Rate-of-strain tensor

For a differentiable velocity field, the rate-of-strain tensor is the symmetric matrix

D(u)=\frac12\bigl(\nabla u+(\nabla u)^{\mathsf T}\bigr), \qquad D(u)_{ij}=\frac12(\partial_j u_i+\partial_i u_j).

Here $\nabla u$ denotes the velocity Jacobian, with $(\nabla u)_{ij}=\partial_j u_i$ .

Infinitesimal deformation

If two nearby particles have infinitesimal separation $\xi$ , then $\dot\xi=(\nabla u)\xi$ and

\frac{d}{dt}|\xi|^2=2\xi^{\mathsf T}D(u)\xi.

The skew-symmetric part of the gradient contributes local rotation but not this change in squared length. Incompressibility implies $\operatorname{tr}D(u)=0$ .

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.