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

Bei der Fluidbewegung unter viskosen Bedingungen ist die Scherspannung direkt proportional zur Fluidverformung. In inkompressiblen newtonschen Fluiden variiert diese Spannung linear mit der Verformungsrate.
Normalspannungen hängen vom Druck und den Verformungsraten in bestimmten Richtungen ab und definieren das Strömungsverhalten des Fluids unter verschiedenen Drücken.
Schubspannungen wirken tangential und beschreiben, wie verschiedene Fluidschichten aneinander vorbeigleiten und Spannungen mit Geschwindigkeitsänderungen zwischen den Schichten verbinden.
Wenn man diese Spannungsbeziehungen in Differentialgleichungen von Bewegungen einsetzt, entstehen die Navier-Stokes-Gleichungen, die Kräfte wie Trägheit, Druck und Schwerkraft in einem viskosen Fluid ausgleichen.
Der Trägheitsterm erfasst die Flüssigkeitsbeschleunigung und zeigt, wie ein sich bewegendes Fluid plötzlichen Änderungen seiner Geschwindigkeit oder Richtung widersteht.
Druckgradienten treiben die Bewegung des Fluids vom Hochdruck- in den Niederdruckbereich an, während viskose Terme die innere Reibung innerhalb des Fluids darstellen.
Jede Richtungsgleichung erfasst Schnittgrößen wie die Viskosität und externe Kräfte wie die Schwerkraft und sagt die Reaktion des Fluids unter verschiedenen Bedingungen voraus.
Die Navier-Stokes-Gleichungen können für stationäre oder laminare Strömungsszenarien vereinfacht werden, was eine einfachere Analyse in kontrollierten Szenarien wie Grenzschicht- und Couette-Strömungen ermöglicht.

navier-stokes-gleichungen

Navier&#8211;Stokes Equations

Bei inkompressiblen newtonschen Flüssigkeiten, bei denen die Dichte konstant bleibt, zeigen Spannungen eine lineare Beziehung zur Verformungsrate, die durch Normal- und Scherspannungen definiert wird. Normalspannungen hängen vom auf die Flüssigkeit ausgeübten Druck und der Verformungsrate in bestimmten Richtungen ab, die bestimmt, wie Flüssigkeit unter unterschiedlichen Drücken fließt. Scherspannungen wirken dagegen tangential über Flüssigkeitsschichten hinweg. Sie erklären, wie benachbarte Flüssigkeitsschichten relativ zueinander gleiten, und verbinden tangentiale Spannung mit Geschwindigkeitsunterschieden zwischen Schichten. Normal- und Scherspannungen definieren die inneren Kräfte in einer Flüssigkeit, die ihre Bewegung beeinflussen.

Die Navier-Stokes-Gleichungen werden abgeleitet, indem diese Spannungsbeziehungen in Differentialgleichungen der Bewegung einbezogen werden.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

Diese Gleichungen gleichen Trägheits-, Druck- und Gravitationskräfte in einer viskosen Flüssigkeit aus und sind grundlegend für die Vorhersage des Flüssigkeitsverhaltens unter verschiedenen Bedingungen. Der Trägheitsterm in den Navier-Stokes-Gleichungen berücksichtigt die Flüssigkeitsbeschleunigung und stellt den Widerstand der Flüssigkeit gegen Geschwindigkeitsänderungen dar. Dieser Term ist entscheidend für die Erfassung der Impulserhaltung einer bewegten Flüssigkeit, da er zeigt, wie Flüssigkeiten plötzlichen Geschwindigkeits- oder Richtungsänderungen widerstehen.

Der Druckgradiententerm in den Navier-Stokes-Gleichungen treibt die Flüssigkeitsbewegung an, indem er eine Nettokraft von Bereichen mit hohem Druck zu Bereichen mit niedrigem Druck erzeugt. Die viskosen Terme stellen unterdessen die innere Reibung dar, die durch molekulare Wechselwirkungen innerhalb der Flüssigkeit entsteht. Abhängig von den vorhandenen Geschwindigkeitsgradienten wirken diese viskosen Kräfte, um die Flüssigkeitsbewegung zu verlangsamen.

Jede gerichtete Navier-Stokes-Gleichung erfasst innere Kräfte wie viskosen Druck und äußere Kräfte wie die Schwerkraft. Diese Gleichungen ermöglichen die Vorhersage der Flüssigkeitsbewegung unter verschiedenen Bedingungen, von einfachen bis hin zu komplexen Strömungen. Bei Szenarien mit gleichmäßiger oder laminarer Strömung, in denen die Strömung glatt und geordnet ist, werden die Gleichungen handhabbarer und ermöglichen eine einfachere Analyse. Diese Vereinfachung ist in kontrollierten Szenarien wie der Grenzschichtströmung nützlich, bei der die Strömung nahe einer Oberfläche eine strukturierte Schicht aufweist, und bei der Couette-Strömung, die eine Flüssigkeit zwischen zwei parallelen Oberflächen beschreibt, die sich mit unterschiedlicher Geschwindigkeit bewegen.

Navier-Stokes-Gleichungen

Bei der Fluidbewegung unter viskosen Bedingungen ist die Scherspannung direkt proportional zur Fluidverformung. In inkompressiblen newtonschen Fluiden ist diese Spannung ...

Education

Lehre

JoVE Core

Civil Engineering

Bauingenieurwesen

Unit 1

Einheit 1

Differential Analysis of Fluid Flow

Differentielle Analyse der Fluidströmung

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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 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

Description

Transcript

Tags