Second-gradient modelling of orientation development and rheology of dilute confined suspensions

•Proposal of the Jeffery’s counterpart in second gradient confined flows.•Consideration of fiber-wall interactions.•Numerical analysis in Poiseuille and squeeze flows.•Impact of confinement on the suspension rheology. We address the extension of Jeffery’s model, governing the orientation of rods immersed in a Newtonian fluid, to confined regimes occurring when the thickness of the flow domain is narrower than the rod length. The main modelling ingredients concern: (i) the consideration of the rod interactions with one or both gap walls and their effects on the rod orientation kinematics; and (ii) the consideration of non-uniform strain rates at the scale of the rod, requiring higher-order descriptions. Such scenarios are very close to those encountered in real composites forming processes and have never been appropriately addressed from a microstructural point of view. We also show that confinement conditions affect the rheology of the suspension.