 Velocity divergence

Velocity divergence measures the local fractional rate of change of a moving material element's volume: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.