Random-Roughness Hydrodynamic Boundary Conditions

We report results of lattice Boltzmann simulations of a high-speed drainage of liquid films squeezed between a smooth sphere and a randomly rough plane. A significant decrease in the hydrodynamic resistance force as compared with that predicted for two smooth surfaces is observed. However, this force reduction does not represent slippage. The computed force is exactly the same as that between equivalent smooth surfaces obeying no-slip boundary conditions, but located at an intermediate position between peaks and valleys of asperities. The shift in hydrodynamic thickness is shown to depend on the height and density of roughness elements. Our results do not support some previous experimental conclusions on a very large and shear-dependent boundary slip for similar systems.

Figure 1

Sketch of the system: a sphere of radius $R$ approaches a rough surface with a fixed area fraction $\varphi$ covered by roughness elements. The separation $h$ is defined on top of the surface roughness at position $x_2=r$.

Figure 2

Hydrodynamic force acting on a sphere with radius $R=16\delta x$ (triangles) and $R=8\delta x$ (circles) driven to a smooth wall with velocity $v={10}^{-3}\delta x/\delta t$ and $v={10}^{-4}\delta x/\delta t$, correspondingly. The solid and dashed curves are calculations of the force expected with no-slip boundary conditions at the wall [Eqs. (1, 2)]. Squares are the results measured for a rough wall ($\varphi =4\%$, $r=10\delta x$).

Figure 3

Hydrodynamic force plotted in different coordinates. (a),(b) the data sets for smooth surfaces reproduced from Fig. 2; (c),(d) show the force for two rough planes with $r=10\delta x$, $\varphi =4\%$ and $\varphi =50\%$. For $\varphi =4\%$ the asymptotic behavior for small $h$ cannot be fitted with a slip, but the assumption of an effective boundary position holds. The values for $\varphi =50\%$ recover the case of a flat surface at $r$. The data for a smooth slip boundary confirms the validity of Eq. (3).

Figure 4

Effective height $r_{\mathrm{eff}}$ normalized by the maximum height $r$ as a function of a density of roughness elements, $\varphi$, for $r=10\delta x$ and $r=20\delta x$ plotted in different scales.