The loss of pressure when a fluid flows through a pipe is given by $P = k ρ^{a} l V^{b} d^{c} μ$ where d and l are diameter $ρ$ and $μ$ length of the pipe respectively, and are the mass density and coefficient of viscosity of the fluid, V is the mean velocity of flow through the pipe and k is a numerical constant. Find the values of a, b and c.

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Solution

$P = k ρ^{a} l V^{b} d^{c} μ d i m e n s i o n o f P = M^{1} L^{- 1} T^{- 2} ρ = M L^{- 3} l = L^{1} V = {L T}^{- 1} d = L μ = M L^{1} T^{- 1} M L^{1} T^{- 2} = M^{a + 1} L^{- 3 a + 1 + c + b} T^{- b - 1}$

Comparing both side
$a + 1 = 1, a = 0 b + 1 = 2, b = 1 b + c + 1 - 3 \times 0 = - 1 b + c + 1 = - 1 c + 2 = - 1 c = - 3. ∴ a = 0, b = 1, c = - 3$

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P=kρalVbdcμdimensionofP=M1L−1T−2ρ=ML−3l=L1V=LT−1d=Lμ=ML1T−1ML1T−2=Ma+1L−3a+1+c+bT−b−1Comparing both side a+1=1,a=0b+1=2,b=1b+c+1−3×0=−1b+c+1=−1c+2=−1c=−3.∴a=0,b=1,c=−3