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Could someone please clarify the difference between flux and normal derivatives?

I am studying PDE's and I came across Neumann Boundary condition (In this condition, the value of the normal derivative is given). I then came across the following: For a steady state thermal process, the Neumann condition describes a prescribed heat flux.

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  • $\begingroup$ This seems similar to the explanation here. $\endgroup$ – Jeffrey J Weimer Nov 24 '18 at 2:01
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The 1-D energy balance on an object with no internal heat generation is

$$k \frac{\partial ^2T}{\partial z^2} = \rho \tilde{C}_p \frac{\partial T}{\partial t} $$

A viable boundary condition (BC) for energy flow into an object is written as

$$-k A\left. \frac{dT}{dz}\right|_{boundary} = \dot{q}$$

Consider an solid object with heat flow entering from the right. The BC says, the heat flow (J/s) that enters is conducted through the object.

This condition applies regardless of steady state or not. In steady state, $\partial T/\partial t = 0$.

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First of all there are a few types of boundary conditions:

  1. Dirichlet BC
  2. Neumann BC
  3. Cauchy BC
  4. mixed BC
  5. Robin BC

The most usual are 1 and 2, while the rest are essentially combinations of the first two. I'll limit the description to the first two. For an example I will use a 2nd order ODE (because most people are more familiar that PDEs), before explaining the flux.

Assuming you have the following equation: $$m\ddot{x}+ c\dot{x} + x =0 $$ then in order to solve it you need two boundary conditions. All the following are valid boundary solutions:

  • $x(t=0) = 0, \; x(t=1)=2$ (two Dirichlet BC)
  • $x(t=0) = 0, \; \dot{x}(t=0)=1$ (one Dirichlet and one Neumann BC)
  • $\dot{x}(t=0)=1, \; \dot{x}(t=1)=0$ (two Neumann BC)

So Neumann BC are associates with derivatives of the variable, while Dirichlet are associated with the values of the variable.

Returning to the concept of the normal derivative and the flux. A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space.

The most easy example understand this is probably heat transfer/diffusion. You can think of a rod or a plaque that you apply some heat (imagine a small torch) at one end. The temperature at that end is not fixed, however the heat supply (or heat flux) is, so in that situation you want to apply a Neumann BC. The normal derivative has to do with the fact that the heat travels perpendicular to the boundary surface (rod crosssection / plaque side).

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