When a charge neutral drop impacts on a flat solid substrate, a small air bubble is always trapped underneath due to the lubrication pressure coming from the viscous stress in the squeezed air film. Herein we find experimentally and numerically that the process of the air entrapment and the initial contact state of the drop with the substrate can be profoundly altered via an external electric field. In an electric field, the induced electric stresses at the bottom of the drop increase drastically right before the drop contacts the substrate, which acts against the lubrication pressure, resulting in reduced initial contact radius and air bubble size. When the external electric field reaches a critical value, the electrical stress accelerates the flow near the bottom of the drop and generates a conical tip quickly instead of a dimple, resulting in a centre contact and eliminating the air bubble entrapment. Based on the dipole mirror charge model, we find the dimensionless strength of critical electric field scales with the square root of capillary number based on the air viscosity. This scaling law of the critical electric field for eliminating the air bubble entrapment is verified experimentally and numerically. This work may offer a new way to mitigate defects caused by air bubble entrapment for inkjet printing and droplet-based additive manufacturing.

Depositing drops on a solid surface without entrapping bubbles is desirable for many spray coating and printing applications. Tian et al. [ J. Fluid Mech. 946, A21 (2022)] reported that an electric field can be applied to eliminate air bubble entrapments for neutral drops. Herein we provide a complete physical picture of the entire process of a drop impacting onto the solid surface under an external electric field. The electrohydrodynamic behavior during the drop impact is divided into three stages: the deformation of the drop in the electric field prior to contact, the initial contact of the drop with the substrate, and the rich postcontact phenomena including spreading, receding, jetting, and fragmentation. The results show that under the increasingly stronger electric fields, the modest drop oscillation transforms into a vertically stretched spindle. As the drop approaches the substrate, the electric stress at the south pole increases rapidly, which sharpens the bottom surface into a conical shape. The cone angle is determined by both the impact velocity and the electric field strength. After the contact, the surface electric stress tends to pull the drop upward, breaking up the drop, forming several jetting modes, and reducing the maximum spreading radius. The various drop deposition modes are summarized in a phase diagram, which sheds light on identifying appropriate electric fields for high-quality drop depositions without air bubble entrapments or jettings.

In this paper, we simulate the process of two-dimensional axisymmetric fluid–structure coupling of a droplet impacting on a flexible disk. The effects of dimensionless disk stiffness (K = 0.1–1000), Weber number (We = 1–500) and contact angle (θ = 130° and 60°) on the dynamics of the droplet impacting on the flexible disk are analysed. The results indicate that there are five typical impact modes for a hydrophobic surface (θ = 130°) and four typical impact modes for a hydrophilic surface (θ = 60°) within the range of considered parameters. The analysis of spreading factor reveals that a part of the energy is transferred to the substrate, which is manifested as a weakening of the droplet spreading, when the substrate deforms downwards due to the droplet impact; the squeezing of the droplet causes a tendency to flow from the centre of the droplet to the edge, which is manifested as an enhancement of the droplet spreading, when the substrate recovers from the downward deformation. The effect of the substrate flexibility on the maximum spreading factor depends on the competition of the two mechanisms above. Based on this, a modified scaling law of β_max has been proposed by introducing the effective Weber number (We_m). The analysis of impact force demonstrates that the peak of the impact force is related to the deflection of the flexible substrate which is different from that of a rigid wall; and three typical processes of impact force variation have been summarised. In addition, unlike the rigid substrate scenario, there is an energy interaction between the droplet and the flexible substrate after impact occurs, which is classified as three typical energy transformation processes.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$ . Three types of $\alpha$ – $Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$ . As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$ . This implies that the ‘walking’ drop instability reported in previous literature does not exist.