At a sudden contraction in a pipe the diameter changes from D 1 to D 2 . The pressure drop, Δ p , which develops across the contraction is a function of D 1 and D 2 , as well as the velocity, V , in the larger pipe, and the fluid density, rho, and viscosity, μ . Use D 1 , V , and μ as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

Velocity in the smaller pipe should not be included as an additional variable.    

Explanation:

The dimensional formula of the variables are

Now using Buchingham's Pi Theorem, 6 - 3 = 3 dimensional parameters are required.

Use, D_1, V, \mu as the repeating variables.

Therefore,

From this

1+c=0

-2+a+b-2c=0

-b+c=0

c=-1, b = -1, a = 1

Now,

For

c = 0

1 + a + b - 2c = 0

-b + c = 0

Therefore, a = -1, b = 0, c = 0

For

1 + c = 0

-4 + a + b - 2c = 0

2-b+c=0

c=-1, b=1, a = 1

Therefore,

Now checking,

Therefore,

From continuity equation

is not independent of

Therefore it should not be included as an additional variable.

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