Please answer these 2 problems and show work.

MAE 242 Fall 2024 Homework #6 Due by Thursday, October 31, 2024, 11:59pm Problem 01 In 1940, fluid dynamicist G. I. Taylor determined from Dimensional Analysis that the radius R of a fireball created by a nuclear explosion is exclusively determined by the released energy E, initial gas density py and elapsed time * since detonation. Note: 1 kiloton = 4 x 10'2 J (Joules). (a) List in a table the primary dimensions of R, E, py and *. How many primary dimensions are there? {b) Find the single Pi group that controls the blast radius R. () In 1945, an experiment was conducted in the desert of New Mexico with a 20 kiloton bomb. On that day, air density was recorded at p; = 1.23kg/m3. The photograph above taken at t* = 0.015s shows a radius R ~ 108 m. Compute the value of the Pi group you have found. (d) Assuming that the Pi group remains constant, find the blast radius R at * = 0.02s. Problem 02 The drag force Fp, on a sphere depends on the relative velocity V', sphere diameter D, the Huid density p, and the fluid viscosity p. (a) Obtain a set of non-dimensional groups that can be used to correlate experimental data. (b) The Stokes solution (obtained in the limit of small relative velocities) states Fp=3npDV Find the exact relationship between drag coefficient Cp and Reynolds number Re. Note Cp = Fp ~D = [1/2)pVi(=D2ja)