Up: Fall 2000

EML 5060 Homework Set 4 Fall 2000

Page HW Class Topic
17 2.19cdg 2.19e Classification
17 2.20   Classification
17 2.21ac 2.21b Classification: assume u=u(x,y,z)
18 2.25 2.24 Canonical form
18 2.26   Canonical form
18 2.22ach 2.22d Characteristics
18 2.27bc 2.27d 2D Canonical form
18 2.28bejo 2.28nml 2D Canonical form
98 7.20 7.19 Unsteady heat conduction in a bar
98 7.21 7.22 Unsteady heat conduction in a bar
98 7.25 7.24 Unidirectional viscous flow
99 7.29 7.28 Acoustics in a pipe
99 7.35 7.36 Steady supersonic flow
99 7.37 7.37 Steady heat conduction in a plate
99 7.39 7.38 Potential flow inside a cylinder
00     Unsteady heat conduction in a disk

Also solve the following problem:

Solve the 3D steady heat conduction inside an infinite circular pipe of radius a if the pipe surface is kept at a constant temperature T0. Inside the pipe, at a radial distance away from the axis, there is a point source of heat putting out Q units of heat per unit time.

Using the results, discuss the temperature distribution in cross sections well away from the point heat source. How is the temperature different from T0? How rapidly does the temperature drop off to T0 in the axial direction?

Hints:

1.
Use cylindrical coordinates .
2.
The governing equations are, of course,

where is the Dirac delta function and is the position of the heat source.
3.
You can align your coordinate system however you want, but does it not make sense to select the axis so that it passes through the source? Then the problem is symmetric both with respect to and with respect to z=0.
4.
Would it not be smart to define a new unknown , new coordinates and , and a nondimensional source strength ?
5.
By definition of the delta function, an integral of the form where function f(r) can be any function, including (hint hint) an eigenfunction. Similar for .
6.
The solution of the following boundary value problem:

is:

(Since immediately before and after z=0, making Czz a delta function.)
7.
Nope, you won't find the solution in any book.
Up: Fall 2000
11/17/00 0:52:15
12/04/00 0:21:32