= Velocity divergence
{title2=$\nabla\mathbin\cdot\mathbf u$}
<Velocity divergence> measures the local fractional rate of change of a moving material element's <volume>:
$$
\nabla\cdot\mathbf v=\operatorname{tr}(\nabla\mathbf v),\qquad \frac{D\delta V}{Dt}=(\nabla\cdot\mathbf v)\delta V.
$$
Positive values indicate local expansion and negative values local contraction. For a uniform dilation $\mathbf v=a\mathbf x$ in three dimensions, it equals $3a$; a rigid <rotation> has zero <velocity divergence>.
<Mass conservation> gives $D\rho/Dt=-\rho\nabla\cdot\mathbf v$, linking local expansion to decreasing <mass density>. An <incompressible flow> has zero <velocity divergence>, but can still stretch one material direction while compressing another, or shear a parcel without changing its <volume>. A narrowing steady pipe illustrates that distinction: fluid accelerates axially while transverse contraction balances that stretching so that the total <velocity divergence> can remain zero.
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