(Solved): Q 1(c) [17 Marks] The pressure drop \\( \\Delta p \\) in a pipe of uniform cross section depends on ...

Q 1(c) [17 Marks] The pressure drop \\( \\Delta p \\) in a pipe of uniform cross section depends on the average flow velocity \\( V \\), the pipe diameter \\( D \\), the pipe length \\( L \\), the roughness height \\( e \\), the fluid density \\( \\rho \\) and viscosity \\( \\mu \\) : \\[ \\Delta p=f(\\rho, \\mu, V, L, D, e) \\] We wish to know how \\( \\Delta p \\) varies with \\( V, L \\) and \\( e \\). Use the pi Buckingham theorem to rewrite this relationship in the following dimensionless form: \\[ \\frac{\\rho D^{2} \\Delta p}{\\mu^{2}}=g\\left(\\frac{\\rho V D}{\\mu}, \\frac{L}{D}, \\frac{e}{D}\\right) \\] To illustrate the method, you should derive the first Pi number \\( \\rho D^{2} \\Delta p / \\mu^{2} \\). You do not have to derive the other two Pi groups but you must describe the whole method in details giving the units or dimensions of all variables. [End of Question1]