Original entry by Joerg Fritz, AP225 Fall 2009
J.M. Skotheim and L. Mahadevan: Physical Review Letters, 2004, 92, pp 245509-1 to 245509-4
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.
Understanding the physics
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.