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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. y b Ice Ice a x Model this situation as a temperature 0(x,y,t) that satisfies the two-dimensional diffusion equation with thermal diffusivity D 00 = DVĀ²0, 0ot together with Dirichlet boundary conditions on the left and right sides = = and Neumann boundary conditions on the other two sides 00 = by (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). = [2] (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). [2] (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. [13] (d) Use the eigenfunctions from part (c) to write down the eigenfunctions and eigenvalues for the V(x,y) eigenproblem. [2] (e) Write down a general solution for the temperature e (x,y,t). [1]

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