Name: reynolds number from buckingham pi
Uploaded: 2025-02-13T15:43:29.000Z
Duration: 1 min 9 s
Channel: JoVE

Considere o fluxo através de um tubo, onde as principais variáveis incluem o diâmetro D do tubo, a velocidade do fluido V, a densidade ρ e a viscosidade μ.
Para analisar o sistema usando o teorema de Buckingham Pi, essas variáveis podem ser combinadas em produtos adimensionais como o número de Reynolds, que caracteriza o comportamento do fluxo.
Nesse caso, o teorema de Buckingham Pi é aplicado identificando primeiro o número de variáveis envolvidas e as dimensões de referência, que são massa M, comprimento L e tempo T.
De acordo com o teorema de Buckingham Pi, o sistema com quatro variáveis e três dimensões de referência (M, L, T) resulta em um termo Pi adimensional, que é o número de Reynolds.
Aqui, o número de Reynolds é um produto adimensional que relaciona as forças inerciais com as forças viscosas no fluido, ajudando a classificar o fluxo como laminar ou turbulento.
O teorema de Buckingham Pi reformula o sistema expressando as variáveis como termos adimensionais, facilitando a generalização e análise do fluxo sob condições variadas.

o teorema pi de buckingham

The Buckingham Pi Theorem

O teorema Pi de Buckingham fornece um método estruturado para simplificar problemas de dinâmica de fluidos reduzindo sistemas complexos de variáveis ​​a termos adimensionais.

<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;">


Ao analisar o fluxo por meio de um tubo, as variáveis ​​que definem o sistema incluem o diâmetro do tubo, a velocidade do fluido, sua densidade e viscosidade. Elas podem ser combinadas em um único termo adimensional, o número de Reynolds (Re), que captura sua relação essencial.

<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;">


O primeiro passo na aplicação do teorema Pi de Buckingham é identificar as dimensões fundamentais de cada variável. Na mecânica dos fluidos, elas são tipicamente massa, comprimento e tempo. Aqui, o diâmetro representa o comprimento, a velocidade combina comprimento e tempo, a densidade abrange massa e volume, e a viscosidade incorpora massa, comprimento e tempo.

<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;">


O teorema sugere que o número de grupos adimensionais independentes, conhecidos como termos Pi, é a diferença entre o número de variáveis ​​k e o número de dimensões fundamentais r. Apenas um termo adimensional é necessário com quatro variáveis ​​e três dimensões, simplificando a análise.

<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;">

Este único termo, o número de Reynolds, compara forças inerciais e forças viscosas no fluido. Estabelecer esta razão revela se o fluxo é provavelmente suave e ordenado, conhecido como fluxo laminar, ou caótico e irregular, conhecido como fluxo turbulento. Valores mais baixos indicam fluxo laminar, onde forças viscosas dominam, e o fluido se move em camadas paralelas. Valores mais altos refletem fluxo turbulento, onde forças inerciais levam a movimento caótico. A aplicação do teorema Pi de Buckingham neste contexto reduz interações complexas a um único termo, facilitando generalizar o comportamento do fluxo sob condições variáveis.

O Teorema Pi de Buckingham

Considere o fluxo através de um tubo, onde as principais variáveis incluem o diâmetro D do tubo, a velocidade do fluido V, a densidade ρ e a viscosidade μ.
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reynolds number from buckingham pi

The Buckingham Pi theorem turns fluid flow variables into dimensionless terms. In a pipe flow example, the key variables are the pipe diameter, fluid velocity, density, and viscosity. These variables can be combined into one dimensionless group, the Reynolds number, which captures their relationship.

Applying the theorem starts with identifying the fundamental dimensions of each variable. In fluid mechanics, these dimensions are mass, length, and time. The pipe diameter represents length, velocity combines length and time, density includes mass and volume, and viscosity includes mass, length, and time.

The theorem also shows how many independent dimensionless groups, or Pi terms, are needed. That number is found by subtracting the number of fundamental dimensions, r, from the number of variables, k. With four variables and three dimensions, only one dimensionless term is needed.

That single term is the Reynolds number, often written as Re. It compares inertial forces to viscous forces in the fluid. Low values point to laminar flow, where the fluid moves in smooth, parallel layers. Higher values point to turbulent flow, where motion becomes chaotic and irregular.

This makes the Buckingham Pi theorem useful for simplifying fluid dynamics problems. It reduces several linked variables to one term that can be used to compare flow behavior under different conditions. That simplification makes it easier to generalize how fluids move through a pipe.

Reynolds Number from Buckingham Pi

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Reynolds Number from Buckingham Pi

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