Normal and oblique droplet impingement dynamics on moving dry walls

Industrial applications that depend on jetting-based technology, such as painting or additive layered manufacturing, involve sequential deposition of droplets onto a moving surface. Spreading and receding dynamics of these impinging drops depend on the momentum transferred by the moving wall to the droplet liquid, which in turn governs the geometric precision and surface finish of the printed outcome. In this work, the impingement dynamics of microdroplets on a flat, smooth, and moving solid surface is computed using a phase-field-based lattice Boltzmann method. Moreover, the motion of the three-phase moving contact line is captured using a geometry-based contact angle formulation. First, we investigate the influence of various process and materials parameters such as wall velocity, droplet viscosity, surface tension, and wettability on the impact behavior of drops. The surface wettability significantly affects the droplet morphology; an elongated tail like structure forms on the rear end of the droplet which becomes sharper as the moving surface becomes more hydrophobic. Furthermore, we examine the underlying flow physics of the symmetry breaking during the spreading and recoiling phases. For a given contact angle, an increase in wall velocity is found to expedite droplet spreading. In addition, for the first time we explore the oblique droplet impingement dynamics on moving dry walls in this work. It is observed that wall momentum affects the structure of the leading edge during the inline impact situations, whereas the moving surface controls the delay in flow reversal inside the droplet for opposing impact scenarios.

Figure 1

A schematic representation of a droplet with diameter $D_i$, impacting on a moving dry wall. The angle of impingement is $\alpha$ and the wall moves with a constant translational velocity $U_w$ along the positive $x$ direction.

Figure 2

Temporal sequence of droplet impingement dynamics illustrating the effect surface wettabilities for two different cases: $\theta ={90}^{\circ }$ (left column) and $\theta ={140}^{\circ }$ (right column). For all cases, the wall velocity is fixed at $U_w=0.075$. The value of the contour density shown in this figure corresponds to $\rho =0.5({\rho }_l+{\rho }_g)$. Rebound is suppressed for $\theta ={90}^{\circ }$, whereas oblique lift-off from the surface is observed for $\theta ={140}^{\circ }$. The scale bar is 50 lattice units.

Figure 3

Temporal evolution of the (a) contact area ($A^{*}$), (b) position of the $X$ center of mass ($X_m$), (c) volume-averaged drop velocity along the wall-normal direction, and (d) surface energy of a droplet impacting onto a target wall moving with a velocity of $U_w=0.075$ for different surface wettabilities. The contact area and surface energy have been nondimensionalized by $\pi D_i^2/4$ and $\pi D_i^2\sigma$, respectively. The line colors are assigned as (i) black (online and printed versions), (ii) red (online version), and gray (printed version).

Figure 4

(a) Instantaneous interface profiles of the impinging droplet along the mid-$y$ plane and (b) the contact line footprint at $T^{*}=6.85$ for different contact angles: $\theta ={70}^{\circ }$ (dash dot dot), ${90}^{\circ }$ (dashed), ${110}^{\circ }$ (solid), and ${140}^{\circ }$ (dash dot). The wall is moving with a constant velocity of $U_w=0.075$. The scale bar is 50 lattice units.

Figure 5

Instantaneous velocity-field plots inside the peripheral rims of the impacting droplet along the mid-$y$ plane during the inertial spreading phase illustrating flow-field symmetry for a stationary wall case with $U_w=0$ inside the (a) left rim at $T^{*}=1.02$. The flow field inside the right rim (not shown here) is symmetrical to that show in (a). Asymmetry in flow field is observed inside the (b) right rim and (c) left rim when a droplet impinges onto a moving surface ($U_w=0.1$) with $\theta ={110}^{\circ }$. The scale bar is 10 lattice units.

Figure 6

Instantaneous velocity-field plots inside the impacting droplet along the mid-$y$ plane during the recoiling phase illustrating symmetry and asymmetry for (a) $U_w=0$ and (b) $U_w=0.1$, respectively, at $T^{*}=1.71$. The surface is characterized with a contact angle of $\theta ={110}^{\circ }$ and moves with $U_w=0.1$. The scale bar is 20 lattice units.

Figure 7

Motion of the capillary wave moving along the free surface of the droplet when it impact a stationary surface (left column) and a moving surface (right column, $U_w=0.125$) at different time instants with $\theta ={110}^{\circ }$. Both the troughs of the capillary are marked in red outline (gray dashed line pattern in printed grayscale version) for the stationary wall case, implying the symmetry of the wave. However, the breakdown in this symmetry is highlighted with different colors and line patterns (blue corresponds to dash-dot-dot pattern and red corresponds to dashed pattern) in the case of a moving wall. The scale bar is 20 lattice units.

Figure 8

Temporal evolution of the (a) spread factor ($D^{*}$), (b) kinetic energy, and (c) surface energy ($E_{\sigma }$) of a droplet impinging onto a surface with contact angle $\theta ={110}^{\circ }$ moving with different wall velocities. The line colors are assigned as (i) black (online and printed versions), (ii) red (online version), and gray (printed version).

Figure 9

Time evolution of (a) surface energy ($E_{\sigma }$) and (b) vertical component of velocity ($U_z$) for different Weber numbers (We) with $\mathrm{Re}=600$, $U_w=0.075$, and surface contact angle $\theta ={90}^{\circ }$.

Figure 10

Temporal evolution of the (a) contact area ($A^{*}$) and (b) volume-averaged vertical component of droplet velocity ($U_z$) for different Ohnesorge numbers ($\mathrm{Oh}$) to investigate the effect of drop viscosity. The Webers number, wall velocity, and surface contact angle were set to be $\mathrm{We}=51.2$, $U_w=0.075$, and $\theta ={90}^{\circ }$, respectively.

Figure 11

(a) Snapshots of oblique inline droplet impingement on substrates with different impact angles: $\alpha ={30}^{\circ }$ (left column) and $\alpha ={60}^{\circ }$ (right column). The scale bar is 50 lattice units. Temporal evolution of the (b) contact area ($A^{*}$) and (c) volume-averaged vertical component of droplet velocity ($U_z$) for different impact angles ($\alpha$) with the inline configuration. For all the cases, the wall velocity and surface contact angle are fixed at $U_w=0.075$ and $\theta ={90}^{\circ }$, respectively. The line colors are assigned as (i) black (online and printed versions), (ii) red (online version), and gray (printed version).

Figure 12

(a) Snapshots of oblique opposing droplet impingement on substrates with different impact angles: $\alpha =-{20}^{\circ }$ (left column) and $\alpha =-{40}^{\circ }$ (right column). The scale bar is 50 lattice units. Time evolution of the (b) contact area ($A^{*}$) and (c) surface energy ($E_{\sigma }$) of the impinging droplet for different impact angles ($\alpha$) with the opposing configuration. For all the cases, the wall velocity and surface contact angle are fixed at $U_w=0.075$ and $\theta ={90}^{\circ }$, respectively. The line colors are assigned as (i) black (online and printed versions), (ii) red (online version), and gray (printed version).