The sedimentation of thin, rigid discs in a viscous fluid: \\ A numerical study using a novel augmented finite-element method

We study the sedimentation of thin, arbitrarily shaped, but (for now) rigid disks, sedimenting under gravity in a quiescent viscous fluid. The sharp edge of such disks creates a singularity in the fluidpressure. At low Reynolds number, this means that a significant contribution of the total drag comes from the vicinity of the disk edge; for a flat disk at zero Reynolds number, 30\% of the total drag is generated by the outermost 5\% of the disk radius. This implies that any under-resolution of the pressure field leads to critical errors in the sedimentation velocity; furthermore, the singularities have a severe impact on the convergence rate of standard finite-element (FE) discretisations under mesh refinement.

 

We present a novel augmented FE method which allows analytic (singular) functions of unknown amplitude to be subtracted from the full solution in a sub-domain around the disk edge, rendering the FE-discretised remainder of the solution regular, thus restoring the standard FE convergence rate. The singular amplitudes are determined via PDE-constrained minimisation of a suitably chosen functional which captures the key signatures of the singularity.