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

Nel movimento di un fluido in condizioni viscose, lo sforzo di taglio è direttamente proporzionale alla deformazione del fluido. Nei fluidi newtoniani incomprimibili, questa sollecitazione varia linearmente con la velocità di deformazione.
Le sollecitazioni normali dipendono dalla pressione e dai tassi di deformazione in direzioni specifiche, definendo il comportamento del flusso del fluido a pressioni diverse.
Le sollecitazioni di taglio agiscono tangenzialmente, descrivendo come diversi strati di fluido scivolano l'uno sull'altro, collegando le sollecitazioni con le variazioni di velocità tra gli strati.
Sostituendo queste relazioni di stress in equazioni differenziali dei moti si formano le equazioni di Navier-Stokes, che bilanciano forze come l'inerzia, la pressione e la gravità in un fluido viscoso.
Il termine inerzia cattura l'accelerazione del fluido, mostrando come un fluido in movimento resiste a cambiamenti improvvisi nella sua velocità o direzione.
I gradienti di pressione guidano il movimento del fluido dalle aree ad alta pressione a quelle a bassa pressione, mentre i termini viscosi rappresentano l'attrito interno all'interno del fluido.
Ogni equazione direzionale cattura le forze interne, come la viscosità, e le forze esterne, come la gravità, prevedendo la risposta del fluido in varie condizioni.
Le equazioni di Navier-Stokes possono essere semplificate per scenari di flusso stazionario o laminare, consentendo un'analisi più semplice in scenari controllati come lo strato limite e i flussi di Couette.

equazioni di navier-stokes

Navier&#8211;Stokes Equations

Per i fluidi newtoniani incomprimibili, dove la densità rimane costante, gli sforzi mostrano una relazione lineare con la velocità di deformazione, definita da sforzi normali e di taglio. Gli sforzi normali, dipendono dalla pressione esercitata sul fluido e dalla velocità di deformazione in direzioni specifiche, che determinano come il fluido scorre sotto pressioni variabili. Gli sforzi di taglio, d'altro canto, agiscono tangenzialmente attraverso gli strati di fluido. Essi spiegano come gli strati di fluido adiacenti scivolano l'uno rispetto all'altro, collegando lo sforzo tangenziale alle differenze di velocità tra gli strati. Gli sforzi normali e di taglio definiscono le forze all'interno di un fluido che ne influenzano il suo movimento.

Le equazioni di Navier-Stokes sono ricavate incorporando queste relazioni di sforzo, in equazioni differenziali del moto.

<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;">

Queste equazioni, bilanciano le forze inerziali, di pressione e gravitazionali in un fluido viscoso e sono fondamentali per prevedere il comportamento del fluido in varie condizioni. Il termine di inerzia nelle equazioni di Navier-Stokes, tiene conto dell'accelerazione del fluido, che rappresenta la resistenza del fluido alle variazioni di velocità. Questo termine, è fondamentale per catturare la conservazione della quantità di moto di un fluido in movimento, poiché dimostra come i fluidi resistono a bruschi cambiamenti di velocità o direzione.

Il termine del gradiente di pressione nelle equazioni di Navier-Stokes, guida il movimento del fluido creando una forza netta diretta dalle aree di alta pressione verso quelle di bassa. D’altro canto, i termini viscosi, rappresentano l'attrito interno derivante dalle interazioni molecolari del fluido. A seconda dei gradienti di velocità presenti, queste forze viscose agiscono per rallentare il movimento del fluido.

Ogni equazione direzionale di Navier-Stokes, cattura forze interne come lo sforzo viscoso e forze esterne come la gravità. Queste equazioni, consentono di prevedere il movimento del fluido in varie condizioni, dai flussi semplici a quelli complessi. Per scenari di flusso costante o laminare, in cui il flusso è regolare e ordinato, le equazioni diventano più gestibili, consentendo un'analisi più semplice. Questa semplificazione è utile in scenari controllati, come il flusso in strato limite, in cui il flusso vicino a una superficie, presenta uno strato strutturato, e nel flusso di Couette, che descrive un fluido tra due superfici parallele, che si muovono a velocità diverse.

Equazioni di Navier-Stokes

Nel movimento di un fluido in condizioni viscose, lo sforzo di taglio è direttamente proporzionale alla deformazione del fluido. Nei fluidi newtoniani incomprimibili, questa sollecitazione ...

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Differential Analysis of Fluid Flow

Analisi differenziale del flusso di fluidi

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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...

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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.

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Parabolic Flow in Circular Tubes

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Pressure Drop in Oil Pipe Design

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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.

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Fluid Stress and Flow Equations

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