We consider the incompressible homogeneous Navier-Stokes (NS) equations on a torus (typically, in dimension 3); we improve previous results of Morosi and Pizzocchero (2014) on the approximation of the solution via an expansion in powers of the Reynolds number. More precisely, we propose this approximation technique in the C-infinity setting of Morosi and Pizzocchero (2015) and present new applications, based on a Python program for the symbolic computation of the expansion. The a posteriori analysis of the approximants constructed in this way indicates, amongst else, global existence of the exact NS solution when the Reynolds number is below an explicitly computable critical value, depending on the initial datum; some examples are given. (C) 2015 Elsevier Ltd. All rights reserved.