Intuitive meaning of Neumann boundary condition

Question

I was revising some notes and found myself not understanding the Neumann boundary condition. I understand it analytically - it's $\frac{∂u}{∂x}|_{x=0}=f(t)$ (let's consider the 1-D example of a metal rod with coordinates $x\in[0,1]$ and time $t\in[0,\infty]$), but I can't understand what it really means. Wikipedia says it is "heat flux" - so I try to imagine heat flux as a heater on the left side of the rod. But if the rod is really hot then the heater accounts for negative heat flux, so the heat flux is the function of the rod's temperature $u|_{x=0}$, which is not in the definition. Another question is the meaning of the derivative $\frac{du}{dx}$. It denotes the steepness of the temperature function at $x=0$. But what if the rod is perfectly insulated inside itself and no thermal conduction is possible? That would only be possible if $\frac{∂u}{∂t}=0$ - a degenerate version of the heat equation. In other words, I don't understand how the $\frac{∂u}{∂x}|_{x=0}=f(t)$ boundary condition does not depend on the properties of the rod (the heat equation itself).

How do you understand this condition to yourself?

Answer 1

The condition says that the rate at which the temperature changes at the end of the rod where $x=0$ is a function of time. The special case $f(t)=k,$ for some constant $k,$ says the temperature at that point is fixed for all time.

Answer 2

I will answer the second part first. You can heuristically derive the heat equation by noting that

• since heat transfer is basically diffusion, the heat flux $\vec q$ is essentially given by the gradient of the temperature $u$ (this is Fourier's law), $$\vec q= - k \vec \nabla u\,,$$
• and the change of temperature at any given point depends of the net flux of heat, $$\frac{\partial u}{\partial t}= A \vec \nabla\cdot\vec q\,.$$

Here, $k$ is the thermal conductivity and $A$ is essentially the specific heat times density. Combining these equations, we get the heat equation, with a combined coefficient $\alpha$.

Now, you ask what happens if the rod is perfectly insulated inside itself. This would mean $k=0$ and thus $\alpha=0$, so indeed there would be nothing interesting happening. Of course, this is also physically obvious.

For the first part: From the general PDE perspective, one can always consider Dirichlet or Neumann BCs. The first, in this case, are physically intuitive -- the temperature at the boundary is kept fixed. The second choice, by Forier's law, means that the heat flux out of the boundary is fixed. If $u_x=0$, this is again simple to interpret -- the sample is insulated, and no heat enters of leaves. For a nonhomogeneous choice, $u_x=f(t)$, it is physically less obvious how to realise such a setting, but it still means that a given amount of heat is entering or leaving the sample. And since the heat equation is linear, it suffices to consider these cases to build general solutions.

However, there is a different choice of BCs (mixed or Robin BCs) that is easier to interpret: You can impose $u + \kappa u_x=f(t)$, with a constant $\kappa$. This says that the heat flux is proportional to the temperature -- just what you expect when the material is ion contact with some other substance, and the heat transfer depends on the relative temperature. Again, by linearity, this type of BCs is already covered if you have dealt with Neumann and Dirichlet BCs.