An incompressible fluid oscillates harmonically (left(V=V_{0} ight.) (sin omega t), where (V) is the velocity) with a

An incompressible fluid oscillates harmonically \(\left(V=V_{0}\right.\) \(\sin \omega t\), where \(V\) is the velocity) with a frequency of \(10 \mathrm{rad} / \mathrm{s}\) in a 4-in.-diameter pipe. A \(\frac{1}{4}\) scale model is to be used to determine the pressure difference per unit length, \(\Delta p_{\ell}\) (at any instant) along the pipe. Assume that

\[ \Delta p_{\ell}=f\left(D, V_{0}, \omega, t, \mu, ho\right) \]

where \(D\) is the pipe diameter, \(\omega\) the frequency, \(t\) the time, \(\mu\) the fluid viscosity, and \(ho\) the fluid density.

(a) Determine the similarity requirements for the model and the prediction equation for \(\Delta p_{\ell}\).

(b) If the same fluid is used in the model and the prototype, at what frequency should the model operate?