What does it mean for a surface to evolve with divergence-free velocity?

Question

Suppose we have an evolving hypersurface which evolves with a velocity field $V$, such that $\nabla_S \cdot V = 0$ where $\nabla_S$ is the surface or tangential gradient.

What does this mean? What does it mean physically for example?

Answer 1

The full divergence of a vector field tells you how the flow of that vector field changes volumes: we have $\mathcal L_V\ d\textrm{Vol} = \operatorname{div} V\ d\textrm{Vol}$. In integral terms this means $$\frac d {dt} \operatorname{Vol}(\Omega(t))=\int_{\Omega(t)} \operatorname{div}V\ d\textrm{Vol}$$ for a volume $\Omega$ flowing along $V$.

The in-surface divergence is the analogous quantity for the (hyper-)surface (hyper-)area of a flowing hypersurface: we have $\mathcal L_V\ dA_S = \operatorname{div}_S V\ dA_S$ for $dA_S$ the surface area form of the surface $S(t)$; i.e.

$$ \frac d {dt} \operatorname{Area}(S(t)) = \int_{S(t)} \operatorname{div}_S V\ dA_S.$$