Understanding Neumann BC

Question

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!

Answer 1

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:

$u_x(x,t)=\frac{k}{\mu}\frac{\partial{P}}{\partial{x}}$

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

Example

Consider a system of 2 grids as shown below:

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

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

$\Rightarrow P_1=P_0$

Answer 2

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:

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.