Due Friday, April 11, 2008 Fall Semester 2008
1] In cold climates, water pipes may freeze and burst if proper precautions are not taken. In such an occurrence, the exposed pipe on the ground ruptures, and water shoots up to 34 m. Estimate the gage pressure of water in the pipe. State your assumptions and discuss if the actual pressure is more or less than the value you predicted.
2] An airplane is flying at an altitude of 12000 m. Determine the gage pressure at the stagnation point on the nose of the plane if the speed of the plane is 200 km/h. How would you solve this problem if the speed was 1050 km/h. Explain.
3] A handheld bicycle pump can be used as an atomizer to generate a fine mist of paint or pesticide by forcing air at a high velocity through a small hole and placing a short tube between the liquid reservoir and the high-speed air jet whose low pressure drives the liquid up through the tube. In such an atomizer, the hole diameter is 3 mm, the vertical distance between the liquid level in the tube and the hole is 10 cm, and the bore (diameter) and the stroke of the air pump are 5 cm and 20 cm respectively. If the atmospheric conditions ate 20 C and 95 kPa, determine the minimum speed that the piston must be moved to initiate the atomizing effect. The liquid reservoir is open to the atmosphere.
4] Consider flow over a small object in a viscous fluid, such as a small particle settling in a glass of water. Analysis of the Navier-Stokes equations show that the inertial terms are much smaller than the viscous and pressure terms in this case. It turns out, then, that fluid density drops out of the Navier-Stokes equations. Such flows are called creeping flows. The only important parameters in creeping flow are the particle velocity U (relative to the fluid), the fluid viscosity μ, and the particle length scale d. For three-dimensional bodies, like spheres, creeping-flow analysis yields very good results. It is uncertain, however, if creeping flow applies to two-dimensional bodies, such as cylinders, since even though the diameter may be very small, the length of the cylinder is infinite. Let us see if dimensional analysis can help. (a) Apply the Pi theorem to generate an expression for the two-dimensional drag force D_{2-D} as a function of the other parameters in the problem. Be careful: two-dimensional drag has dimensions of force per unit length, not simply force. (b) Is your result in part (a) physically plausible? If not, explain why not. (c) It turns out that fluid density ρ cannot be neglected in analysis of creeping flow over two-dimensional bodies. Repeat the dimensional analysis, this time including ρ included as a parameter. Find the resulting nondimensional relation between the parameters in this problem.