Viscous flow around spherical macroscopic cavities in a granular material is investigated. The Stokes equation inside and the Darcy–Brinkman equation outside the cavities are considered. In particular, the interaction of two equally sized cavities positioned in tandem is examined in detail, where the asymptotic effect of the other cavity is taken into account. The present analysis gives a reasonable estimate on the volume flow into the cavity and the local enhancement of stresses. This is applicable to predict the microscale waterway formation in that material, onset of landslides, collapse of cliffs and river banks, etc.
Viscous flow through a symmetric wavy channel filled with anisotropic porous material is investigated analytically. Flow inside the porous bed is assumed to be governed by the anisotropic Brinkman equation. It is assumed that the ratio of the channel width to the wavelength is small (i.e. δ 2 1). The problem is solved up to O(δ 2) assuming that δ 2 λ 2 1, where λ is the anisotropic ratio. The key purpose of this paper is to study the effect of anisotropic permeability on flow near the crests of the wavy channel which causes flow reversal. We present a detailed analysis of the flow reversal at the crests. The ratio of the permeabilities (anisotropic ratio) is responsible for the flow separation near the crests of the wall where viscous forces are effective. For a flow configuration (say, low amplitude parameter) in which there is no separation if the porous media is isotropic, introducing anisotropy causes flow separation. On the other hand, interestingly, flow separation occurs even in the case of isotropic porous medium if the amplitude parameter a is large.
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.
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