P384_Assignment_5_SOLUTION - 1 Physics 384 2016 Assignment 5 SOLUTION Problem I Compute the steady-state temperature distribution in a

P384_Assignment_5_SOLUTION - 1 Physics 384 2016 Assignment...

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1 Physics 384 2016 Assignment 5 SOLUTION Problem I Compute the steady-state temperature distribution in a thermally-conducting thin wedge of radius a and angle β , illustrated above. The rim is held at temperature u 0 , while the edges of the wedge are held at zero temperature. (a) Work through the separation of variables in 2D plane polar coordinates 1 r ∂r r ∂r + 1 r 2 2 ∂θ 2 u ( r, θ ) = 0 where u ( r, θ ) = R ( r )Θ( θ ) taking note that you cannot use the usual argument that the azimuthal basis functions are periodic in θ θ + 2 π . Show that the boundary conditions rule out exponential functions in θ , so that one still ends up with trigonometric functions Θ( θ ) = A cos + B sin where k is a separation constant. Be sure to identify the two linearly independent solutions for k = 0 . To find the radial functions, try the simplest possible ansatz ;)! (b) Taking the bottom edge of the wedge to be at polar angle θ = 0 , determine the eigenvalues that are allowed by the boundary conditions Θ( θ = 0) = Θ( θ = β ) = 0 Which, if any, of the k = 0 solutions satisfy these boudary conditions? (c) Find the linear combination of basis functions that satisfies the remaining boundary condition u ( r = a, θ ) = u 0 , θ [0 , β ] This equation should take the form of a Fourier series, but in this case the trigonometric functions are orthogonal over the interval θ [0 , β ] , which is just another case of Sturm- Liouville orthogonality with Dirichlet boundary conditions. You will need to compute the normalization integral that is required to invert the Fourier series, and then find a closed expression for the coefficients of the expansion.
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2 Solution I First, a note regarding my original, incorrect specification of the boundary conditions and angular range of the wedge. The configuration of the wedge in the original statment of the problem was a disk with a “cut” of angular width β (giving the conducting region the shape of a “pacman”). I suggested to take the lower edge of the cut to lie at θ = 0 , and the upper edge at θ = β , and that the rim of the the wedge was to be taken between θ = β and θ = 2 π ; however, because of the cut, one cannot identify the angular coordinates θ = 0 and θ = 2 π . The correct specification for that pacman configuration would have been for the rim to run from
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