Basset-Boussinesq History Force Acting on a Drop in an Oscillatory Flow

The total force experienced by a droplet embedded in a uniform oscillating flow can be divided into four contributions: steady drag force, inertial or pressure gradient force, added-mass force, and Basset-Boussinesq history force. The effect of the history force is often neglected, due to numerical difficulties for its time integration, but also because no analytic expression is available in the time domain for droplets. The contribution of the history force acting on a spherical droplet in an oscillatory flow is determined using direct numerical simulation. Variation of the viscosity ratio makes the analysis relevant to bubbles, droplets, and solid particles. By changing the flow oscillation frequency, we can determine the range of physical parameters for which the contribution of the history force is significant. Additionally, the relevance of the kernel expression recently proposed by Legendre et al. [Phys. Rev. Fluids 4, 073603 (2019)] is discussed and compared to theoretical predictions in the frequency domain. We demonstrate that this kernel can be used to model history effects for droplets, especially since the contribution of the history force to the total force is important. The proposed history force expression provides a significant improvement for droplet trajectory prediction when the history force contribution cannot be neglected.