Homework #3 ME 363 - Fluid Mechanics
Due Feb 13, 2008 Spring Semester 2008
1] For steady laminar (not turbulent) flow through a long tube, the axial velocity distribution is given by , where R is the tube outer radius and C is a constant. Integrate u(r) to find the total volume flow Q through the tube in terms of C.
2] A theory proposed by S. I. Pai in 1953 gives the following velocity values u(r) for turbulent airflow in a 4-cm-diameter tube:
Compare these data to the laminar flow velocity distribution given in problem 1, e.g., by plotting both on the same set of axes. Discuss the differences. Estimate, as best you can, the
total volume flow Q through the tube, in m3/s.
3] When a gravity-driven liquid jet issues from a slot in a tank, an approximation for the exit velocity distribution is where h is the depth of the jet centerline. Near the slot, the jet is horizontal, two-dimensional, and of thickness 2L, as shown. Find a general expression for the total volume flow Q issuing from the slot; then take the limit of your result if Lh.
4] Three pipes steadily deliver water at 20°C to a large exit pipe. The velocity V2 = 5 m/s, and the exit flow rate Q4 = 120 m3/h. Find (a) V1; (b) V3; and (c) V4 if it is known that increasing Q3 by 20% would increase Q4 by 10%.
5] A pipe flow fills a cylindrical surge tank. At time t = 0, the water level is 30 cm above the bottom of the tank. Estimate the time required to fill the remainder of the tank.
6] An incompressible fluid flows past an impermeable flat plate with a uniform inlet profile u=U0 and a cubic polynomial exit profile . Compute the volume flow rate Q across the top surface of the control volume.
problem 1: no figure / problem 2: no figure /
Problem 1:
Problem 2:
Problem 3:
The mathematical reduction can be understood as follows:
where the first 2 terms from the Taylor series:
are used to convert
Problem 4:
Problem 5:
Problem 6: