Analytical solution for heat equation with simple boundary conditions

  • Thread starter uqjzhang
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  • #1
uqjzhang
4
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I am trying to solve the following heat equation ODE:
d^2T/dr^2+1/r*dT/dr=0 (steady state) or
dT/dt=d^2T/dr^2+1/r*dT/dr (transient state)

The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2.
I have searched the analytical solution for this kind of ODEs in polar coordinate systems, found a lot of solutions for more difficult problems :devil:

Can anybody help me with this one please?
 

Answers and Replies

  • #2
JJacquelin
801
34
The stady state problem :
d^2T/dr^2+1/r*dT/dr=0 is a simple linear ODE. With boundary conditions T(r1)=T1, T(r2)=T2, did you find the solution ?
The transiant problem :
dT/dt=d^2T/dr^2+1/r*dT/dr is a linear PDE. Boundary conditions T(r1)=T1, T(r2)=T2 are not enough. A unique solution requires more conditions related with time (for exemple what is the state at time 0, and/or how the T(r1), T(r2) are estabished as a function of time, or...)
Various methods are possible, with more or less difficulties depending on the initial conditions.
If the initial conditions are not stated, one can find the general solutions on the form of a series of particular solutions T(r,t)=Sum(Ak*Fk(r)*Gk(t)) where Ak are constants, Fk are Bessel functions or r and Gk is exponential functions of t. Other representations are possible.
So, first you have to clarify the initial conditions.
 
  • #3
JJacquelin
801
34
I found it difficult to solve the equation?
Anyone of you can help me?
Steady state solution in attachment :
 

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  • #4
JJacquelin
801
34
I found it difficult to solve the equation?
Anyone of you can help me?
Transient solutions in attachment :
 

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    Transient.JPG
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