Evolution During Forming

Paper has a layered structure; how are the layers related?

Two extreme cases:

Independent Layers:
Cov(pq) =0 so Var(p+q) = Var(p) + Var(q), and hence, for statistically similar layers we conclude that

Laminated papers or forming from dilute, well-mixed suspensions.

Reinforcing Layers:
q=p so Cov(pq)= Var(p) and hence

Precisely cloned successive layers.

Hydrodynamic Smoothing Modeled And Detected

Fibres flow towards lower density regions of earlier layers. Smoothing appears as negative covariance of local areal densities in successive layers

In practice, we expect:
Smoothing Layers: Cov(pq) < 0.
If Cov(pq) < 0, then define the
smoothing parameter s:
$Cov(pq)=-s\, Var(p),$ for $0 \leq s < 1.$
Then

See Sampson et al. [36] for experimental detection and verification of smoothing.