Gas-cushioned droplet impacts with a thin layer of porous media

Abstract

The pre-impact gas cushioning behaviour of a droplet approaching touchdown onto a thin layer of porous substrate is investigated. Although the model is applicable to droplet impacts with any porous substrate of limited height, a thin layer of porous medium is used as an idealized approximation of a regular array of pillars, which are frequently used to produced superhydrophobic- and superhydrophilic-textured surfaces. Bubble entrainment is predicted across a range of permeabilities and substrate heights, as a result of a gas pressure build-up in the viscous-gas squeeze film decelerating the droplet free-surface immediately below the centre of the droplet. For a droplet of water of radius 1 mm and impact approach speed 0.5 m s$^{-1}$, the change from a flat rigid impermeable plate to a porous substrate of height $5~\upmu$m and permeability $2.5~\upmu$m$^2$ reduces the initial horizontal extent of the trapped air pocket by $48~\%$, as the porous substrate provides additional pathways through which the gas can escape. Further increases in either the substrate permeability or substrate height can entirely eliminate the formation of a trapped gas pocket in the initial touchdown phase, with the droplet then initially hitting the top surface of the porous media at a single point. Droplet impacts with a porous substrate are qualitatively compared to droplet impacts with a rough impermeable surface, which provides a second approximation for a textured surface. This indicates that only small pillars can be successfully modelled by the porous media approximation. The effect of surface tension on gas-cushioned droplet impacts with porous substrates is also investigated. In contrast to the numerical predictions of a droplet free-surface above flat plate, when a porous substrate is included, the droplet free-surface touches down in finite time. Mathematically, this is due to the regularization of the parabolic degeneracy associated with the small gas-film-height limit the gas squeeze film equation, by non-zero substrate permeability and height, and physically suggests that the level of surface roughness is a critical parameter in determining the initial touchdown characteristics.

Appendices

Appendix

Gas cushioning with alternative forms of the Beavers–Joseph boundary condition

In addition to the full Beavers–Joseph boundary condition on the surface of a shallow porous substrate (30), two simplified versions of this expression are also considered. These correspond to no slip in the substrate ($\delta = 0$) and the absence of velocity shear in the gas film ($\delta = 0$ and $\alpha \rightarrow \infty$). In these cases, the Reynolds lubrication equation (36) simplifies to give

$$\begin{aligned} \frac{\partial f}{\partial t} = \frac{1}{12} \frac{\partial }{\partial x} \left[ \frac{f^2 \left( \alpha f^2 + 4 k^{1/2} f\right) }{\alpha f + k^{1/2}} \frac{\partial p}{\partial x}\right] + k h\frac{\partial ^2 p}{\partial x^2} \end{aligned}$$

(55a)

and

$$\begin{aligned} \frac{\partial f}{\partial t} = \frac{1}{12} \frac{\partial }{\partial x} \left[ f^3 \frac{\partial p}{\partial x}\right] + k h\frac{\partial ^2 p}{\partial x^2}, \end{aligned}$$

(55b)

respectively, while the boundary integral (13) and the initial and far-field conditions remain unchanged.

Fig. 9

Droplet free-surface and pressure evolutions in gas-cushioned impacts for $k = 4$ and $h = 3$, with simplified forms of the Beavers–Joseph boundary condition

Comparative results for these alternative boundary conditions are presented in Figure 9 alongside the full Beavers–Joseph condition for the cases $k = 4$ and $h = 3$. The full Beavers–Joseph condition (shown in Fig. 9a) results in a smaller pocket of trapped gas compared to the other two boundary conditions; this is because this case has the greatest amount of gas velocity slip at the substrate interface, and so the gas is more able to escape from beneath the oncoming droplet. The trapped gas pocket in Fig. 9c where $\alpha \rightarrow \infty$, $\delta =0$ is the largest gas pocket of the three, as this boundary condition corresponds to no slip for the gas on the substrate surface, and hence the gas is less able to escape from underneath the oncoming droplet. The variations between the different forms of the boundary condition diminish as the value of k falls. In cases where the substrate permeability is well known, a detailed comparison of the radii of the air pockets, both experimentally and using the models described herein, would enable an accurate determination of the parameters in the Beavers–Joseph condition.

