I understand what Neumann BC means physically and how to imply them. However, I am not able to perfectly understand it the way it is represented mathematically as $\partial u / \partial n$ where $n$ is the outward normal vector to the boundary and $u$ is the solution.

I understand that the normal vector is essential for calculation over surface integral. But what are we doing here? Traction is outcome of partial derivative of the solution vector w.r.t to the normal vector? What does this mean?

Can any one try to simplify this briefly?

Thanks a lot!

  • $\begingroup$ To me the mathematical sense is clearer than the "physical" one, in part because your application's meaning for $u$ is unstated. $\endgroup$
    – hardmath
    Nov 22, 2015 at 18:02
  • 1
    $\begingroup$ Say 'u' is the displacement of the surface as a wave passes through it. $\endgroup$
    – CRG
    Nov 23, 2015 at 4:32
  • $\begingroup$ The normal derivative gives you the flux, i.e., the "amount" that is going in or out of your domain. $\endgroup$
    – nicoguaro
    Nov 24, 2015 at 23:05

2 Answers 2


Basic Concept

The normal vector corresponds to a physical parameter such as cartesian coordinate $x$. For example, a flux along $x$ direction can be defined by Darcy's law as following:


According to Neumann's BC, the flux should be constant or zero at the boundary i.e. $u(0,t)=u(L,t)=0$

In other words, $\frac{k}{\mu}\frac{\partial{P(0,t)}}{\partial{x}}=\frac{k}{\mu}\frac{\partial{P(L,t)}}{\partial{x}}=0$


Consider a system of 2 grids as shown below:

enter image description here

If we apply the above concept for this system, we can get the following equations:

$\Rightarrow \frac{P_i-P_{i-1}}{\Delta x}=\frac{P_1-P_{0}}{\Delta x}=0$

$\Rightarrow P_1=P_0$


When we write $\partial{u}/\partial{n}$ this is just a short-hand for $\nabla{u}\dot{}n$. Thus we are taking the dot product of the gradient and the unit normal. In the case of a rectangular region, for example:

enter image description here

the unit normal on the left wall is (1,0) (or (-1,0) depending on your definition). Thus the Neumann condition would be $\partial{u}/\partial{x}=0$ on the left hand wall. Similarily for the bottom edge the unit normal is (0,1) and so the Neumann condition would be $\partial{u}/\partial{y}=0$.

In terms of the surface integral I think the Wikipedia explaination is pretty good. It says in part Wikipedia:

Alternatively, if we integrate the normal component of the vector field, the result is a scalar. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S per unit time. This illustration implies that if the vector field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S, and neither in nor out.

  • $\begingroup$ Hi James, thanks for your reply. So, the gradient of the solution 'u' is the partial derivative of it in 3 directions and we are doing a dot product of it with normal vector. So, $\nabla u . n = \nabla u * n * cos \theta$ for each $ \theta$ between particular $u$ and $n$? $\endgroup$
    – CRG
    Nov 19, 2015 at 19:47
  • $\begingroup$ And what do they mean when they say zero (homogenious Neumann BC) , i.e., ∂u/∂n = 0, i.e., traction is zero? $\endgroup$
    – CRG
    Nov 19, 2015 at 20:14
  • 3
    $\begingroup$ Zero Neumann condition, i.e. $\nabla{u}\dot{}n=0$, simple means that the gradient of u is parallel to the surface S. For example if the variable $u$ is temperature then the Neumann condition corresponds to an insulated boundary - i.e. there is no flux of temperature through the surface/boundary S. $\endgroup$
    – James
    Nov 21, 2015 at 21:08

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