Coupling of translation and rotation in the motion of finite-length rods near solid boundaries

The motion of finite-length cylindrical rods moving near a planar rigid surface is a scenario common across many engineering and natural settings. We study the low-Reynolds-number flow around finite rods that are allowed to rotate or translate in directions perpendicular or parallel to the plane. We develop a three-dimensional lubrication theory to characterize the pressure and hydrodynamic resistances of the cylinders through a special consideration of the cylinder's end effects. In addition, we use three-dimensional numerical simulations to solve these Stokes flows for cylinders of varying lengths and with varying gap sizes between the cylinder and plane, and the numerical results support the developed analytical descriptions. We also use visualizations of the flow to provide qualitative insights and rationalize the effect of the ends on the dynamics of the cylinders. The numerical simulations and theoretical predictions show good agreement in the long (isolated ends) and short (disk-like) limits.