CEE 341 Fluid Mechanics for Civil Engineers
Homework # 1, due 29 Jan 2015 in class
1. Show that for ideal gases
v Ep =
2. Assume air consists of oxygen and nitrogen only. The ideal gas law can be applied separately to the two gases, as
2 2 2 O O O p R T ρ = and 2 2 2 N N N p R T ρ = The total atmospheric pressure is simply the sum of the partial pressures, i.e.
2 2 2 2 air O O N N p f p f p = +
where 2 Of and 2 Nf are the molar fraction of oxygen and nitrogen in the air, respectively. Use gas densities and gas constants in Table A.2, show that air can be considered as consisting of roughly 20% of oxygen and 80% of nitrogen.
3. Speed of horizontal wind near the ground can be approximated using “log wind profile” as ( ) * 0 ln uyuy y κ =
where y is the vertical coordinate (y = 0 at the ground surface and positive upwards); u* is the friction velocity; κ = 0.4 is the von Karman constant; and y0 is the roughness length. Assume o *0 0.3 m/s, 10 mm and 25 C u y T = = = . (a) At what elevation does the “non-slip” condition, i.e. u = 0, occur for log wind profile? (b) What is the shear stress at y = y0? (c) Plot the horizontal wind speed u(y) and the shear stress τ (y) versus y for 0 < y < 100 m.
4. A cylinder of diameter d, length l, and density ρ is falling inside a pipe (diameter D) filled with oil, at a constant velocity Vfall. The small space between the cylinder and the pipe is lubricated with the oil of viscosity µ . Assume a linear velocity profile for a thin film. (a) Find the velocity of descent Vfall of the cylinder. (b) Given d = 50 mm, D = 50.5 mm, l = 100 mm, ρ = 2400 kg/m3, and µ = 0.1 Ns/m2, compute Vfall.
5. A drop of water at 20 oC is forming as a hemisphere under a solid surface as shown. What is the radius of the hemisphere just before separation?