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Research Papers: Biotribology

# A Potential Elastohydrodynamic Origin of Load-Support and Coulomb-Like Friction in Lung∕Chest Wall Lubrication

[+] Author and Article Information
James P. Butler

Molecular and Integrative Physiological Sciences, Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115; Department of Geriatric and Respiratory Medicine, Tohoku University School of Medicine, Sendai 980, Japan; Harvard Medical School, 25 Shattuck Street, Boston, MA 02115

Stephen H. Loring1

Department of Anesthesia and Critical Care, Beth Israel Deaconess Medical Center, 330 Brookline Avenue, Boston, MA 02215; Harvard Medical School, 25 Shattuck Street, Boston, MA 02115sloring@bidmc.harvard.edu

1

Corresponding author.

J. Tribol 130(4), 041201 (Aug 01, 2008) (7 pages) doi:10.1115/1.2958076 History: Received April 06, 2007; Revised March 14, 2008; Published August 01, 2008

## Abstract

During normal breathing, the mesothelial surfaces of the lung and chest wall slide relative to one another. Experimentally, the shear stresses induced by such reciprocal sliding motion are very small, consistent with hydrodynamic lubrication, and relatively insensitive to sliding velocity, similar to Coulomb-type dry friction. Here we explore the possibility that shear-induced deformation of surface roughness in such tissues could result in bidirectional load-supporting behavior, in the absence of solid-solid contact, with shear stresses relatively insensitive to sliding velocity. We consider a lubrication problem with elastic blocks (including the rigid limit) over a planar surface sliding with velocity $U$, where the normal force is fixed (hence the channel thickness is a dependent variable). One block shape is continuous piecewise linear (V block) and the other continuous piecewise smoothly quadratic (Q block). The undeformed elastic blocks are spatially symmetric; their elastic deformation is simplified by taking it to be affine, with the degree of shape asymmetry linearly increasing with shear stress. We find that the V block exhibits nonzero Coulomb-type starting friction in both the rigid and the elastic case, and that the smooth Q block exhibits approximate Coulomb friction in the sense that the rate of change of shear force with $U$ is unbounded as $U→0$, shear force $∝U1∕2$ in the rigid asymmetric case and $∝U1∕3$ in the (symmetric when undeformed) elastic case. Shear-induced deformation of the elastic blocks results in load-supporting behavior for both directions of sliding. This mechanism could explain load-supporting behavior of deformable surfaces that are symmetrical when undeformed and may be the source of the weak velocity dependence of friction seen in the sliding of lubricated, but rough, surfaces of elastic media such as the visceral and parietal pleural surfaces of the lung and chest wall.

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## Figures

Figure 1

(Top panel) Geometry of the linear V block. The channel height d is a linear (continuous) function in both of its two segments and is normalized by the surface roughness δ, given by max(d)−min(d). The degree of asymmetry is ξ=0.2. This is the point x∕L at which the channel height is minimal. The dimensionless minimal channel height is given by D0=0.3. The bottom surface is flat and is sliding in the positive x direction with velocity U. (Bottom panel) Geometry of the quadratic Q block. The channel height is a quadratic function in both of its two segments and horizontal where they join. The Q block geometry is shown for the same degree of asymmetry and minimal channel height D0.

Figure 2

Coefficient of sliding friction FT∕FN versus normalized velocity, U*=μU∕rFN, for rigid V blocks (light lines with V label) and rigid Q blocks (bold lines with Q label). These are shown as a family in degree of asymmetry of the blocks given by ξ=0.1 (solid line), 0.2 (dotted line), and 0.3 (dashed line). Surface roughness is r=0.2. At fixed U*, the friction coefficient decreases with increasing asymmetry.

Figure 3

Dimensionless minimal gap thickness D0 versus normalized velocity, U*=μU∕rFN. These families are for the same degree of asymmetry (ξ=0.1, 0.2, and 0.3) and surface roughness r=0.2 as in Fig. 2.

Figure 4

Coefficient of sliding friction FT∕FN versus normalized velocity, U*=μU∕rFN, for elastic V blocks (light lines with V label) and elastic Q blocks (bold lines with Q label). These are shown as a family in material stiffness Φ of the blocks given by Φ∕FN=3.0 (solid line), 1.0 (dotted line), and 0.3 (dashed line). Surface roughness is given by r=0.2. At fixed U*, the friction coefficient increases with increasing material stiffness.

Figure 5

Coefficient of sliding friction FT∕FN versus normalized velocity, U*=μU∕rFN, each axis scaled by the ratio of geometric asymmetry ξ to surface roughness r. This scaling reduces all data, for both rigid and elastic materials, to universal curves. Light line, V block; bold line, Q block.

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