The falling-ball viscometer is a device where a spherical particle falls along the axis of a circular cylinder filled with viscous fluid. The various classical results for this device are developed under the assumption that the Reynolds number of the flow is zero, i.e., Stoke's flow. Inertial effects are not taken into account. To better understand the dynamics of the particle sedimentation process and the role of inertia in this process, we implemented a numerical simulation.
The ADI (Alternating Direction Implicit) scheme is widely used to solve the vorticity-stream function formulation of the Navier-Stokes equation in axisymmetric geometries. However, a severe timestep restriction for low Reynolds flow makes application of this method cumbersome for simulating the falling-ball viscometer. Through a study of a classical 2-D cavity problem, the cause of the instability requiring the restricted time step is identified. For this cavity problem, a modification of the usual treatment of the boundary condition for the stream function relaxes the restriction on the time step. Unfortunately, the complexity of the falling-ball viscometer simulation makes it difficult to efficiently implement this modification.
A penalty method is used here to handle the moving boundary associated with the sedimenting particles. Fluid is allowed to flow through the particle but it encounters resistance proportional to a paxameter (or penalty number) that can be viewed as a porosity. For large values of the penalty number the flow converges to that for a impenetrable paxticle. The validity of this method is demonstrated. The results of the numerical simulation are compared with previous work. The relationships among the sedimentation speed Used, the Reynolds number Re, the radius of cylinder R0, the particle permeability β and the dimensionless mass δ are studied. Multiparticle dynamics are easily simulated using our penalty method with virtually no additional numerical resources needed. The breaking of the timereversal symmetry by inertia is observed and studied.