1. An incompressible fluid of density p and viscosity µ flows through a long horizontal section of a round pipe of diameter D. You would like to determine the shear stress on the walls of the tube (7) given an average velocity V. Assume some constant average roughness height & along the inside wall of the pipe. (a) Choosing μ, D, and p as your repeating parameters, develop a non-dimensional relationship between shear stress and the other parameters. Tw Π2 PV2, use PVD (b) Using different repeating parameters results in the following II groups: II₁ = 2, II₂ = 5, and II3 = μ these Pi groups for the rest of the problem. If you are attempting to model the shear stress on a proposed new section of Water Tunnel #3 feeding New York City which must potentially supply 64.8 m³/s of water per second. Given the future tunnel parameters given in the table, choose the experimental parameters left blank in the table to ensure similarity, (make sure you show your work and prove that you have met the requirements for similarity. Additionally, when choosing your working fluid, clearly denote the name of the fluid and it's properties. ). Parameter Velocity ρ μ D € Prototype 1.5 m/s 1000 kg/m³ 1.0x10-3kg/ms 7.3 m 0.25 mm Experiment 0.0045 mm (c) Explain your working fluid choice. Use engineering arguments, i.e. cite rational design concerns (physical proper- ties, cost, corrosiveness, toxicity,...) (d) If your experiment measures a shear stress of 1000.0 N/m², what do you predict the shear stress on the walls of Tunnel #3 to be? (e) What is the meaning of each of these Pi groups. I.e. describe in words what are they the ratios of.

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A liquid of density ? and viscosity ? flows by gravity through a hole of diameter d in the bottom of a tank of diameter DFig. . At the start of the experiment, the liquid surface is at height h above the bottom of the tank, as sketched. The liquid exits the tank as a jet with average velocity V straight down as also sketched. Using dimensional analysis, generate a dimensionless relationship for V as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. (Hint: There are three length scales in this problem. For consistency, choose h as your length scale.) except for a different dependent parameter, namely, the time required to empty the tank tempty. Generate a dimensionless relationship for tempty as a function of the following independent parameters: hole diameter d, tank diameter D, density ? , viscosity ? , initial liquid surface height h, and gravitational acceleration g.

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