Impact of the history force on the motion of droplets in shaken liquids

Abstract

Droplets, solid particles, and gas bubbles in unsteady flows experience the Basset-Boussinesq history force (BBH) in addition to steady viscous drag, added mass, and buoyancy. Although physically relevant, the BBH term is often neglected because its inclusion is analytically and numerically demanding. To assess when this approximation fails, we revisit unsteady Stokes flows around spherical droplets of finite viscosity and derive, from first principles, the velocity fields and hydrodynamic forces, including both the classical rigid-particle limit and the free-slip (zero-viscosity) bubble limit. The resulting expressions also encompass cases with time-dependent bubble radii. We further illustrate how the BBH force arises from transient, diffusion-driven vortex structures around accelerating particles. Applying these results to droplets or particles in horizontally shaken liquids (periodically accelerated flows), we find that in the transition regime between the quasi-steady Stokes limit and the inertia-dominated regime, BBH can lead to a reduction of the droplet deflection amplitude by more than 60\% compared to predictions that neglect memory effects. We also derive a characteristic scaling of the displacement amplitude in the low-frequency limit, providing an unambiguous, experimentally verifiable signature of the BBH. For light particles and gas bubbles, the BBH contribution becomes more significant (relative to the other hydrodynamic forces) compared to that o heavy particles, such as droplets in air.

Figures (10)

• Figure 1: Sketch of the system. A container filled with a liquid of mass density $\rho_f$ and kinematic viscosity $\nu_f$ (gray) performs oscillatory motion described by the position $\vb s(t)$ of the center of its bottom. Studied is the dynamics of a small droplet of density $\rho_d$, kinematic viscosity $\nu_d$ and radius $R_0$ (white) immersed in the much larger and liquid-filled container. The droplet position is given by $\vb r_d$ in the container-fixed coordinate system, or is given in the laboratory frame by $\vb r_l = \vb s + \vb r_d$.
• Figure 2: Drag force acting on a solid particle executing oscillations of the form $\vb r_d=-X_0\sin(\omega t)\vb e_z$ with respect to a quiescent liquid. The solid red line represents the drag force including the BBH force as calculated using Eq. \ref{['eq_harmonic_drag']}. The dashed line shows the instantaneous Hadamard-Rybczynski viscous drag alone. The black dots at (1)-(5) indicate the times at which the flow profiles around a particle are shown in Fig. \ref{['fig:flow_field_illustration']}. Other parameters include: $\omega=4rad/s,\, R_0=1mm,\,X_0=0.4m,\,\mu_f=0.001Pas,\,\mu_d=0.01Pas,\,\rho_f=\rho_d=1000kg\per\cubic m$.
• Figure 3: Snapshots of the fluid flow in the laboratory system (arrows), and qualitative heat maps of the stream function, around a solid particle at different phases of the oscillation cycle in Fig. \ref{['fig:illustration_force']} at times (1) - (5). The upper panels show the velocity field in a radial cross-section, while the lower panels display the unsteady profile of the vertical velocity component $u_z(r)$ in the equatorial plane (solid lines), compared with the corresponding steady velocity (dashed lines). The quasi-steady Stokes profile assumes instantaneous flow relaxation, valid only for $\delta \ll 1$. The unsteady profiles illustrate how a finite $\delta$ causes vortices, increasing the viscous drag force. The parameters were chosen as in Fig. \ref{['fig:illustration_force']}. Times: $t=0$ in (1), $t=0.2T$ in (2), $t=0.25T$ in (3) $t=0.3T$ in (4) and $t=0.7T$ in (5).
• Figure 4: Sketch of an experimental setup to study the effects of the added mass and the BBH force on a droplet (gray) in a horizontally shaken container filled with an immiscible carrier liquid. Part (a) shows a spherical droplet sedimenting along the red arrow in a stationary liquid container of height $D$. Part (b) depicts the container undergoing horizontal harmonic shaking with amplitude $A$ and frequency $\omega$. The periodic horizontal motion of the liquid induces a corresponding periodic acceleration of the droplet via viscous and buoyancy forces, deflecting it from its straight sedimentation path. The dependence of the droplet deflection on shaking amplitude and frequency provides insight into the forces acting on the particle in unsteady flows. The position of the center of the shaken container in the laboratory frame is indicated by the red arrow at the bottom of part (b), $s_x(t) = A \sin(\omega t)$.
• Figure 5: $\delta$-dependence of $X_0/(A\delta)$ with ($H=1$, red) and without ($H=0$, black) the BBH force. Panel (a) shows the overall suppression of the amplitude by the BBH force across a wide frequency range. Panel (b) focuses on the small-$\delta$ regime, comparing the asymptotic results in Eqs. \ref{['eq:X0whist']} and \ref{['eq:X0ohist']} (dashed) with the full expression in Eq. \ref{['eq:Sigma2']} (solid). Here $\rho_d/\rho_f=2$.