Teorema de Bridgman

O teorema de Bridgman afirma que as únicas funções que podem ter argumentos dimensionais são produtos de potências das grandezas de base de um determinado sistema de unidades:

[F]= Mα . Lβ . Tγ

Onde:

M = dimensão de massa
L = dimensão de comprimento
T = dimensão de tempo

Veja a tabela de dimensões e alguns exemplos sobre este teorema:

Quantidade símbolo Dimensões
Comprimento (l) (L)
Tempo (t) (T)
Massa (m) (M)
Velocidade (V) (M0 . L1 . T-1)
Aceleração (a) (M0 . L1 . T-2)
Força (F) (M1 . L1 . T-2)
Frequência (f) (T-1)
Gravidade (g) (M0 . L1 . T-2)
Vazão (Q) (M0 . L3 . T-1)
Fluxo de massa (∅m) (M1 . L0 . T-1)
Pressão (p) (M1 . L-1 . T-2)
Tensão (τ) (M1 . L-1 . T-2)
Massa Específica (ρ) (M1 . L-3 . T0)
Peso Específico (γ) (M1 . L-2 . T-2)
Viscosidade (μ) (M1 . L-1 . T-1)
Viscosidade Cinemática (μc) (M0 . L2 . T-1)
Trabalho (W) (M1 . L2 . T-2)
Fluxo de calor ∅Q (M1 . L2 . T-3)
Tensão Superficial (σ) (M1 . L0 . T-2)
Módulo da Elasticidade Volumétrica (B) (M1 . L-1 . T-2)


Exemplo:

Determine que a vazão "Q", através de um tubo capilar horizontal, depende da queda de pressão por unidadede comprimento "∆P/L", do diâmetro do capilar "d" e da viscosidade absoluta do fluido "µ".

Resposta:

Sabendo que: Q≡L3 . T-1
P≡M . L-1 . T-2
L≡L
d≡L
μ≡M . L-1 . T-1

Então:

Q= K . (P/L)α . dβ . µγ (TEOREMA DE BRIDGMAN)
(M0 . L3 . T-1)= K . (M . L-1 . T-2 . L-1)α . (L)β . (M . L-1 . T-1)γ
(M0 . L3 . T-1)= K . (M . L-2 . T-2)α . Lβ . (M . L-1 . T-1)γ

Vamos agora trabalhar com as potências:

Para as potências de M teremos:
0 = α + γ
Para as potências de L teremos:
3 = -2α + β - γ
Para as potências de T teremos:
-1 = -2α - γ
Agora temos três equações e três icógnitas
0 = α + γ
3 = -2α + β - γ
-1 = -2α - γ
Resolvendo a mesma encontraremos:
α=1; β=4; γ=-1;
Substituindo na equação inicial teremos:
Q= K . (P/L)1 . d4 . µ-1 (TEOREMA DE BRIDGMAN)

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