A numerical method for a second-gradient theory of incompressible fluid flow

This work concerns the development of a finite-element method for discretizing a recent second-gradient theory for the flow of incompressible fluids. The new theory gives rise to a flow equation involving higher-order gradients of the velocity field and introduces an accompanying length scale and boundary conditions. Finite-element methods based on similar equations involving fourth-order differential operators typically rely on C1-continuous basis functions or a mixed approach, both of which entail certain implementational difficulties. Here, we examine the adaptation of a relatively inexpensive, non-conforming method based on C0-continuous basis functions. We first develop the variational form of the method and establish consistency. The method weakly enforces continuity of the vorticity, traction, and hypertraction across interelement boundaries. Stabilization is achieved via Nitsche’s method. Further, pressure stabilization scales with the higher-order moduli, so that the classical formulation is recovered as a particular limit. The numerical method is verified for the problem of steady, plane Poiseuille flow. We then provide several numerical examples illustrating the robustness of the method and contrasting the predictions to those provided by classical Navier–Stokes theory.