  We consider the Virtual Element method (VEM) introduced by Beir\~ao da Veiga,
Lovadina and Vacca in 2016 for the numerical solution of the steady,
incompressible Navier-Stokes equations; the method has arbitrary order $k \geq
2$ and guarantees divergence-free velocities. For such discretization, we
develop a residual-based a posteriori error estimator, which is a combination
of standard terms in VEM analysis (residual terms, data oscillation, and VEM
stabilization), plus some other terms originated by the VEM discretization of
the nonlinear convective term. We show that a linear combination of the
velocity and pressure errors is upper-bounded by a multiple of the estimator
(reliability). We also establish some efficiency results, involving lower
bounds of the error. Some numerical tests illustrate the performance of the
estimator and of its components while refining the mesh uniformly, yielding the
expected decay rate. At last, we apply an adaptive mesh refinement strategy to
the computation of the low-Reynolds flow around a square cylinder inside a
channel.

Numerical Analysis;Numerical Analysis