Similitude and Physical Modeling

In a flow through a small orifice discharging freely into atmosphere under a constant head (H), the flow discharge (Q) depends on diameter of pipe (d), constant head, dynamic viscosity (\(\mu\)), density of fluid (\(\rho\)) and acceleration due to gravity (g). Using Rayleigh’s methods develop the relation in terms of non-dimensional terms.

The pressure difference \(\Delta p\) in a pipe of diameter ‘D’ and length ‘L’ due to turbulent flow depends on the velocity ‘V’, viscosity ‘\(\mu\)’, density ‘\(\rho\)’ and roughness height ‘K’. Using Buckingham’s \(\pi\)-theorem, obtain an expression for \(\Delta p\).

Using Buckingham’s \(\pi\)-method, show that the resistance ‘F’ to the motion of sphere of diameter ‘D’ moving with a uniform velocity ‘V’ through a real fluid of density ‘\(\rho\)’ and viscosity ‘\(\mu\)’ is given by: \(F=\rho D^2V^2 f\left(\frac{\mu}{\rho VD}\right)\).