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The nonconforming Morley-type virtual element method for the incompressibleNavier-Stokes equations formulated in terms of the stream-function on simplyconnected polygonal domains (not necessarily convex) is designed. A rigorousanalysis by using a new enriching operator is developed. More precisely, byemploying such operator, we provide novel discrete Sobolev embeddings, whichallow to establish the well-posedness of the discrete scheme and obtain optimalerror estimates in broken $H^2$-, $H^1$- and $L^2$-norms under minimalregularity condition on the weak solution. The velocity and vorticity fieldsare recovered via a postprocessing formulas. Furthermore, a new algorithm forpressure recovery based on a Stokes complex sequence is presented. Optimalerror estimates are obtained for all the postprocessed variables. Finally, thetheoretical error bounds and the good performance of the method are validatedthrough several benchmark tests.

The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes