Virtual Element methods for non-Newtonian fluid flow problems
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In this work, we develop and analyze a Virtual Element Method (VEM) for the numerical approximation of non-Newtonian Stokes flow problems. Our analysis addresses a general class of fluids governed by a nonlinear viscosity law that satisfies assumptions involving a Sobolev exponent r and a degeneracy parameter delta, encompassing classical models such as the Carreau–Yasuda fluid. This framework ensures the well-posedness of both the continuous and discrete problems for shear-thinning (pseudoplastic, r < 2) and shear-thickening (dilatant, r > 2) behaviors. We derive a priori error estimates that explicitly depend on the rheological exponent r and the degeneracy parameter, thereby capturing the influence of the fluid’s non-Newtonian characteristics. Moreover, under additional regularity assumptions, we show that for non-degenerate (delta > 0) pseudoplastic fluids, the convergence rate becomes optimal and matches that of the Newtonian case (r = 2). Numerical experiments are presented to validate the theoretical bounds and to demonstrate the practical performance and flexibility of the proposed VEM formulation.