An extension to the Beavers–Joseph boundary condition (8) due to Jones [43]

$$\begin{aligned} \frac{K^{1/2}}{\alpha } \left( \frac{\partial \tilde{u}_{\text{ g }}}{\partial \tilde{y}} + \frac{\partial \tilde{v}_{\text{ g }}}{\partial \tilde{x}}\right) = \tilde{u}_{\text{ g }} - \delta \tilde{u}_{\text{ s }} \end{aligned}$$

(56)

is often applied to non-parallel flows where (as herein) $\tilde{v}_\text{ g }\left( x,\,0,\,t \right) \ne 0$. This formula, which has never been explicitly verified experimentally [32], includes shear stress instead of just velocity shear. However, because of the disparate length and velocity scales (15–16) in the gas film, the second term on the left-hand side is $\text {O}\left( \varepsilon ^2 \right)$ smaller than the first term. Consequently, if this extra term was included in the analysis, then it would be neglected at leading order leaving the Beavers–Joseph condition (8) as used.

Numerical solution of the free-surface integral with surface tension

The gas pressure and the deviation of the droplet free-surface from its undisturbed position are expressed as complex Fourier series

$$\begin{aligned}&p_{\text{ g }}\left( x,\,t \right) = \sum _{n=-\infty }^\infty P_n\left( t \right) \exp \left( \frac{\text {i}n \pi x}{L} \right) , \end{aligned}$$

(57a)

$$\begin{aligned}&f\left( x,\,t \right) - \frac{x^2}{2} + t = \sum _{n=-\infty }^\infty F_n\left( t \right) \exp \left( \frac{\text {i}n \pi x}{L} \right) , \end{aligned}$$

(57b)

where the Fourier coefficients are given by

$$\begin{aligned} P_n\left( t \right) =\,&\frac{1}{2L} \int _{-L}^L p_{\text{ g }}\left( x,\,t \right) \exp \left( -\frac{\text {i}n \pi x}{L} \right) \,\text {d}{x}, \end{aligned}$$

(57c)

$$\begin{aligned} F_n\left( t \right) =\,&\frac{1}{2L} \int _{-L}^L \left( f\left( x,\,t \right) - \frac{x^2}{2} + t\right) \exp \left( -\frac{\text {i}n \pi x}{L} \right) \,\text {d}{x}. \end{aligned}$$

(57d)

Here the droplet free-surface repeats with period 2L, although, by choosing a spatial domain $-L < x < L$ and L suitably large, edge effects are negligible. The deviation of the droplet free-surface from its undisturbed position is approximated by the Fourier series rather than the free-surface position itself in order to increase the smoothness of the Fourier series at $x=\pm L$.

The integral

(58)

is the Hilbert transform of some function $g\left( x \right)$. Consequently, upon substituting the Fourier series (57) into the integral equation (48)

$$\begin{aligned} \sum _{n=-\infty }^\infty \frac{\text {d} ^2 F_n}{\text {d} t^2}\left( t \right) \exp \left( \frac{\text {i}n \pi x}{L} \right) =&\sum _{n=-\infty }^\infty \frac{\text {i}n \pi }{L} P_n\left( t \right) {\mathscr {H}}\left[ \exp \left( \frac{\text {i}n \pi x}{L} \right) \right] \nonumber \\&+ \sigma \sum _{n=-\infty }^\infty -\frac{\text {i}n^3 \pi ^3}{L^3} F_n\left( t \right) {\mathscr {H}}\left[ \exp \left( \frac{\text {i}n \pi x}{L} \right) \right] . \end{aligned}$$

(59)

Using properties of the Hilbert transform [44]

$$\begin{aligned} {\mathscr {H}}\left[ \exp \left( \frac{\text {i}n \pi x}{L} \right) \right] =&-\text {i}\, \text {sgn}\left( n \right) \exp \left( \frac{\text {i}n \pi x}{L} \right) , \end{aligned}$$

(60)

and hence upon collecting coefficients of matching exponential terms, the Fourier coefficients are related through

$$\begin{aligned} \frac{\text {d} ^2 F_n}{\text {d} t^2}\left( t \right) + \sigma \frac{\left| n \right| ^3 \pi ^3}{L^3} F_n\left( t \right) =&\,\, \frac{\left| n \right| \pi }{L} P_n\left( t \right) . \end{aligned}$$

(61)

The Fourier series (57) are truncated and sufficient terms are retained to achieve convergence. Given a pressure profile $p_{\text{ g }}\left( x,\,t \right)$, the Fourier pressure coefficients are obtained by means of a Fast Fourier Transform. Subsequently, (61) can be discretized in time and used to calculate the evolution of the corresponding free-surface Fourier coefficients, which (using an inverse Fast Fourier Transform), allow the free-surface profile to be recovered.