# Soft lubrication

Original entry by Joerg Fritz, AP225 Fall 2009

## Contents

## Source

J.M. Skotheim and L. Mahadevan: *Physical Review Letters*, 2004, 92, pp 245509-1 to 245509-4

## Keywords

Lubrication theory, Elsticity, Soft interface, Poroelasticity, Adhesion

## Summary

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.

## An exemplary scaling

A simple way to derive the scaling for the pressure as a function of the gap height is to balance the only two forces existing in the layer according to the lubrication approximation: pressure and viscosity. This means <math>p/l \approx \mu V / h^2</math> or

<math>p \approx \frac{\mu V R^{1/2}}{h^{3/2}}</math>

for definitions of the quatities, refer to figure 1.

It is possible to split up this pressure into an "inelastic" component that cannot contribute to the lift as outlined before and an "elastic" component that will. The lift itself will then scale like

<math>F \approx \frac{\mu V R}{h_o^2} \Delta h</math>

where <math>\Delta h</math> is the solution of an elasticity problem. For the geometry drawn in figure 1, a thin compressible layer the normal strain scales like <math>\Delta h/H_l \approx p/G</math> where *G* is the elasticity of the layer. We can plug this into our general scaling for the lift to get

<math>F_{t,c} = \frac{\mu^2 V^2}{G} \frac{H_l R^{3/2}}{h_0^{7/2}}</math>

## Conclusion

In a very similar manner a large class of other geometries and elastic models can be treated, which are summarized in the table in figure 2. This scalings in this table govern in principle all elastic lubrication problems with non-conforming contacts, a class of problems with a truly astounding breadth. A possible extension to this problem formulation is possible. In a large class of biological problems the goal is not create the maximal normal force possible, but to create a high enough normal force while keeping the viscous forces on the particle small. The framework presented in this paper would certainly allow such an extension.