Approximate Analytical Model for the Squeeze-Film Lubrication of the Human Ankle Joint with Synovial Fluid Filtrated by Articular Cartilage

The aim of this article is to propose an analytical approximate squeeze-film lubrication model of the human ankle joint for a quick assessment of the synovial pressure field and the load carrying due to the squeeze motion. The model starts from the theory of boosted lubrication for the human articular joints lubrication (Walker et al., Rheum Dis 27:512–520, 1968; Maroudas, Lubrication and wear in joints. Sector, London, 1969) and takes into account the fluid transport across the articular cartilage using Darcy’s equation to depict the synovial fluid motion through a porous cartilage matrix. The human ankle joint is assumed to be cylindrical enabling motion in the sagittal plane only. The proposed model is based on a modified Reynolds equation; its integration allows to obtain a quick assessment on the synovial pressure field showing a good agreement with those obtained numerically (Hlavacek, J Biomech 33:1415–1422, 2000). The analytical integration allows the closed form description of the synovial fluid film force and the calculation of the unsteady gap thickness.

Geometric Integrability of the Camassa-Holm Equation. II

It is known that the Camassa–Holm (CH) equation describes pseudo-spherical surfaces and that therefore its integrability properties can be studied by geometrical means. In particular, the CH equation admits nonlocal symmetries of “pseudo-potential type”: the standard quadratic pseudo-potential associated with the geodesics of the pseudo-spherical surfaces determined by (generic) solutions to CH, allows us to construct a covering π of the equation manifold of CH on which nonlocal symmetries can be explicitly calculated. In this article, we present the Lie algebra of (first-order) nonlocal π-symmetries for the CH equation, and we show that this algebra contains a semidirect sum of the loop algebra over sl(2,R) and the centerless Virasoro algebra. As applications, we compute explicit solutions, we construct a Darboux transformation for the CH equation, and we recover its recursion operator. We also extend our results to the associated Camassa–Holm equation introduced by J. Schiff.

New reynolds equation for line contact based on the carreau model modification by bair

This paper presents a new form of the one-dimensional Reynolds equation for lubricants whose rheological behaviour follows a modified Carreau rheological model proposed by Bair. The results of the shear stress and flow rate obtained through a new Reynolds–Carreau equation are shown and compared with the results obtained by other researchers.