Deriving Heat Equation from change of variables.

[renegade05]

Struggling with this (simple?) problem. Part (a) is simple, but part b I am kinda lost. I am not sure what is meant by 'natural time scale'. My attempt was:

\(\displaystyle \alpha^2 = \frac{\kappa}{\rho c} \)

Then we have :

\(\displaystyle \frac{\partial \theta}{\partial t}=\alpha ^2 \frac{\partial^2 \theta}{\partial x^2}\)

I tried to solve for x, so x = L*Ksi and t = Eta * B and to do some chain rules.... but I don't think that is what the problem is after.

I also don't really know how to incorporate the u = function given.

Any tips and guidance would be appreciated!

 

[Deleted member 4993]

View attachment 4883Struggling with this (simple?) problem. Part (a) is simple, but part b I am kinda lost. I am not sure what is meant by 'natural time scale'. My attempt was:

\(\displaystyle \alpha^2 = \frac{\kappa}{\rho c} \)

Then we have :

\(\displaystyle \frac{\partial \theta}{\partial t}=\alpha ^2 \frac{\partial^2 \theta}{\partial x^2}\)

I tried to solve for x, so x = L*Ksi and t = Eta * B and to do some chain rules.... but I don't think that is what the problem is after.

I also don't really know how to incorporate the u = function given.

Any tips and guidance would be appreciated!

Start with

\(\displaystyle du = u_{\xi} d\xi + u_{\eta} d\eta\)

Lot of tedious algebra....

 

[renegade05]

Start with

\(\displaystyle du = u_{\xi} d\xi + u_{\eta} d\eta\)

Lot of tedious algebra....

What's
\(\displaystyle u_{\xi}, u_{\eta}\)

How do incorporate the two functions?