  The paper introduces a geometrically unfitted finite element method for the
numerical solution of the tangential Navier--Stokes equations posed on a
passively evolving smooth closed surface embedded in $\mathbb{R}^3$. The
discrete formulation employs finite difference and finite elements methods to
handle evolution in time and variation in space, respectively. A complete
numerical analysis of the method is presented, including stability, optimal
order convergence, and quantification of the geometric errors. Results of
numerical experiments are also provided.
