  The paper is aimed at analysing a singular perturbation of the Navier-Stokes
equations on a compact closed manifold. The case of compact smooth manifolds
with boundary under the Dirichlet conditions is also included. Global existence
and uniqueness is established for the weak solutions of the Cauchy problem. The
solution of the regularised system is shown to converge to the solution of the
conventional Navier-Stokes equations provided it is uniformly bounded in
parameter.
