Squeeze-film flow between a flat impermeable bearing and an anisotropic porous bed

Abstract

We consider a theoretical model of the squeeze film in the presence of a porous bed. The gap between the porous bed and the bearing is assumed to be filled with a Newtonian fluid. We use the Navier-Stokes equation in the fluid region and the Darcy equation in the fluid filled porous region. Lubrication approximation is used to derive the corresponding evolution equation for the film thickness. We use G. S. Beavers and D. D. Joseph [“Boundary conditions at a naturally permeable wall,” J. Fluid. Mech. 30, 197–207 (1967)] and M. Le Bars and M. G. Worster [“Interfacial conditions between a pure fluid and a porous medium: Implications for binary alloy solidification,” J. Fluid. Mech. 550, 149–173 (2006)] condition at the liquid porous interface and present a detailed analysis on the corresponding impact. We assume that the porous bed is anisotropic in nature with permeabilities K2 and K1 along the principal axes. Accordingly, the anisotropic angle ϕ is taken as the angle between the horizontal direction and principal axis with permeability K2. We show that the anisotropic permeability ratio and the anisotropic angle make a significant influence on the contact time, flux, velocity, etc. Contact time to meet the porous bed when a bearing approaches under a constant prescribed load is estimated. We present some important findings (relevant to the knee joint) based on the anisotropic properties of the human cartilage. For a prescribed constant load, we have estimated the time duration, during which a healthy human knee remains fluid lubricated.

Mathematical analysis of hydrodynamics and tissue deformation inside an isolated solid tumor

“…For example, some of the biological tissues and cells display anisotropic permeability [46]. In particular, articular cartilage typically shows anisotropic nature [47,49]. Further, the variations in the porosity sometimes may induce nonlinear permeability [48].…”

Viscous backflow from a model fracture network: influence of a permeable boundary

“…knee and hip, Knox et al. 2015; Karmakar & Raja-Sekhar 2018), soft robotics (Matia & Gat 2015) and various microfluidic applications (Christov et al. 2018; Wang & Christov 2019).…”

“…The limit where the wall is rigid and the particle has a flat front is analogous to the problem presented here. Venerus (2018), Karmakar & Raja-Sekhar (2018) and Knox et al (2015Knox et al ( , 2017 investigated squeeze flows in liquid films in a confined channel bounded by a porous disk in contact with either a uniform-pressure reservoir or a sealed edge. However, they all considered the case of a constant load driving the flow.…”

“…Such cases include hydrodynamic load and thrust bearings (Venerus 2018), hydraulic fractures (Holditch 2007), diarthrodial joints (e.g. knee and hip, Knox et al 2015;Karmakar & Raja-Sekhar 2018), soft robotics (Matia & Gat 2015) and various microfluidic applications (Christov et al 2018;Wang & Christov 2019). However, many such applications involve not only a pressure-driven or boundary-driven motion along the channel, but can involve permeation into either a rigid or compressible porous layer.…”

Ferro-nanofluid squeeze film lubrication with heat transfer

“…Many authors have investigated various factors such as friction reduction, self-lubrication properties and cost-effective manufacturing of the porous bearings. 1–6 A porous squeeze film geometry can be extensively seen in the bearings of mechanical industries, ceramic engineering 5,6 and articular cartilages of human synovial joints. 7–10…”