Math Help

Heat Conduction in a Semi-Circular Plate

Example: Consider steady state heat conduction in a semicircular plate of radius a shown

below. The temperature along the edge, y = 0, u(x, 0) = 0 and the temperature distribution

along the edge, y = b, is u(x, b) = f(x). See the figure below.

R’’θ + (1/r) R' θ + (1/r²)Rθ’’ = 0 (1)

u(r, 0) = u(r, π) = 0

u(a, θ) = f(θ) (prescribed temperature distribution on r = a)

Strategy: Separate the variables by assuming u(r,θ) = R(r) θ (θ), by putting this expression

into eq. 1 above. The result is:

( r² R'' + rR' ) / R = ˗ θ '' / θ = separation constant = λ

So r² R'' + rR' ˗ λ R = 0

and θ ’’ + λ θ = 0

The separation constant, λ, can take on three possible cases -- such as

λ = 0, λ > 0, and λ < 0 . You need to evaluate each case.

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