Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

En el flujo incompresible bidimensional, hay componentes de velocidad horizontales y verticales.
La ecuación de continuidad asegura la conservación de la masa al requerir que la suma de las tasas de cambio de estos componentes sea cero, lo que significa que el flujo neto de entrada y salida en cualquier región está equilibrado.
Para satisfacer esta condición, se introduce la función de flujo.
La velocidad horizontal es la derivada parcial de la función de corriente con respecto a la dirección vertical, mientras que la velocidad vertical es la derivada parcial negativa con respecto a la dirección horizontal. Esto garantiza que la conservación de la masa se mantenga automáticamente.
Una característica clave de la función de corriente es que permanece constante a lo largo de las líneas de corriente, que representan las trayectorias de las partículas de fluido y son tangentes a las velocidades en el campo de flujo.
Las líneas aerodinámicas, a menudo representadas como curvas de nivel, permiten una interpretación intuitiva del comportamiento de los fluidos.
La función de flujo simplifica los cálculos de movimiento de fluidos. La diferencia en los valores de la función de corriente entre dos líneas de corriente representa el caudal volumétrico entre ellas.
Esto elimina la necesidad de resolver los componentes de velocidad por separado, lo que lo convierte en una herramienta eficiente para analizar el flujo bidimensional.

función de corriente

Stream Function

En el flujo bidimensional de un fluido incompresible, la ecuación de continuidad es fundamental para garantizar la conservación de la masa. Esto implica que cualquier cambio en la cantidad de fluido que entra o sale de una región se compensa con un cambio equivalente en otra parte del sistema. En el caso del flujo incompresible, donde la densidad permanece constante, esta condición se reduce a que la divergencia del campo de velocidad sea igual a cero. Matemáticamente, esto se expresa como: <img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;"> donde u y v representan los componentes de velocidad horizontal y vertical, respectivamente. Para satisfacer inherentemente esta ecuación, introducimos la función de corriente ψ, que permite definir los componentes de velocidad en términos de ψ como: <img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;"> Estas definiciones satisfacen automáticamente la ecuación de continuidad, ya que las derivadas parciales mixtas de ψ se cancelan. La función de corriente ψ es constante a lo largo de las líneas de corriente, que son los caminos que siguen las partículas de fluido dentro del flujo. Por lo tanto, las líneas de corriente se pueden visualizar como líneas de contorno de ψ, lo que proporciona una representación clara del movimiento del fluido. Cada línea de corriente es tangente al vector de velocidad en cualquier punto a lo largo de su trayectoria, lo que ilustra la dirección del flujo sin la necesidad de calcular las velocidades en cada punto individualmente. Además, la diferencia en los valores de ψ\psiψ entre dos líneas de corriente representa el caudal volumétrico por unidad de profundidad entre ellas, lo que permite el cálculo directo de los caudales. Esta propiedad simplifica significativamente el análisis de flujos incompresibles bidimensionales al eliminar la necesidad de resolver ecuaciones separadas para u y v. Como resultado, la función de corriente ψ es una herramienta poderosa para modelar e interpretar el comportamiento de los fluidos en aplicaciones que involucran flujo incompresible y constante.

Función de corriente

En el flujo incompresible bidimensional, hay componentes de velocidad horizontales y verticales.
La ecuación de continuidad asegura la conservación de la masa ...

Education

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JoVE Core

Civil Engineering

Ingeniería Civil

Unit 1

Unidad 1

Differential Analysis of Fluid Flow

Análisis Diferencial del Flujo de Fluidos

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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 to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible Flow

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

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...

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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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steady-laminar-flow-between-parallel-plates

Laminar Flow Profile Between Parallel Plates

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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

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design example: flow of oil through circular pipes

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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 the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible Flow | 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

Stream Function for Incompressible Flow

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