Spherically symmetric Heat Equation

[throneoo]

Homework Statement

A ball of radius a, originally at T₀, is immersed to boiling water at T₁ at t=0. From t≥0, the surface (of the ball) is kept at T₁
Define u(r,t)=R(r)Q(t)=T(r,t)-T₁
ΔT=T₀-T₁<0

r,t≥0

Homework Equations

∇²u=r^-2 ∂/∂r ( r² ∂u/∂r ) =D^-1∂u/∂t
D>0

The Attempt at a Solution

Boundary Condition: u(r=a,t)Ξ0
Initial Condition: u(r,t=0)=ΔT*Top(0,a) where Top(0,a)=Θ(r)-Θ(r-a) is the top hat function with end-points 0 and a

This is not a precise formulation as I want u(0,0) to be ΔT and u(a,0)=0. However it conveniently represents u(0≤r<a,0)=ΔT and u(r>a,0)=0

Let u(r,t)=R(r)Q(t)

Separation of variables give
(Rr²)^-1d/dr (r² R')=(D*Q)^-1Q'=-μ² where μ∈R

The choice of the separation constant is such that the boundary condition can be satisfied

rR''+2R'+μ²rR=0
Q'+μ²DQ=0

The radial equation is simplified by the substitution S=rR, reducing it to S''+μ²S=0

Thus S=sin(μr) or cos(μr) and Q=exp(-μ²D*t) and u=SQ/r

Since I expect u to be finite at r=0, S(r=0)=0 is needed to create an indeterminate expression, ruling out the cosine solution.

S(r=a)=sin(μa)=0⇒μ_na=nπ ; n∈ℕ

So the solution has the form
u(r,t)=r^-1∑A_n sin(μ_n r)*exp(-μ_n²Dt)
u(r,0)=r^-1∑A_n sin(μ_n r)
u(r,0)*r=∑A_n sin(μ_n r)-----------------------(@)

Now multiply both sides with sin(μ_m r) and integrate both sides of (@) w.r.t. r from 0 to a
I found that A_n=2ΔT/(π*n/a) * (-1)ⁿ⁺¹ ;
I'd appreciate it if someone could point out where I made a mistake...

 

[mighty2000]

There are a few potential issues with your solution:

1. Boundary condition: The boundary condition you have chosen, u(r=a,t)=0, does not match the given condition of u(r=a,t)=ΔT. This may be the cause of some of the discrepancies in your solution.

2. Initial condition: The initial condition you have chosen, u(r,t=0)=ΔT*Top(0,a), does not match the given condition of u(r,t=0)=ΔT. This may also be contributing to the discrepancies in your solution.

3. Separation of variables: When using separation of variables, it is important to ensure that the resulting equations are consistent with the boundary and initial conditions. In this case, the boundary and initial conditions do not match the solutions you have obtained for R(r) and Q(t), so this may be another source of error.

4. Integration: In your final step, where you integrate both sides of (@) with respect to r, you should also include the term r in the integration. This will result in a different solution for An.

I recommend going back and carefully checking your boundary and initial conditions, as well as your separation of variables method and integration steps. You may also want to try a different approach to solving the problem, such as using Fourier series or a numerical method.