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

Viskoz koşullar altında akışkan hareketinde, kayma gerilimi akışkan deformasyonu ile doğru orantılıdır. Sıkıştırılamaz Newton sıvılarında, bu gerilme deformasyon hızı ile doğrusal olarak değişir.
Normal gerilmeler, farklı basınçlar altında akışkan akış davranışını tanımlayan belirli yönlerdeki basınç ve deformasyon oranlarına bağlıdır.
Kayma gerilmeleri teğetsel olarak hareket eder, farklı akışkan katmanlarının birbirinin yanından nasıl geçtiğini açıklar ve gerilimi katmanlar arasındaki hız değişiklikleriyle ilişkilendirir.
Bu gerilme ilişkilerini hareketlerin diferansiyel denklemlerine koymak, viskoz bir sıvıda atalet, basınç ve yerçekimi gibi kuvvetleri dengeleyen Navier-Stokes denklemlerini oluşturur.
Atalet terimi, hareket eden bir sıvının hızındaki veya yönündeki ani değişikliklere nasıl direndiğini gösteren sıvı ivmesini yakalar.
Basınç gradyanları, sıvının yüksek basınçtan düşük basınçlı alanlara hareketini sağlarken, viskoz terimler sıvı içindeki iç sürtünmeyi temsil eder.
Her yönlü denklem, viskozite gibi iç kuvvetleri ve yerçekimi gibi dış kuvvetleri yakalayarak çeşitli koşullarda sıvı tepkisini tahmin eder.
Navier-Stokes denklemleri, kararlı veya laminer akış senaryoları için basitleştirilebilir ve sınır tabakası ve Couette akışları gibi kontrollü senaryolarda daha basit analizlere olanak tanır.

navier-stokes denklemleri

Navier&#8211;Stokes Equations

Yoğunluğun sabit kaldığı sıkıştırılamaz Newton akışkanları için, gerilmeler normal ve kayma gerilmeleriyle tanımlanan deformasyon oranıyla doğrusal bir ilişki gösterir. Normal gerilmeler, akışkana uygulanan basınca ve belirli yönlerdeki deformasyon oranına bağlıdır ve bu da akışkanın değişen basınçlar altında nasıl aktığını belirler. Öte yandan kayma gerilmeleri, akışkan katmanları arasında teğetsel olarak etki eder. Bunlar, bitişik akışkan katmanlarının birbirlerine göre nasıl kaydığını açıklayarak teğetsel gerilmeyi katmanlar arasındaki hız farklılıklarına bağlar. Normal ve kayma gerilmeleri, akışkanın hareketini etkileyen akışkan içindeki iç kuvvetleri tanımlar.

 Navier-Stokes denklemleri, bu gerilme ilişkilerinin diferansiyel hareket denklemlerine dahil edilmesiyle türetilir.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

 Bu denklemler, viskoz bir akışkandaki eylemsizlik, basınç ve yerçekimi kuvvetlerini dengeler ve çeşitli koşullar altında akışkan davranışını tahmin etmek için temeldir. Navier-Stokes denklemlerindeki eylemsizlik terimi, akışkanın hızdaki değişikliklere karşı direncini temsil eden akışkan ivmesini hesaba katar. Bu terim, akışkanların hız veya yöndeki ani değişimlere nasıl direndiğini gösterdiği için hareket eden bir akışkanın momentum korunumunu yakalamada kritik öneme sahiptir.

 Navier-Stokes denklemlerindeki basınç gradyanı terimi, yüksek basınçlı alanlardan düşük basınçlı alanlara net bir kuvvet oluşturarak akışkan hareketini yönlendirir. Bu arada, viskoz terimler akışkan içindeki moleküler etkileşimlerden kaynaklanan iç sürtünmeyi temsil eder. Mevcut hız gradyanlarına bağlı olarak, bu viskoz kuvvetler akışkan hareketini yavaşlatmak için hareket eder.

 Her yönlü Navier-Stokes denklemi, viskoz stres gibi iç kuvvetleri ve yerçekimi gibi dış kuvvetleri yakalar. Bu denklemler, basit akışlardan karmaşık akışlara kadar çeşitli koşullarda akışkan hareketini tahmin etmeyi sağlar. Akışın düzgün ve düzenli olduğu sabit veya laminer akış senaryoları için, denklemler daha yönetilebilir hale gelir ve daha basit analiz sağlar. Bu basitleştirme, bir yüzeyin yakınındaki akışın yapılandırılmış bir katmana sahip olduğu sınır tabakası akışı ve farklı hızlarda hareket eden iki paralel yüzey arasındaki sıvıyı tanımlayan Couette akışı gibi kontrollü senaryolarda faydalıdır.

Navier-Stokes Denklemleri

Viskoz koşullar altında akışkan hareketinde, kayma gerilimi akışkan deformasyonu ile doğru orantılıdır. Sıkıştırılamaz Newton sıvılarında, bu stres ...

Education

JoVE Core

Civil Engineering

İnşaat Mühendisliği

Unit 1

Ünite 1

Differential Analysis of Fluid Flow

Akışkan Akışının Diferansiyel Analizi

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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...

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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 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

Fluid Stress and Flow Equations

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