Constants of Heart Diffusion Equation

[Mike87]

Hi all,

Please could someone explain me how to get constants of Heat Diffusion Eq. for cylindrical coordinates step by step.

\(\displaystyle \dfrac{1}{r}\, \dfrac{\partial}{\partial r}\, \left(r\, \dfrac{\partial T}{\partial r}\right)\, =\, 0\)

Thanks a lot!

 

[HallsofIvy]

I don't understand what you mean by "constants" here. Are you asking how to derive this equation or how to solve it?

How you would derive this equation depends upon what facts of "heart diffusion" you are starting with.

Solving it is relatively simple. First multiply both sides by r to get \(\displaystyle \frac{\partial}{\partial r}\left(\frac{r \partial T}{\partial r}\right)= 0\). Since the derivative with respect to r is 0, the function must not depend on r but, in cylindrical coordinates might depend on \(\displaystyle \theta\) and z: \(\displaystyle r\frac{\partial T}{\partial r}= F(\theta, z)\) where F is an unknown function of \(\displaystyle \theta\) and z.
So \(\displaystyle \frac{\partial T}{\partial r}= \frac{1}{r}F(\theta, z)\) and, integrating with respect to r again, \(\displaystyle T(r,\theta, z)= -\frac{F(\theta, z)}{r^2}+ G(\theta, z)\) where G is another unknown function of \(\displaystyle \theta\) and z.

 

[Deleted member 4993]

Hi all,

Please could someone explain me how to get constants of Heat Diffusion Eq. for cylindrical coordinates step by step.

\(\displaystyle \dfrac{1}{r}\, \dfrac{\partial}{\partial r}\, \left(r\, \dfrac{\partial T}{\partial r}\right)\, =\, 0\)

Thanks a lot!

Start with:

https://www.youtube.com/watch?v=KRklpKkuMx0