Add a phantom to the contents of \overbrace; here's a command \xoverbrace that takes an optional argument (default a big fraction) for finer adjustments. I also added a handier way to typeset partial derivatives.

\documentclass{article}
\usepackage{amsmath}

\newcommand{\xoverbrace}[2][\vphantom{\dfrac{A}{A}}]{\overbrace{#1#2}}
\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}

\begin{document}


\begin{align}
\xoverbrace{\pder{\rho\mathbf{U}}{t}}^{\text{time derivative}}+
\xoverbrace{\mathbf{U}\cdot \nabla \rho\mathbf{U}}^{\text{advection}}
&amp;=
\xoverbrace{-\nabla P}^{\text{pressure gradient}}+
\xoverbrace{\mu\nabla^2\mathbf{U}}^{\text{diffusion}}+
\xoverbrace{\rho \mathbf{g}}^{\text{body force}},
&amp;&amp;\text{\emph{momentum eq.}}\label{eq:mom}
\\
\pder{\rho}{t}+\nabla\cdot \rho\mathbf{U}
&amp;=
0
&amp;&amp;\text{\emph{continuity eq.}}\label{eq:cont}
\end{align}

\end{document}




Overbrace height adjustment
I just switched from natbib to biber + biblatex, and the closest bibliography style to my preferred style is "ieee" (by Joseph Wright).  I am using it with the citation style "authoryear".  This makes the numbers in the bibliography useless; how do I remove them?
