Application of Fourier law of conduction

[QuArK21343]

Consider a wire of length l. At time t=0, one end of the wire is at temperature T_0, while the rest of the wire is at temperature T_1, T_1<T_0 (for example, one end of the wire is kept in contact with an external hot body, at temperature T_0). After how much time the temperature near the second end of the wire will begin to rise? I tried to use Fourier law of conduction, in particular I wrote. starting from the diffusion equation, that the Fourier transform of the function T(x,t) that gives the temperature along the wire is f(k)e^(-k^2Dt), D is the diffusion constant and f is to be determined from the initial conditions. From now on, I am not sure how to go on...

 

[haruspex]

The classical diffusion law implies the temperature would start to rise everywhere in the conductor immediately. Diffusion doesn't have a 'speed' in that sense. Of course, it can't really happen faster than light, but that doesn't help much. The resolution is that it only starts to rise very slowly at first at remote points. So if you want the speed of diffusion to mean something you have to find another way to define it.
You could specify a threshold temperature rise; or you could define the 'wavefront' as being the location at which the temperature gradient is a maximum.

 

[QuArK21343]

So, is it impossible to extract some kind of heat-front velocity directly from the heat equation (as in the classical wave equation)? And why is it so? Is there some way to see that the velocity of propagation is infinite from the equation itself?

Also, is there some simple way to estimate the velocity of the wavefront along which the temperature gradient is maximum?

Thanks

 

[haruspex]

Actually I'm not at all sure that the point of maximum gradient reasonably matches the intuitive concept of wavefront. Rising above a temperature threshold would be better.
In the set up you state, there is zero length of wire at T_0, so I think you'll find the temperature will become T_1 everywhere immediately. You need more of a heat reservoir, or some external source maintaining one end at T_0.
Once you've clarified the set up you can try solving the equation and then seeing if some reasonable definition of wavefront provides an algebraic solution, but I don't hold out much hope.

 

[vanhees71]

The reason for this discrepency between the heat-conduction (diffusion) equation and special relativity is very simple: The diffusion equation is not relativistic. If you want to consider heat conduction within a relativistic framework you have to use relativistic transport theory.

 

[haruspex]

More thoughts...
At http://www.mth.pdx.edu/~marek/mth510pde/notes%202.pdf there's a useful equation (2.4.12) for the infinite uniform bar case. For t > 0, substituting x+2y√t for y inside the integral gives u(x, t) = (1/√π)∫e^(-y^2).g(x+2y√t).dy.
That can be read as indicating that the influence of an initial temperature g(x) at point x spreads out as the square root of time.

 

[QuArK21343]

Thank you, haruspex, for your link. I'll read it with care (I've had a quick look and seems good). Yet, I still don't see if and why the velocity of heat transfer should be infinite (I mean, is there a way to look at the heat equation and see it?).

 

[haruspex]

Looking at eqn 2.4.12, consider some very small t > 0 and an initial distribution with a very high temp over a smal range around x =0, 0 elsewhere.
The integral gives a positive result for all x.
I don't think you need to invoke relativity to explain why this doesn't happen in practice. It's probably limited by the velocity of sound in the object.