MTH 444/544 Homework #6

Due at the beginning of class, Thursday April 6, 2006.
WARNING to GRAD STUDENTS: You may find problem 2 quite difficult and long. Get started ASAP.

1. (a) Using the experimental data mailed to you by Josh: 

(laptime,t) ((seconds),(minutes:seconds))
(1.83, 28:00)
(2.16, 28:09)
(4.62, 28.21)
(6.52, 28:43)
(10.10, 29:06)
(19.27, 29:46)
(29.31, 30:36)
(51.21, 31:44)

estimate how long it took for the fluid in the bucket 4cm in from the outside wall (where the ring is) to slow down to 10% of its initial speed.
(b) Using the Rayleigh Impulsive Flow solution, determine (using Maple's "solve" or "fsolve" if necessary) how long it takes for the 4cm layer of water closest to the plate to speed up to 90% of U (or equivalently, slow down to 10% of U if viewed in the reference frame of the plate).
(c) Compare the two times obtained in (a) and (b) and compare them. (The RIF in (b) could be considered a crude model of the flow observed in (a), if we ignore the bottom of the bucket.)

2.
UNDERGRADS Estimate the decay time in the slowing to rest of an infinite pool of fluid of uniform depth h, initially moving at uniform velocity U parallel to the (co-moving) floor, where the motion of the floor is suddenly halted. Use separation of variables to solve the momentum balance equation. (You do not need to determine the coefficients to satisfy the initial condition: just obtain the decay rate of the slowest-decaying mode.)
GRADS Estimate the decay time in the slowing to rest of an infinite cylinder of fluid of radius R, initially rotating rigidly, where the motion of the cylinder wall is suddenly halted. Use separation of variables to solve the momentum balance equation. (You do not need to determine the coefficients to satisfy the initial condition: just obtain the decay rate of the slowest-decaying mode.) (You will find the radial equation is a Bessel equation. You may use Maple to find the appropriate root of the appropriate Bessel function: type "?bessel" for information.)

3. Read Segel, sections 4.1 and 4.2.

4. Given the displacement field

(a) Calculate the Lagrangian strain at the point A=(1,0,2).
(b) What is the extension of a line of initial length and orientation (da1,0,0) at this point?
(c) What is the rotation of this line?