Stochastic Departures from Randomness

Abstract

So far we have considered only random fibre networks where fibre centres are distributed according to a point Poisson process in the plane and fibre axes have a uniform distribution of orientations to any given direction. When discussing the fractional between-zones variance in Section 4.2.3, we remarked that industrially manufactured fibre networks often exhibit a greater variance of local areal density than a random fibre network formed from the same constituent fibres and that this arises as a consequence of fibre interactions during the forming process. A competing effect occurs during the evolution of a fibre network during filtration of a suspension and results as a consequence of the distribution of permeabilities discussed in the previous section: fibres are deposited preferentially in high permeability regions of the evolving network. Since permeability regions with high permeability will typically have low areal density and thickness, the network is ‘smoothed’ [145] or ‘self-heals’ [62, 95, 111] during its evolution. Of course, if the first layers of an evolving structure are highly non-uniform as a result of clumping, or fibre flocculation, then we expect the contribution of smoothing to be greater. Experimental evidence suggests that this is indeed the case, but that the smoothing is not a sufficiently strong process to overcome the influence of fibre clumping [145].