From dilute to entangled fibre suspensions involved in the flow of reinforced polymers: A unified framework

Abstract

A B S T R A C TMost suspension descriptions nowadays employed are based on Jeffery model and some of its phenomenological adaptations that do not take into account the possible existence of a relative velocity between the fibres and the suspending fluid when the fibre interactions increase. It is expected that at very low density of contacts, as predicted by standard suspension models, fibres move with the suspending fluid velocity. When the density of fibre interactions becomes extremely high and a percolated network of fibre contacts is established within the suspension, fibres cannot move anymore and then the fluid flows throughout the rigid or moderately deformable entangled fibre skeleton, like a fluid flowing through a porous medium. In between these two limit cases, one could expect that fibres move but with a velocity lower than the one of the suspending fluid. Thus, two contributions are expected, one coming from standard suspension theory in which fibres and fluid move with the same velocity, and the other resulting in a Darcy contribution consisting of the relative fibre/fluid velocity. In this paper, we elaborate a general model able to adapt continuously to all these flow regimes.

Advanced modeling and simulation of sheet moulding compound (SMC) processes

“…The fluid velocity is denoted by v, the one of the fibres that defines the suspension velocity by v s , and the relative velocity of the fluid with respect to the fibres by v d (the Darcy contribution). As discussed in [25] we have the equalities:…”

“…Lubrication approximation states that the velocity gradients in the thickness direction are much larger than in-plane gradients, and that the throughthe-thickness velocity component can be neglected with respect to the in-plane components. Following [25], pressure becomes independent on the thickness coordinate, i.e. P = P (x, y) and by enforcing no-slip boundary conditions at both gap walls the velocity field reads…”

“…The flow model must be complemented with the one governing the evolution of the fluid phase field needed for defining the flow domain at each time Ω f (t) [25] (a deep discussion on the flow domain updating was performed in [7]),…”

Stable and contact‐free time stepping for dense rigid particle suspensions

“…Fluid flow equations (Stokes or Navier‐Stokes) are coupled with evolution equations for the orientation tensors. In order to solve the resulting equations, models for interaction between particles and closure approximations have to be specified externally . This is in contrast to direct numerical simulations where individual particles are explicitly represented.…”