We present an implementation of the finite-element method with hierarchical basis functions of arbitrary order for simulating incompressible steady-state fluid flow governed by Navier-Stokes equations (NSE). To verify the proposed formulation, we solved a problem of the fluid flow around a rigid sphere considering different velocity magnitudes. For a low-Reynolds-number flow the result coincides with the analytical solution for the Stokes flow, while for a higher Reynolds number a vortex is

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... ed behind the sphere. The developed framework can be used to study the fluid flow in contact interfaces between solids with rough surfaces (e.g. in fractures) and compare the effect of the model describing the fluid flow (Reynolds, Stokes, or full NSE) on the transmissivity of these interfaces. The development is undertaken in MoFEM.