Generalized slip condition

Using a homogenisation technique, we generalize the well-known Navier slip condition in the form: \begin{equation*}u_i=-{W}_{ij}\partial_j {p} + {E}_{ilk}(\partial_k u_l + \partial_l u_k).\end{equation*} This condition may be applied to any flow over a rough or only partially wetted surface, without any limitation on the flow regime. The macroscopic velocity depends on $W_{ij}$, a ``wettability'' tensor formally analogous to a permeability, and on the tensorial slip length $E_{ijk}$. Components of these tensors are obtained as solutions of microscopic problems arising during the derivation of the above condition. We validated the latter by performing DNS of the flow about a rough sphere under various conditions. This rational framework clarifies the missed analogy between the microscopic characterization of Navier's classical slip length and its applicability to a macroscopic boundary condition.