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

In fluid motion under viscous conditions, shear stress is directly proportional to fluid deformation. In incompressible Newtonian fluids, this stress varies linearly with the deformation rate.
Normal stresses depend on pressure and deformation rates in specific directions, defining fluid flow behavior under different pressures.
Shear stresses act tangentially, describing how different fluid layers slide past one another, connecting stress with velocity changes across layers.
Substituting these stress relationships into differential equations of motions forms the Navier-Stokes equations, which balance forces like inertia, pressure, and gravity in a viscous fluid.
The inertia term captures fluid acceleration, showing how a moving fluid resists sudden changes in its speed or direction.
Pressure gradients drive fluid movement from high-pressure to low-pressure areas, while viscous terms represent internal friction within the fluid.
Each directional equation captures internal forces, such as viscosity, and external forces, like gravity, predicting fluid response in various conditions.
The Navier-Stokes equations can be simplified for steady or laminar flow scenarios, allowing simpler analysis in controlled scenarios like boundary layer and Couette flows.

navier-stokes-vergelijkingen

Navier&#8211;Stokes Equations

Voor onsamendrukbare Newtoniaanse vloeistoffen, waarbij de dichtheid constant blijft, vertonen spanningen een lineaire relatie met de vervormingssnelheid, gedefinieerd door normale en schuifspanningen. Normale spanningen zijn afhankelijk van de druk die op de vloeistof wordt uitgeoefend en de snelheid van vervorming in specifieke richtingen, wat bepaalt hoe vloeistof stroomt onder wisselende druk. Schuifspanningen werken daarentegen tangentieel over vloeistoflagen. Ze verklaren hoe aangrenzende vloeistoflagen ten opzichte van elkaar glijden, waarbij tangentiële spanning wordt gekoppeld aan snelheidsverschillen tussen lagen. Normale en schuifspanningen definiëren de interne krachten in een vloeistof die de beweging ervan beïnvloeden. De Navier-Stokes-vergelijkingen worden afgeleid door deze spanningsrelaties op te nemen in differentiële bewegingsvergelijkingen.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

Deze vergelijkingen balanceren traagheids-, druk- en zwaartekrachtkrachten in een viskeuze vloeistof en zijn fundamenteel voor het voorspellen van vloeistofgedrag onder verschillende omstandigheden. De traagheidsterm in de Navier-Stokes-vergelijkingen houdt rekening met vloeistofversnelling, wat de weerstand van de vloeistof tegen veranderingen in snelheid weergeeft. Deze term is cruciaal bij het vastleggen van het behoud van impuls van een bewegende vloeistof, omdat het laat zien hoe vloeistoffen plotselinge veranderingen in snelheid of richting weerstaan. De drukgradiëntterm in de Navier-Stokes-vergelijkingen drijft vloeistofbeweging aan door een nettokracht te creëren van gebieden met hoge druk naar lage druk. Ondertussen vertegenwoordigen de viskeuze termen interne wrijving die voortkomt uit moleculaire interacties in de vloeistof. Afhankelijk van de aanwezige snelheidsgradiënten, werken deze viskeuze krachten om de vloeistofbeweging te vertragen. Elke directionele Navier-Stokes-vergelijking legt interne krachten vast zoals viskeuze spanning en externe krachten zoals zwaartekracht. Deze vergelijkingen maken het mogelijk om vloeistofbeweging te voorspellen in verschillende omstandigheden, van eenvoudige tot complexe stromingen. Voor scenario's met stabiele of laminaire stroming, waarbij de stroming soepel en ordelijk is, worden de vergelijkingen beter beheersbaar, wat een eenvoudigere analyse mogelijk maakt. Deze vereenvoudiging is nuttig in gecontroleerde scenario's zoals grenslaagstroming, waarbij stroming nabij een oppervlak een gestructureerde laag heeft, en in Couette-stroming, waarbij vloeistof tussen twee parallelle oppervlakken met verschillende snelheden beweegt.

Navier-Stokes-vergelijkingen

In fluid motion under viscous conditions, shear stress is directly proportional to fluid deformation. In incompressible Newtonian fluids, this stress ...

Education

JoVE Core

Civil Engineering

Civiele techniek

Unit 1

Differential Analysis of Fluid Flow

Differentiële analyse van vloeistofstroming

KEY TERMS AND CONCEPTS

KERNTERMEN EN CONCEPTEN

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

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