Numerical study of droplet impact on a flexible substrate

Droplets interacting with deformable moving boundaries is ubiquitous. The flexible boundaries may dramatically affect the hydrodynamic behavior of droplets. A numerical method for simulating droplet impact on flexible substrates is developed. The effect of flexibility is investigated. To reduce the contact time and increase the remaining upward momentum in the flexible cases, the Weber number should be larger than a critical value. Moreover, the ratio of the natural frequency of the plate to that of the droplet $F_r$ should approximately equal to the reciprocal of the contact time of droplets impact on the rigid surfaces $\left(t_{\text{ctr}}\right)$ at the same We, e.g., $F_r\approx 1/t_{\text{ctr}}$. Only under this circumstance would the kinetic energy convert into the surface energy of the droplet and the elastic energy of the plate simultaneously, and vice versa. Moreover, based on a double spring model, we proposed scaling laws for the maximal deflection of the plate and spreading diameter of the drop. Finally, the droplet impact under different wettability is qualitatively studied. We found that the flexibility may contribute to the droplet bouncing at a smaller contact angle.

Figure 1

Schematic diagram for a droplet impacting on the flexible plate. The droplet with diameter $D_0$ is placed above the flexible plate at a distance $H=0.2D_0$. The initial downward velocity is $U$.

Figure 2

A moving deformable boundary (solid red line) immersed in fluid. Circles and squares represent fluid and solid points, respectively. Filled black circles and squares are fluid and solid nodes closest to the boundary. $e_{\overline{\alpha }}$ represents the opposite direction of $e_{\alpha }$. Filled red circles represent the Lagrangian points on the moving deformable boundary. ${\mathit{x}}_w$ is the intersection of the link from ${\mathit{x}}_f$ to ${\mathit{x}}_s$ and the boundary.

Figure 3

(a) A droplet with contact angle ${\theta }_{\mathrm{eq}}$ rests on a flat surface. The dash-dotted and solid lines $\left(\varphi =0.5\right)$ represent the drop's initial and equilibrium $\left(t=50T_{\text{ref}}\right)$ shapes, respectively. (b) The ratio of height to initial diameter as a function of contact angle.

Figure 4

A droplet with contact angle ${\theta }_{\mathrm{eq}}$ rests on a membrane at $t=100T_{\text{ref}}$. The red, blue, and green solid lines represent membrane profile, the fitted circle of the droplet profile, and the fitted circle of the bulge, respectively.

Figure 5

(a) The droplet contact angle as a function of $|{\mathit{T}}_{\text{in}}|/|\mathit{\sigma }|$. The red dash-dotted line represents the equilibrium contact angle ${\theta }_{\mathrm{eq}}={81}^{\circ }$. (b) The bulge contact angle as a function of $|{\mathit{T}}_{\text{in}}|/|\mathit{\sigma }|$.

Figure 6

Snapshots of the water droplet impact on (a) a rigid surface and (b) a flexible surface with $K_B=1.0\left(F_r=0.99\right)$. In both cases $\text{We}=16$. The droplet profile is described by the contour line $\varphi =0.5$. $t_{\text{spr}}$ is the spreading time at which the droplet reaches $D_{\max }$ and $t_{\text{ct}}$ is the contact time at which the droplet leaves off the plate.

Figure 7

(a) Contact time $t_{\text{ct}}$ as a function of We for different $K_B$. (b) $t_{\text{ct}}$ as a function of $F_r$ for different We. Cases of $F_r=10$ represent the cases with rigid substrate. Black, blue, green, and red circles represent the cases at $F_r=10$, 1.4, 0.77, 0.2, respectively.

Figure 8

(a) The coefficients of restitution $\frac{M_y}{M_0}$ as a function of We for different $K_B$. $M_0$ is the initial momentum of the droplet for each We. (b) Remaining upward momentum as a function of $F_r$ for different We, and $F_r=10$ represents the cases of droplets impacting on the rigid substrate.

Figure 9

(a) The evolution of the kinetic energy of the droplet for different $F_r$. (b) The evolution of total energy of the flexible substrate, including the kinetic energy, the bending energy, and the stretching energy. (c) The evolution of $M_y$. (d) The evolution of the displacement of the center of the plate in the $y$ direction (solid lines) and the evolution of $D_{\text{max}}$ (dash-dotted lines). The short vertical arrows on each line in (a, b, c, d) represents $t=t_{\text{ct}}$. The shaded areas in (d) represent the “spread” and “retract” stages of a droplet when it impacts on a surface. In all cases $\text{We}=16$.

Figure 10

(a) The elastic energy conversion efficiency $\eta$ as a function of the $F_r$. (b) The ratio of the stretching energy to the bending energy $\frac{E_s}{E_b}$ as a function of $F_r$. (c) Definition of the phase according to the movement of the center of the substrate. (d) The bouncing-off phase $\varphi$, at which the droplet leaves the surface, $t_{cr}$ and $M_y$ as functions of $F_r$. In all cases $\text{We}=16$.

Figure 11

The maximal spreading diameter ratio as a function of We. The solid black line represents the equivalent gravitational acceleration scaling rate ${\text{We}}^{1/2}$. In all cases, the surfaces are rigid.

Figure 12

(a) The maximal deflection of the center of the substrate as a function of $K_B$ for different We. (b) Normalized deflection as a function of $K_B$. (c) ${\beta }_{\max }$ as a function of $K_B$ for different We. (d) Normalized diameter ratio as a function of $K_B$.

Figure 13

“bounce off” and “adhere” phase diagram for different We and wettability, characterized by the contact angle $\theta$. (a) Droplets impact on a rigid surface. (b) Droplets impact on a flexible surface. In all cases of (b) $K_B=0.6\left(F_r=0.77\right)$.

Figure 14

Schematic of the least square method. ${\mathit{x}}_{\text{BI}}$ is the projection of ${\mathit{x}}_s$ on the beam (the red line). A local coordinates $(\xi ,\eta )$ is established with origin at ${\mathit{x}}_{\text{BI}}$. The large dotted circle denote a range of $|\mathit{x}-{\mathit{x}}_{\text{BI}}|<R$, where $R$ is a specified radius. Blue circle points are inside the range. Discs $(•)$ denotes the fluid nodes inside the beam (suppose the droplet is inside).