Lubrication of textured surfaces: A general theory for flow and shear stress factors

We report on a mean field theory of textured surface lubrication. We study the fluid flow dynamics occurring at the interface as a function of the texture characteristics, e.g. texture area density, shape and distribution of microstructures, and local slip lengths. The present results may be very important for the investigation of tailored microtextured surfaces for low-friction hydrodynamic applications.

Figure 1

Schematic (not in scale). A surface-structured solid (block) in contact with a rigid solid (substrate) with a flat surface. The substrate moves with the velocity ${\mathbf{v}}_0$ while the block is stationary.

Figure 2

Schematic. Effective medium theories allow us to describe, at the macroscopic scale, the disorder in a physical system, e.g., fluctuations in the interfacial separation $u\left(\mathbf{x}\right)$ or in the slip length $l\left(\mathbf{x}\right)$. For a $n$-component system (e.g., in the case of an interfacial separation $u$ constituted by $n$ different discrete values) the flow in the effective medium should be the same as the average fluid flow obtained when circular regions of the $n$-components are embedded in the effective medium.

Figure 3

A generic elliptical homogeneous inclusion immersed in a homogeneous infinite medium. The medium hydraulic conductivity $\Lambda$ is principally valued in the reference, whereas the medium sliding conductivity $\Gamma$ is not. $\phi$ is the ellipse semi-major axis inclination angle. The macroscopic flow driving terms $\nabla p_0$ and $U_m$ are also showed. A Peklenik number $\gamma =a/b$ can be defined as the ratio between the ellipse semi-major axis $a$ and the semi-minor axis $b$.

Figure 4

Pressure flow factor $\tilde{\Lambda }$ for circular inclusions embedded in an isotropic medium, as a function of the dimensionless hole ratio, and for different values of hole density $p_h$.

Figure 5

Shear flow factor $\tilde{\Gamma }$ for circular inclusions embedded in an isotropic medium, as a function of the dimensionless hole ratio, and for different values of hole density $p_h$.

Figure 6

BEM-homogenized fluid pressure (the smooth blue curve) as a function of $x$ coordinate (sliding direction), compared with the full scale numerical predictions partially adapted from Ref. [13].