On the generation of lift forces in random soft porous media

Abstract

In this paper, we examine the generation of pressure and lift forces in a random soft fibrous media layer that is confined between two planar surfaces, an infinite horizontal lower boundary and a horizontal inclined upper boundary, in the lubrication limit where the characteristic thickness of the fibre layerH≪Lthe length of the inclined surface. The model for the fibre layer is a Brinkman equation and the Darcy permeabilityKpis described by the widely used Carman–Kozeny equation for random porous media. Two cases are considered: (a) an inclined upper boundary which slides freely on top of a stationary fibre layer which is firmly attached to the lower boundary and (b) an inclined stationary upper boundary with an attached fibre layer in which the horizontal lower boundary slides freely in its own plane beneath it. Superficially, the problems appear equivalent to the classical problem for a slider bearing where the solutions for the pressure distribution and lift force are independent of which boundary is moving. In this problem there is an optimum compression ratiok=h1/h2= 2.2, whereh1andh2are the heights at the leading and trailing edges, for maximum lift force. However, this symmetry is lost if the intervening space is filled with a soft porous fibrous material since the Brinkman equation is not invariant under a transformation of coordinates in which the inherently unsteady problem in case (a) is transformed to a steady reference frame in which the inclined upper boundary is stationary and the horizontal boundary with the adhered fibre layer moves below it. Although in the steady reference frame case (a) now appears to resemble case (b), the solutions are strikingly different and depend critically on the value of the dimensionless fibre interaction layer thickness$\alpha\,{=}\, H/\sqrt{K_p}$. For α ≪ 1 the solutions for both cases approach the classical solution for a slider bearing. For α ≫ 1 we show, using asymptotic analysis that the solutions diverge dramatically. In case (a) the pressure and lift force increase as α2and asymptotically approach a limiting behaviour for large values of α, first predicted in Feng and Weinbaum (J. Fluid Mech., vol. 422, 2000, p. 288), while in case (b) the pressure and lift force decay as α−2since the inclined upper boundary is screened by the fibre layer and the amount of fluid dragged through the fluid gap decreases as α increases and vanishes for α ≫ 1. The solution in case (a), where the inclined upper boundary moves, is of particular interest since it reveals the potential to generate enormous lift forces using commercially available inexpensive soft porous materials provided the lateral leakage at the edge of the planform can be eliminated through the use of a channel with impermeable sidewalls as first proposed in the work by Wu, Andreopolous and Weinbaum (Phys. Rev. Lett., vol. 93, 2004, p. 194501). The behaviour is illustrated for both a toboggan sliding in such a channel and a larger planform that might be useful in commercial transportation.