Eulerian Numerical Study of the Sedimentation of Fibre Suspensions

Abstract:  Sedimenting suspensions exist in a varity of natural phenomena and industrial applications. It is already observed in experiments that the dilute fibre suspensions experience a hydrodynamics instability under gravity at low Reynolds numbers. Initially well-mixed suspensions become inhomogeneous and anisotropic due to this instability.The main goal of this work is to understand the instability in a dilute fibre suspension by means of an Eulerian approach which is based on the Navier-Stokes equations coupled to Fokker-Planck equation for the PDF of fibres.Using a linear stability analysis, we show that inertia and hydrodynamic translational diffusion damp perturbations at long wavelengths and short wavelengths, respectively, leading to a wavenumber selection. For small, but finite Reynolds number of the fluid bulk motion, the most unstable wavenumber is a finite value which increases with Reynolds number, and where the diffusion narrows the range of unstable wavenumbers. With periodic boundary conditions, numerical simulations of the full non-linear evolution in time of a normal mode perturbation show that the induced flow may either die or saturate on a finite amplitude. The character of this long time behaviour is dictated by the wavenumber and the presence or absence of the translational and rotational diffusivities.In a simulation domain confined by vertical walls, a series of alternating structures of risers and streamers emerge continuously from the walls until they meet in the middle of the domain. For moderate times, this agrees qualitatively with experimental and theoretical results. Moreover, our simulation in a vessel of infinite height obtained an increasing wavelength evolution due to the congregation of the streamers or risers. In the end, there is constantly only one streamer left, and it drifts randomly to one side of the container until the evolution reaches a steady state. It is also found that the perturbations added to the initial conditions can induce more high density regions whose sizes and velocities are strongly linked to the initial perturbations of the number density or the flow field. In addition, the maximum number of streamers increases with Reynolds number, volume fraction and channel width.