(Solved): Open channel flows (such as river flows) have quadratic velocity profiles along the gravitation ...

Open channel flows (such as river flows) have quadratic velocity profiles along the gravitational direction, with non-slip velocity at the bottom: An experimental system, with a weight tank and a pitot tube, has been designed to calibrate the coefficients $$a$$ and $$b$$ in the equation above. The Pitot tube, with a U-tube manometer containing mercury, is used to measure the surface velocity $$U_3$$. The channel has a rectangular cross-section, with a width $$\mathrm{D}$$. The flow in-depth $$\mathrm{H}$$ is measured by the in-depth scale. (1) Determine the surface velocity $$U_s$$ from the measurement of liquid density $${?}_1$$, mercury density $${?}_{\mathrm{m}}$$, and height difference in mercury column $$(?\mathrm{h})$$ of Pitot tube; (2) Derive the volumetric flow rate $$Q$$ as a function of a, b, D, and $$H$$; (3) Express $$\mathrm{a}$$ and $$\mathrm{b}$$ as functions of weight tank measurements ( $$?\mathrm{W}$$ and $$?\mathrm{t}$$ ), liquid density $${?}_1,\mathrm{D},\mathrm{H}$$, and $${\mathrm{U}}_3$$;