Name: irrotational flow near boundaries
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

Il flusso irrotazionale si riferisce al movimento di particelle fluide che non ruotano lungo il percorso del flusso, caratterizzato da un campo di velocità con curvatura zero e nessuna vorticità.
Un flusso uniforme con una sola componente diversa da zero della velocità, una costante u=U, è un esempio di flusso irrotazionale perché manca di gradienti di velocità.
Quando un oggetto solido viene introdotto in un flusso uniforme, il flusso rimane irrotazionale tranne che vicino ai contorni. Vicino alla sua superficie, la velocità aumenta bruscamente da zero, creando gradienti ripidi e una significativa sollecitazione di taglio.
Nel caso di un flusso veramente invisibile, il flusso scivolerebbe oltre il confine, preservando il movimento irrotatorio. Tuttavia, nei fluidi reali, la viscosità forma uno strato limite vicino a superfici fisse, producendo effetti rotazionali e talvolta causando la separazione a valle.
Nel flusso del tubo, la regione di ingresso presenta inizialmente un nucleo irrotazionale con flusso uniforme, ma quando uno strato limite si forma lungo la parete del tubo e si ispessisce, passa in una regione completamente sviluppata dominata da forze viscose.
Supponendo che il flusso sia irrotazionale, incomprimibile, stazionario e invisibile, l'equazione di Bernoulli può essere applicata nel campo del flusso.

flusso irrotazionale

Irrotational Flow

Il flusso irrotazionale, è caratterizzato da un movimento del fluido, in cui le particelle non ruotano attorno ai propri assi, con conseguente vorticità zero. Affinché un flusso sia irrotazionale, il rotore del campo di velocità, deve essere zero. Ciò impone condizioni specifiche sui gradienti di velocità. Ad esempio, per mantenere una rotazione zero attorno all'asse z, la condizione del gradiente sarà:

<img alt="Equation 1" src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;">

questa, deve essere soddisfatta, insieme a condizioni simili per altri assi:

<img alt="Equation 2" src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;">

Un flusso uniforme, in cui u è costante e v e w sono entrambi zero, soddisfa queste condizioni e funge da semplice esempio di flusso irrotazionale. Tuttavia, quando viene introdotto un oggetto solido, l'irrotazionalità persiste solo lontano dall'oggetto. Vicino al confine, la velocità aumenta bruscamente da zero (a causa della condizione di non slittamento), al valore del flusso libero, creando una significativa sollecitazione di taglio e uno strato limite nei fluidi viscosi.

Quando il fluido entra in un tubo da un grande serbatoio, il flusso iniziale è quasi uniforme, con conseguente nucleo irrotazionale, in cui la distribuzione della velocità è piatta. Mentre il fluido si sposta a valle, si forma uno strato limite lungo le pareti del tubo a causa della viscosità, causando lo sviluppo di gradienti nel profilo di velocità, che alla fine portano a un profilo parabolico completamente sviluppato. In questa regione completamente sviluppata, gli effetti rotazionali dominano vicino alle pareti, mentre un nucleo centrale, può rimanere approssimativamente irrotazionale.

Nella regione irrotazionale iniziale del flusso del tubo, l'equazione di Bernoulli

<img alt="Equation 3" src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;">

può essere applicata tra punti lungo una linea di flusso, poiché valgono le condizioni di incomprimibilità, flusso inviscidio e irrotazionalità. Tuttavia, man mano che lo strato limite cresce e gli effetti rotazionali diventano significativi vicino alle pareti, le ipotesi dell'equazione di Bernoulli crollano. Ciò significa che l’equazione di Bernoulli, è applicabile solo nella regione di ingresso e nel nucleo irrotazionale, ma non nella regione rotazionale completamente sviluppata, dove dominano gli effetti della viscosità.

Flusso Irrotazionale

Il flusso irrotazionale si riferisce al movimento di particelle fluide che non ruotano lungo il percorso del flusso, caratterizzato da un campo di velocità con curvatura zero e ...

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Differential Analysis of Fluid Flow

Analisi differenziale del flusso di fluidi

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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 far from the object. Near the surface, the fluid speed rises sharply from zero because of the no-slip condition. This creates strong shear stress and a boundary layer in viscous fluids.

A similar change happens when fluid enters a pipe from a large reservoir. At first, the flow is nearly uniform, so the center region forms an irrotational core with a flat velocity profile. Farther downstream, viscosity causes boundary layers to grow along the pipe walls.

As the flow continues, the velocity profile develops more gradient and becomes fully developed and parabolic. In that region, rotational effects are strongest near the walls, while the center may still be approximately irrotational. Bernoulli's equation can apply in the entrance region and in the irrotational core, where the flow is incompressible, inviscid, and irrotational. It no longer applies well in the fully developed region, where viscosity and rotation dominate.

Irrotational Flow Near Boundaries

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...

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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 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...

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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

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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...

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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...

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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.

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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...

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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 irrotational flow behaves near solids and in pipe entrance flow, including boundary layers, rotational effects, and when Bernoulli's equation applies.

Irrotational Flow Near Boundaries | 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.

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Irrotational Flow Near Boundaries

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