# How do you interpret this symbol within the following Navier-Stokes equations?

I was reading this fluid simulation paper(http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/ns.pdf) by Jos Stam and encountered these Navier-Stokes equations.

1.

2.

Now, I know that

However, how should I interpret from the second equation? Because dot products are interchangeable, does it mean that it's zero by the first equation? That would not make any sense though.

Any help would be appreciated! Thanks in advance.

• – jacob1729 Mar 24 '19 at 22:09
• Navier-Stokes in eq. n.2. From your question, I would deduce that your problem is not Navier-Stokes equation but some vector field analysis ( en.wikipedia.org/wiki/Vector_calculus ). If $\nabla$ is looked as a vector operator of components $\frac{\partial{}}{\partial{x}},\frac{\partial{}}{\partial{y}},\frac{\partial{}}{\partial{z}}$, $u \cdot \nabla = u_x \frac{\partial{}}{\partial{x}} +u_y \frac{\partial{}}{\partial{y}} +u_z \frac{\partial{}}{\partial{z}}$. Each cartesian component of N-S equation contains this operator applied in turn to $u_x$,$u_y$, and $u_z$. – GiorgioP Mar 24 '19 at 22:22

You should interpret $$\left(u \cdot \nabla\right)$$ as the operator such that for any function $$f\left(x, y, z\right),$$ $$\left(u \cdot \nabla\right)f = u_x\frac{\partial f}{\partial x} + u_y\frac{\partial f}{\partial y} + u_z\frac{\partial f}{\partial z}.$$