Numerical investigation of lubrication force on a spherical particle moving to a plane wall at finite Reynolds numbers

► Correlations of lubrication forces are valid over a range of Reynolds numbers. ► The similar correlation is derived for a sphere freely falling to a wall. ► An IB pressure correction method is used to investigate the lubrication forces. A direct-forcing immersed-boundary (IB) pressure correction method is used to investigate numerically the total hydrodynamic force for a solid sphere in normal approach to a plane wall. For constant approaches, Cox and Brenner’s classic asymptotic formula in the lubrication regime was extended to cover particle Reynolds number Re⩽50. An explicit force formula was proposed by fitting the current simulation results. We also consider gravity-induced decent of an immersed sphere towards a wall and an equation of motion was proposed containing quasi-steady viscous drag and two unsteady components-added mass force and history force. Possible effects from liquid inertia at finite-Re and wall at small gaps are described by least-square fitting of the simulation results. The proposed formula agrees to existing low-Re theories.