At a sudden contraction in a pipe the diameter changes from

At a sudden contraction in a pipe the diameter changes from D_1 to D_2. The pressure drop. Delta 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, and viscosity, mu. Use Du K and n 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?

Solution

Let DP= F(D1,d2,v,RHO,MU)

units dims pressure F/L^2

L

V =L/t

D = L

rho = FL^-4T^2

Mu= F L^-2T

First dimensionless group using D1 V mu are recurring variables

Pi1 = DelP * D1^a V^b MU^c

since this is non dim, obtains Pi1 = DELP *D1/(V*MU)

Similarly for P2: new entering variable D2, obtain D2/D1

Similarly Pi3, using rho as entering variable,

Pi3 = Rho*D1*V/Mu