A Note on the Vector Form of the Navier-Stokes Equations

The Navier-Stokes equations, written in conservative vector form, are

\frac{\partial \rho \mathbf{u}}{\partial t} = -\nabla\cdot\mathbf{u}\rho\mathbf{u} – \nabla\cdot\boldsymbol{\tau} – \nabla p

I always liked to use this form of the Navier-Stokes equations. I think it is simple and convenient. But, today I realized that it could easily lead to trouble, specifically when interpreting the convective term. For example, if one needs to expand the convective term things could easily get confusing. The reason is that the convective term, as written here, is known as a diadic – when two vectors are stacked next to each other $\mathbf{u}\rho \mathbf{u}$.  The divergence of a diadic is given by
\nabla\cdot\mathbf{A}\mathbf{B} = \mathbf{A}\cdot\nabla\mathbf{B} + \mathbf{A}\nabla\cdot\mathbf{B}
But this may be confusing for the convection term since you can easily lose track of what A and B are. Here’s how I do it – but first let me remind you of a few things:

  1. It is essential to distinguish which quantity is being transported in the convective term. In the case of a conserved scalar $\rho\phi$, the convective term is $\nabla\cdot\mathbf{u}\rho\phi$. Here, $\mathbf{u}$ is the advective velocity and the quantity being transport is $\rho\phi$. While it is not wrong to write the convective term as $\nabla\cdot\rho\mathbf{u}\phi$ – using the former approach makes things a bit clearer.
  2. On to the momentum equations, and virtue of item 1, the convective term is $\nabla\cdot\mathbf{u}\rho\mathbf{u}$, the transported quantity is $\rho\mathbf{u}$ and the advective velocity is $\mathbf{u}$. But which velocity component does $\mathbf{u}$ correspond to in $\rho\mathbf{u}$?

To resolve this last issue, I use the following mixed tensor-vector notation

\frac{\partial \rho u_i}{\partial t} = -\nabla\cdot\mathbf{u}\rho u_i – \nabla\cdot\boldsymbol{\tau}_i – \frac{\partial p}{\partial x_i}

I use this form whenever I want to derive the weak form of the momentum equations to avoid using diadics. When all is said and done, I switch back to the vector form for convenience.

Cite as: Tony Saad, “A Note on the Vector Form of the Navier-Stokes Equations,” in Dr. Saad’s Notes, retrieved on April 15, 2015,

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