= Momentum balance for a material fluid element
{title2=$\rho D\mathbf u/Dt=-\nabla p$}
For an inviscid <material volume element>, neglecting body forces, <Newton's second law> gives $D(\rho\mathbf u\delta V)/Dt=-\nabla p\,\delta V$. Since <mass conservation> keeps $\rho\delta V$ constant, the result is $\rho D\mathbf u/Dt=-\nabla p$. Cancelling $\delta V$ before differentiating would incorrectly omit its changing volume. A fixed control volume instead requires the advected <momentum flux>, yielding $\partial_t(\rho\mathbf u)+\nabla\cdot(\rho\mathbf u\otimes\mathbf u)=-\nabla p$.
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