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

점성 조건의 유체 모션에서 전단 응력은 유체 변형에 정비례합니다. 비압축성 뉴턴 유체에서 이 응력은 변형 속도에 따라 선형적으로 변합니다.
수직 응력은 특정 방향의 압력과 변형 속도에 따라 달라지며, 서로 다른 압력에서 유체 흐름 거동을 정의합니다.
전단 응력은 접선으로 작용하여 서로 다른 유체 레이어가 서로 미끄러지는 방식을 설명하고 응력을 레이어 전체의 속도 변화와 연결합니다.
이러한 응력 관계를 운동의 미분 방정식으로 대체하면 점성 유체에서 관성, 압력 및 중력과 같은 힘의 균형을 맞추는 Navier-Stokes 방정식이 형성됩니다.
관성 항은 유체 가속도를 포착하여 움직이는 유체가 속도나 방향의 급격한 변화에 어떻게 저항하는지 보여줍니다.
압력 구배는 고압에서 저압 영역으로 유체 이동을 유도하는 반면, 점성 항은 유체 내 내부 마찰을 나타냅니다.
각 방향 방정식은 점도와 같은 내부 힘과 중력과 같은 외부 힘을 포착하여 다양한 조건에서 유체 반응을 예측합니다.
Navier-Stokes 방정식은 정상 또는 층류 시나리오에 대해 단순화할 수 있으므로 경계층 및 Couette 흐름과 같은 제어된 시나리오에서 더 간단하게 해석할 수 있습니다.

navier-stokes 방정식

Navier&#8211;Stokes Equations

밀도가 일정하게 유지되는 비압축성 뉴턴 유체의 경우 응력은 수직 및 전단 응력으로 정의되는 변형률과 선형 관계를 보입니다. 수직 응력은 유체에 가해지는 압력과 특정 방향의 변형률에 따라 달라지며, 이는 다양한 압력 하에서 유체가 흐르는 방식을 결정합니다. 반면 전단 응력은 유체 층에 접선 방향으로 작용합니다. 이는 인접한 유체 층이 서로에 대해 어떻게 미끄러지는지 설명하며, 접선 응력을 층 간의 속도 차이와 연결합니다. 수직 및 전단 응력은 유체의 움직임에 영향을 미치는 유체 내부의 내부 힘을 정의합니다.

나비에-스토크스 방정식은 이러한 응력 관계를 운동의 미분 방정식에 통합하여 도출됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

이러한 방정식은 점성 유체에서 관성력, 압력력 및 중력력을 균형 있게 조절하며 다양한 조건에서 유체 거동을 예측하는 데 기본이 됩니다. 나비에-스토크스 방정식의 관성 항은 유체 가속도를 설명하며, 이는 유체의 속도 변화에 대한 저항을 나타냅니다. 이 항은 움직이는 유체의 운동량 보존을 포착하는 데 중요한데, 유체가 속도나 방향의 갑작스러운 변화에 어떻게 저항하는지 보여주기 때문입니다.

나비에-스토크스 방정식 내의 압력 기울기 항은 고압 영역에서 저압 영역으로 순 힘을 생성하여 유체 이동을 주도합니다. 한편, 점성 항은 유체 내의 분자 상호 작용으로 인해 발생하는 내부 마찰을 나타냅니다. 존재하는 속도 기울기에 따라 이러한 점성력은 유체 이동을 늦추는 역할을 합니다.

각 방향 나비에-스토크스 방정식은 점성 응력과 같은 내부 힘과 중력과 같은 외부 힘을 포착합니다. 이러한 방정식을 사용하면 간단한 흐름에서 복잡한 흐름에 이르기까지 다양한 조건에서 유체 운동을 예측할 수 있습니다. 흐름이 ​​매끄럽고 질서 있는 정상 또는 층류 시나리오의 경우 방정식이 더 관리하기 쉬워져 더 간단한 분석이 가능합니다. 이러한 단순화는 경계층 흐름과 같은 제어된 시나리오에서 유용합니다. 경계층 흐름은 표면 근처의 흐름이 구조화된 층을 갖는 경우이고, 쿠에트 흐름은 서로 다른 속도로 움직이는 두 평행 표면 사이의 유체를 설명하는 경우입니다.

Navier-Stokes 방정식

점성 조건의 유체 모션에서 전단 응력은 유체 변형에 정비례합니다. 비압축성 뉴턴 유체에서 이 응력은 ...

Education

교육

JoVE Core

Civil Engineering

토목공학

Unit 1

제1단원

Differential Analysis of Fluid Flow

유체 흐름의 미분 분석

KEY TERMS AND CONCEPTS

핵심 용어 및 개념

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...

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 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 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

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Fluid Stress and Flow Equations

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