Drag force on a sphere moving in low-reynolds-number pipe flows

In this paper, the drag force on a sphere moving constantly along the centerline of a circular pipe filled with viscous fluid (the falling-sphere problem) under low Reynolds number condition is investigated via numerical calculation. The incompressible Navier-Stokes equations are formulated in a pseudo-compressibility form. The numerical scheme makes use of finite-volume method and the numerical flux terms are evaluated using the Total-Variation Diminishing (TVD) strategy commonly applied to the compressible flow. Steady solution is obtained by marching (iterating) in time until the artificial time derivative of pressure term in the continuity equation drops to zero. In the calculation, six different Reynolds number (Re) ranging from 0.1 to 1 and seven different pipe-to-sphere diameter ratios (D/d) ranging from 5 to 40 are selected to study the pipe-wall effect. In each case, the drag force on the sphere is evaluated and the results are compared with the existing approximate theoretical values derived from correcting the Stokes' formula. Both results agree in trend, but with noticeable deviation in values, particularly for cases with large pipe-to-sphere diameter ratios. The deviation is due to the fact that theoretical values were based on the solution to the linearized Navier-Stokes equations (Stokes' creeping-flow equations), while the fully nonlinear form of the Navier-Stokes equations are adopted in the present calculations. Finally, a least-square regression technique is applied to collapse the calculated results into a single expression exhibiting the functional relationship between the drag force, Reynolds number (Re), and the pipe-to-sphere diameter ratio (D/d).