The Solution of Fourier's Equation for a Sphere

  Fourier's presentation of the solution of his equation for the changes in the temperature a spherical body is difficult to follow and could benefit from a little cleanup. One can arrive at the boundary equation Fourier used by combining Newton's law of cooling with his law relating the change in heat content with the radial change in temperature.

Fourier reduces his temperature equation to a simpler one involving a factor and an auxiliary function. The factor can be found using a procedure similar to that for finding an integrating factor.

One next looks for the form of the elementary functions which are solutions of the auxiliary function U which one can do by arbitrarily setting U=e^φ. Substituting into the differential equation for U gives one for φ. The trick is to find something as simple as possible so we set φ equal to a linear function of r and t. The differential equation for φ then gives a relation connecting two of the coefficients and we can write one in terms of the other. Requiring that U → 0 as t → ∞ tells us that the coefficient of r is complex. So we get a class of functions that fit a simple form.

This gives us the general form of the elementary functions of the differential equation for T and like Fourier we can replace the complex exponential function with cosine and sine functions. Requiring that the functions to be bounded forces us to eliminate the cosine term. Next we consider the effect of the boundary equation on the form for T. We can eliminate the constant term involving the ambient temperature T_a which Fourier ignores by noting that T can be expressed as the sum of a specific solution involving just T_a and a general solution T_g which satisfies the equation dT_g/dr+hT_g=0. Finally, substituting the general form of T for T_g we get Fourier's constraint on the permissible functions for T, μR/tan(μR)=1-hR or, with ε=μR and λ=1-hR, ε/tanε= λ.

Note we can write T as a sum of the permissible functions ψ, that is, T_g=Σ_kT_kψ_k. The values for the coefficients T_k are determined by the expansion of the initial distribution of T=T(r,0) in terms of the permitted functions. One can use the constraint condition to find a table of permitted values for ε=μR as a function of λ.

Supplemental (Mar 20): I noticed a minor inconsistency in the argument above. It was assumed that U → 0 as t → ∞ but that apparently like Fourier ignores the ambient temperature T_a. There are to ways of correcting this, first one could assume that T is measured relative to T_a or alternatively one could assume the condition only applies to the general elementary functions for T_g.