EML 5060 Analysis in Mechanical Engineering 12/13/95
Closed book Van Dommelen 12:30-2:30pm

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

(33 points). Consider steady supersonic flow at a Mach number 3.33 under a 45 degree angle with the x-axis. Small changes in the horizontal velocity component u(x,y) satisfy

4 u_xx + 10 u_xy + 4 u_yy = 0.

Find new coordinates $\xi$ and $\eta$ such that the equation satisfied by $u(\xi,\eta)$ takes the form

$$u_{\xi\eta} = 0.$$

Use this result to show that the solution for u must be of the form

u(x,y) = f(2x-y)+g(2y-x),

where f and g are arbitrary functions. Solution

2.

(33 points). Solve heat conduction in a semi-infinite bar which is initially at zero temperature. Heat is removed through the end at a rate which increases linearly in time:

$$T_t = T_{xx} \qquad T(x,0)=0 \qquad T_x(0,t) = t.$$

Note: you will probably not succeed in simplifying the final result for u. Solution

3.

(34 points). Solve steady heat conduction in a quarter circle. The straight sides are insulated:

$$\nabla^2 T = 0 \qquad T_\theta(r,0) = T_\theta(r,\pi/2) = 0 \qquad T(1,\theta)= f(\theta).$$

Write the solution in terms of $f(\theta)$.Also, if $f(\theta)=\cos(4\theta)$, what is u? Solution a Solution b