Question

Compute the steady-state temperature distribution in an infinitely long cylindri cal wedge of radius a and angle B, whose cross-section is illustrated below. The two straight sides of the wedge are held at zero temperature, while the curved edge is at uniform temperature uo uo Here are a few points to consider in r solution to this problem (a) In polar coordinates, the steady-state temperature satisfies You are required to use the usual approach of separation of variables and to solve the eigenvalue problem for the basis functions R(r) and Φ(d) and finally to compute the linear combination that satisfies the required boundary conditions. Note that you cannot use the usual argument that Φ(d) is periodic under φ → φ+ 2π when choosing the sign of the separation constant, by which one is ordinarily led to trigonometric functions as the appropriate basis Remember that an exponential form for ф(d) would be generated by the other choice of sign of the separation constant. Nevertheless, the trigono- metric basis is still the only acceptable form in this problem. You must find an argument that rules out the Also be sure to find the two linearly independent solutions for R(r) and φ(d) in the case where k = 0 exponential form. (b) If our systern occupied all polar angles φ, then the allowed eigenvalues k would be integers. In this case however, the acceptable values of k must be determined by imposing the two boundary conditions Determine the allowed eigenvalues k. Which, if any, of the k satisfy these boundary conditions? 0 solutions (c) Once you have the allowed eigenvalues, write a general solution by superpo- sition, and solve for the expansion coefficients from the remaining boundary condition This condition should take the form of a Fourier trigonometric functions are orthogonal over the interval φ-10, β], thanks to the boundary conditions, which make for a 'standard' Sturm-Liouville orthogonality scenario. You w need to compute the normalization integral that is required to invert the Fourier series but in this case the

Answer 1