Question

A rectangular lamina of width a and height b is sandwiched between two slabs of ice, so that the lower and upper sides, of width a, are at a temperature Θ 0, The other two sides are thermally isolated. Fix coordinates so that the origin below is the lower left corner, as shown in the figure се се Model this situation as a temperature e(x, y,t) that satisfies the two-dimensional diffusion equation with thermal diffusivity D, namely ot together with Dirichlet boundary conditions on the upper and lower sides given by 6(z, 0, t)-0, θ(z, b,t)-0, 0 < x < a, and Neumann boundary conditions on the other two sides given by (0, y, t)0, (a, y, t)-0, 0<b (a) Show that applying the method of separation of variables using and the (x, y,t)- V(x, y)T(t) gives the solution T(t) Ce eigenproblem V2V(x, y)-HV(x, y), where C and u are constants. V(x,y)- X(x) Y(y), gives the two further eigenproblems X"-XX and Translate the boundary conditions for 9 into boundary conditions for (b) Applying the method of separation of variables again, using X(x) and Y (y) (c) The X (x) eigenproblem has been analysed in Unit 11, and the eigenfunctions were found to be (z) = cos(kmz), where km-mm/a, m = 0, 1, 2, . . . , with corresponding eigenvalues A- Find the eigenfunctions and eigenvalues for the Y(y) eigenproblem. TL (12) (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 (x, y,t)

Answer 1

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