The unsteady motion of a fully immersed solid sphere at low to moderate particle Reynolds number, from 1 to about 1600, has been modeled by considering the quasi-steady viscous drag, the added mass force, and the history force. The last force results from unsteady boundary layer growth and is often neglected in multiphase flow applications due to its complex formula. However, it has been shown that this unsteady viscous force is more crucial than the widely employed added mass force to describe an unsteady sphere motion. Thus, targeting the flow regime when the quasi-steady viscous drag is dominating, this work proposes two simple formulae to approximate the errors of neglecting either the history force or the added mass force in describing a fully immersed spherical pendulum motion. The proposed formulae are shown to capture the actual error when the numerical solution of a partial model that omits one force component is compared to the full model prediction when the sphere density is not too close to the ambient liquid.
