Question 5 - 20 marks
A rectangular lamina of width a and height b is sandwiched between two
slabs of ice, so that the left and right sides, of height b, are at a temperature
e = 0. The other two sides are thermally isolated. Fix coordinates so that
the origin is the lower left corner, as shown below.
Model this situation as a temperature 0(x,y,t) that satisfies the
two-dimensional diffusion equation with thermal diffusivity D
= DV²0, 0ot
together with Dirichlet boundary conditions on the left and right sides
and Neumann boundary conditions on the other two sides
(a) Show that applying the method of separation of variables using
0(x,y,t) = V(x,y) T(t) gives the solution T(t) = euDt and the
eigenproblem V2V(x,y)=V(x,y). =
(b) Applying the method of separation of variables again, using
V(x,y) = X (x) Y(y), gives the two further eigenproblems
X" = - =
Translate the boundary conditions for e into boundary conditions for
X(x) and Y(y).
(c) The Y(y) eigenproblem has been analysed in the text, and the
eigenfunctions were found to be
Yn(y) = cos(kny), where kn = nz/b, n = 0,1,2,
with corresponding eigenvalues X = kn.
Find the eigenfunctions and eigenvalues for the X(x) eigenproblem.
(d) Use the eigenfunctions from part (c) to write down the eigenfunctions
and eigenvalues for the V(x,y) eigenproblem.
(e) Write down a general solution for the temperature e (x,y,t).
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