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 pipe of diameter 2.5 m is required to transport an oil of sp.gr. 0.86 & viscosity 3×10- 2 poise at 3750 li/s. Tests were conducted on a 20cm diameter pipe using water at 20oC Find the velocity and rate of flow in the model. Viscosity of water at 20 m= 0.01poise.

Give example of shear thinning fluids. An infinite plate is moved over a second plate on a layer of liquid as shown. For small gap width, d, we assume a linear velocity distribution in the liquid. The liquid viscosity is 0. R1 centipoise and its specific gravity is 0. R2. Determine: (a) The absolute viscosity of the liquid, (b) The kinematic viscosity of the liquid, where, R1 is 88 °C and R2 is 1 °C

fluid mechanics

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.

Ex. The force required to tow a 1:30scale model of a motor boat in a lake at a speed of 2m/s is 0.5 N, Assuming that the viscosity resistance due to water and air is negligible in comparison with the wave resistance, calculate the corresponding speed of the prototype for dynamically similar conditions. What would be the force required to propel the prptype at that velocity in the same lake? Ans.: 10.95m/s, 13500N

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HW1.1: Equation 2.7 in the text provides the velocity profile, u(y) for a narrow, infinitely long, slit-like microchannel. Use the no-slip boundary conditions (channel walls not moving) to obtain 1 dp - hy} 2n dx the standard solution, u(y)= For an incompressible, Newtonian fluid, with fluid properties as described in the text, shear stress du is typically given by t =n· dy Write-out an expression for the shear stress as a function of y. Calculate shear stress at: (a) y = 0; (b) y = h; and (c) y = h/2.