equations of Motion

流体力学では、せん断応力は粘性から生じます。粘性は、流体の変形に対する内部抵抗を表します。水などの低粘性流体の場合、これらの応力は最小限であるため、流体を非粘性、つまり摩擦のない流体として扱うことができるため、流れの解析が簡素化されます。非粘性流体では、せん断応力は存在せず、流体要素に対して垂直に作用する法線応力のみが残ります。特に、法線応力の負として定義される圧力は、さまざまな方向にわたって均一であり、応力の方向の影響を受けません。

 非粘性流体の運動は、流体力学に適用されるニュートンの第 2 法則から派生したオイラー方程式によって決まります。これらの方程式は、速度の変化率を流体要素に作用する力に関連付けることで、運動量保存を記述します。非粘性流体の場合、関連する力には、圧力勾配と重力などの体積力が含まれます。オイラー方程式は、粘性項を除いたこれらの力のバランスを次のように表します。

<img alt="Equation 1" src="/files/ftp_upload/18057/18057_Equation_1.svg" style="text-align: center; height: 30px;">

 ここで、ρ は流体の密度、v は速度ベクトル、p は圧力、g​ は重力加速度を表します。

 オイラー方程式は粘性を無視しますが、流体粒子の対流加速を考慮した非線形項があるため複雑です。これらの方程式を流線に沿って積分すると、ベルヌーイ方程式が得られます。この方程式は、非粘性流の圧力、速度、および高度を結び付ける流体力学の基本的な関係を表します。ベルヌーイ方程式は、流体の流れ内のエネルギー分布を分析するための強力なツールであり、流体輸送、空気力学、および流体力学の応用を支えます。

移動する流体では、粘度によってせん断応力が発生します。粘度の低い水の場合、これらの影響は無視でき、流体の動きを簡素化します...

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

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 same in different directions and does not depend on how the stress is oriented.

Euler's equations come from Newton's second law applied to fluid motion. They describe momentum conservation by linking the change in velocity to the forces on a fluid element. For inviscid flow, those forces are pressure gradients and body forces such as gravity. The equations include density, velocity, pressure, and gravitational acceleration, but they leave out viscous terms.

Even without viscosity, Euler's equations are still nonlinear. That means they are complex because they include convective acceleration, which follows the motion of fluid particles. When the equations are integrated along a streamline, they lead to Bernoulli's equation. Bernoulli's equation connects pressure, velocity, and elevation in inviscid flows and is useful for fluid transport, aerodynamics, and hydrodynamics.

Pressure and Velocity in Inviscid Flow

euler&#39;s equations of motion

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

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 pressure, velocity, and gravity shape inviscid fluid motion through Euler's equations, and how they lead to Bernoulli's equation along a streamline.

Pressure and Velocity in Inviscid Flow | 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

Pressure and Velocity in Inviscid Flow

Description

Transcript