Droplet rebounds off a fluid bath at low Weber numbers

Abstract

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution, and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e., neglecting that of the droplet. The method's extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime.

Figures (10)

• Figure 1: ($a$) A rendering of the experimental setup. ($b$) Height of the centre of mass $h$ against time $t$ for a bouncing drop, where $W e _d\,=1.285$, $O h_d\,=0.0238$, and $B o\,=0.049$. The dashed lines are parabolic fits to the incoming and outgoing data, and the dotted line denotes the drop radius $R_{d} = 0.328$ mm. ($c$) Select experimental images corresponding to panel ($b$). The dashed line indicates the height of the undisturbed free surface at the point of contact.
• Figure 2: Schematic of the problem. An undeformed droplet (thick solid black line) of radius $R_d$ and fluid properties $(\sigma_d, \rho_d, \nu_d)$ is depicted above the surface of a fluid bath with properties $(\sigma, \rho, \nu)$, shown with a solid grey line. The surface of the droplet is described in an non-inertial spherical reference frame by $\xi'(t, \theta)$, whose origin is the centre of mass of the droplet. The height of the bath's surface, denoted by $\eta(r, t)$, is described in a fixed cylindrical reference frame whose origin coincides with the initial point of impact.
• Figure 3: Schematic of deformations during impact. The surface of the bath is shown with thin grey solid lines outside the pressed surface $S(t)$, and with a thick grey solid line in the pressed surface $S(t)$. The droplet interface is shown with a thick black line. The orthogonal projection of $S(t)$ onto the $z = 0$ plane is $C(t)$, shown with a thick dark grey dashed line. Variables $h(t)$, $\eta(r, t)$ and $r_c(t)$ correspond to the height of the centre of mass of the droplet, the elevation of the free-surface of the bath and the radius of $C(t)$, respectively. The origin of the $(x',z')$ system of reference is attached to the centre of mass of the droplet. The separation between the droplet and the bath over the pressed surface is introduced solely for the purpose of better visualisation of the two, as droplet and bath interfaces are predicted to fully coincide within $S(t)$.
• Figure 4: Dependence of the angle of the point of application of the pressure on the deformation of the droplet. Panels show the angle $\theta$ that corresponds to $r=\delta_r$ when the droplet is (a) undeformed and (b) significantly deformed. The lower height of the centre of mass in panel b) results in a larger value of $\theta$.
• Figure 5: Simulation of a water droplet impacting a water bath at $V_0 = 38$ cm/s, corresponding to $W e _d\, = 0.7, B o\, = 0.017, O h_d\,=0.006$ and $M= 20$. Columns represent snapshots of the impact. From top to bottom, the rows show the spatial reconstruction of the impact, the dimensionless pressure distribution in azimuthal spherical coordinates at the droplet's surface, and the amplitude of the pressure distribution in Legendre's decomposition as a function of the mode number.