(Solved): A4. State Bernoulli's Equation for a steady flow, making sure you define all the quantities involv ...

A4. State Bernoulli's Equation for a steady flow, making sure you define all the quantities involved. Water, which can be treated as an inviscid fluid of constant density \( \rho \), flows steadily through a long piece of drainpipe. The pipe consists of a horizontal section of length \( l \) followed by a right-angle bend and a section vertically downwards of length \( l / 2 \). The pipe is of slowly varying cross-section, and narrows from area \( 2 A \) at the inlet to area \( A \) at the outlet. The pressure at the outlet is atmospheric pressure \( p_{0} \), whilst the inlet pressure is \( p_{0}+3 \rho U^{2} \) where \( U \) is a constant. Derive the version of Bernoulli's equation appropriate to this situation, and use this equation together with the conservation of mass to find the inlet speed in terms of the constants \( U, l \) and \( g \) (where \( g \) is the gravitational acceleration). [Hints: 1. You should assume that the flow is uniform across each cross-section of the pipe. 2. Conservation of mass means that the mass flow rate \( (\rho \times \) flow speed \( \times \) cross-sectional area) is the same at each cross-section of the pipe.] [8 marks]