Effective slip boundary conditions for flows over nanoscale chemical heterogeneities

We study slip boundary conditions for simple fluids at surfaces with nanoscale chemical heterogeneities. Using a perturbative approach, we examine the flow of a Newtonian fluid far from a surface described by a heterogeneous Navier slip boundary condition. In the far field, we obtain expressions for an effective slip boundary condition in certain limiting cases. These expressions are compared to numerical solutions which show they work well when applied in the appropriate limits. The implications for experimental measurements and for the design of surfaces that exhibit large slip lengths are discussed.

Figure 1

Illustration of the flow over a highly heterogeneous hydrophobic surface characterized by length scale $L$. At low pressures, the liquid is in the Cassie state, where it does not penetrate into the surface, leading to large effective slip lengths (left). At intermediate pressures, the liquid begins to penetrate the surface (described by radius of curvature $R$) and the effective slip decreases (center). Finally, at sufficiently high pressures the liquid penetrates the surfaces, which will drastically reduce the effective slip length (right).

Figure 2

Effective slip length as a function of stripe width $\beta$ as given by Eq. (18) (solid line) and by finite difference solutions of the Laplace equation (symbols) for the case of flow over parallel stripes for ${\delta }_1=0.1-10L$ and ${\delta }_2={10}^{2}L$.

Figure 3

Effective slip length for $\beta =0.1$ as a function of ${\delta }_2$ for several values of ${\delta }_1$ as given by Eq. (18) (solid line) and by finite difference solutions of the Laplace equation (symbols) for the case of flow over parallel stripes.