Velocity gradient ($\nabla \mathbf{u}$)

The velocity gradient describes how a velocity field varies between nearby points. With components $L_{ij}=\partial v_i/\partial x_j$, nearby material points separated by $\delta \mathbf{x}$ have relative velocity $L\delta \mathbf{x}$ to first order. It splits into symmetric $D=(L+L^{\mathsf{T}})/2$, the rate-of-strain tensor, and antisymmetric $W=(L-L^{\mathsf{T}})/2$, describing local spin.

A rigid rotation has $D=0$ despite a nonzero velocity gradient: neighbouring points move differently without changing their separation lengths. In simple shear $v_x=\dot{\gamma }y$, both $D_{xy}=\dot{\gamma }/2$ and spin are nonzero. The trace of $L$ is velocity divergence, determining local fractional volume change. These pieces separate stretching, rotation and dilation, making the velocity gradient more informative than a single strain rate in a general three-dimensional flow.

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.