Linear stability of free shear flows of fibre suspensions

Abstract

A linear stability analysis of the mixing layer in the presence of fibre additives is presented. Using a formulation based on moments of the probability distribution function to determine the particle orientation, we extend the classical linear stability theory and derive a modified Orr–Sommerfeld equation. It is found that, for large Reynolds numbers, the flow instability is governed by two parameters: a dimensionless group H, analogous to a reciprocal Reynolds number representing the importance of inertial forces versus viscous forces associated with the anisotropic elongational viscosity of the suspension; and a coefficient CI that accounts for inter-particle hydrodynamic interactions. A parametric study reveals that both parameters can induce an important attenuation of the flow instability. Furthermore, we show that the stabilizing effects arise from the orientation diffusion due to hydrodynamic interactions, and not from the anisotropy induced by the presence of fibres in the flow, as speculated before. The examination of profile contours of different perturbation terms and the analysis of the rate of production of enstrophy show clearly that the main factor behind the reduction of the flow instability is associated with the fibre shear stress disturbance. This disturbance acts as a dissipative term as the fibres, due to the orientational diffusivity arising from hydrodynamic interactions, deviate from the fully aligned anisotropic orientation. On the other hand, fibre normal stresses act as a destabilizing factor and are important only in the absence of hydrodynamic interactions.