Water flows down a long, straight, inclined pipe of diameter D and length L (Fig. P9-128). There is no forced pressure gradient between points 1 and 2; in other words, the water flows through the pipe by gravity alone, and P 1 = P 2= P atm. The flow is steady, fully developed, and laminar. We adopt a coordinate system in which x follows the axis of the pipe. ( a) Use the control volume technique of Chap. 8 to generate an expression for average velocity V as a function of the given parameters ρ, g, D, Δ z, µ, and L. ( b) Use differential analysis to generate an expression for V as a function of the given parameters. Compare with your result of pan ( a) and discuss. ( c) Use dimensional analysis to generate a dimensionless expression for V as a function of the given parameters. Construct a relationship between your Π’s that matches the exact analytical expression.