I am currently working through Alexandre Chorin's Mathematical Introduction to Fluid Mechanics. In the first chapter, he treats the change in Kinetic energy of a fluid region $W\subset D$ subject to the fluid flow map $\varphi_t:\mathbf{x}\mapsto\varphi(\mathbf{x},t)$ in the following manner ($\frac{D}{Dt}$ denoting the material derivative):
\begin{aligned} \frac { d } { d t } E _ { \text { kinetic } } & = \frac { d } { d t } \left[ \frac { 1 } { 2 } \int _ { W _ { t } } \rho \| \mathbf { u } \| ^ { 2 } d V \right] \\ & = \frac { 1 } { 2 } \int _ { W _ { t } } \rho \frac { D \| \mathbf { u } \| ^ { 2 } } { D t } d V \\ & = \int _ { W _ { t } } \rho \left( \mathbf { u } \cdot \left( \frac { \partial \mathbf { u } } { \partial t } + ( \mathbf { u } \cdot \nabla ) \mathbf { u } \right) \right) d V \end{aligned}
As best as I can determine, an implicit assumption seems to be made that the fluid has a density constant with time in this derivation. Specifically, it appears that the assumption $\frac{\partial}{\partial t}(\rho\| \mathbf { u } \| ^ { 2 }) = \rho\frac{D}{Dt}(\| \mathbf { u } \| ^ { 2 })$ is being made rather than obeying the typical product rule. I am not quite sure why. I would greatly appreciate it if someone could help shed some light on what is going on here!
If more context is needed, I can provide it upon request, or you may reference the presentation which is given on page 12 of Chorin's book.