1. Mathematics
2. Calculus and Analysis
3. 30546

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L .M. Chiappetta and D.R. Sobel ("Temperature distribution within a hemisphere exposed to a hot gas stream," SIAM Review 26, 1984, p. 575?577) analyze the steady- state temperature in the rounded tip of a combustion-gas sampling probe. The tip is approximately hemispherical iii shape, with radius a. Its outer surface is exposed to hot gases at a temperature Tc, and its base is cooled by water at temperature Tw circulating inside the probe. Let the temperature inside the tip he T. Because of the symmetry of the problem, there is no azinnitha.1 variation in T. In other words, if we are using spherical coordinates, T = T(r, 0), where U is the colatitude. So, T is governed by the differential equation
....
inside the domain ... (the upper hemisphere with radius a). Let us use a Dirichlet condition at the base of the tip.
....
and a Robin boundary condition at the outer surface of the tip,
....
As usual, we will also assume that tile temperature is bounded at the origin and along the north pole. Let .... and solve for it using separation of variables.
Hints: You will need to figure out the orthogonality condition for the eigenfunctions, but you can assume that tile eigenfunctions will he orthogonal with respect to the appropriate weight function because the eigenvalue problem is in Sturm-Liouville form.